Quantum Walk in Position Space with Single Optically Trapped Atoms

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氢原子量子力学模型

氢原子量子力学模型英文回答：The quantum mechanical model of the hydrogen atom is a fundamental concept in physics that describes the behavior of a single hydrogen atom. This model is based on the principles of quantum mechanics, which is the branch of physics that deals with the behavior of particles at the atomic and subatomic levels.In the quantum mechanical model, the hydrogen atom is treated as a system consisting of a single electron orbiting a nucleus. The electron is described by a wave function, which is a mathematical function that determines the probability of finding the electron at a particular position in space. The wave function is governed by the Schrödinger equation, which is a differential equation that describes the behavior of quantum systems.The wave function of the hydrogen atom can be solvedanalytically, resulting in a set of wave functions called the hydrogen atom orbitals. These orbitals describe the different energy levels and spatial distributions of the electron in the hydrogen atom. The lowest energy level is called the ground state, while higher energy levels are called excited states.Each orbital is characterized by a set of quantum numbers, which specify the energy, shape, and orientation of the orbital. The principal quantum number (n) determines the energy level of the orbital, with larger values of n corresponding to higher energy levels. The azimuthal quantum number (l) determines the shape of the orbital, with different values of l corresponding to different shapes such as s, p, d, and f orbitals. The magnetic quantum number (m) determines the orientation of theorbital in space.For example, the 1s orbital is the ground state orbital of the hydrogen atom, with n=1, l=0, and m=0. This orbital is spherically symmetric and has the lowest energy level. The 2s and 2p orbitals are examples of excited stateorbitals, with n=2. The 2s orbital is spherically symmetric like the 1s orbital, while the 2p orbitals have different shapes and orientations.The quantum mechanical model of the hydrogen atom provides a detailed understanding of the behavior of electrons in atoms. It explains phenomena such as the quantization of energy levels, the stability of atoms, and the formation of chemical bonds. This model has been successful in predicting and explaining a wide range of experimental observations in atomic physics.中文回答：氢原子的量子力学模型是物理学中的一个基本概念，描述了单个氢原子的行为。

Quantum walks on the hypercube

Quantum Walks on the HypercubeC RISTOPHER M OOREComputer Science DepartmentUniversity of New Mexico,Albuquerque and the Santa Fe Institute,Santa Fe,New Mexicomoore@A LEXANDER R USSELL Department of Computer Science and Engineering University of ConnecticutStorrs,Connecticutacr@November12,2001AbstractRecently,it has been shown that one-dimensional quantum walks can mix more quickly than clas-sical random walks,suggesting that quantum Monte Carlo algorithms can outperform their classicalcounterparts.We study two quantum walks on the n-dimensional hypercube,one in discrete time andone in continuous time.In both cases we show that the instantaneous mixing time isπ4n steps,fasterthan theΘn log n steps required by the classical walk.In the continuous-time case,the probabilitydistribution is exactly uniform at this time.On the other hand,we show that the average mixing timeas deﬁned by Aharonov et al.[AAKV01]isΩn32in the discrete-time case,slower than the classical walk,and nonexistent in the continuous-time case.This suggests that the instantaneous mixing time is amore relevant notion than the average mixing time for quantum walks on large,well-connected graphs.Our analysis treats interference between terms of different phase more carefully than is necessary for thewalk on the cycle;previous general bounds predict an exponential average mixing time when applied tothe hypercube.1IntroductionRandom walks form one of the cornerstones of theoretical computer science.As algorithmic tools,theyhave been applied to a variety of central problems,such as estimating the volume of a convex body[DFK91,LK99],approximating the permanent[JS89,JSV00],andﬁnding satisfying assignments for Boolean for-mulae[Sch99].Furthermore,the basic technical phenomena appearing in the study of random walks(e.g.,spectral decomposition,couplings,and Fourier analysis)also support several other important areas such aspseudorandomness and derandomization(see,e.g.,[AS92,(9,15)]).The development of efﬁcient quantum algorithms for problems believed to be intractable for classicalrandomized computation,like integer factoring and discrete logarithm[Sho97],has prompted the investi-gation of quantum walks.This is a natural generalization of the traditional notion discussed above where,roughly,the process evolves in a unitary rather than stochastic fashion.The notion of“mixing time,”theﬁrst time when the distribution induced by a random walk is sufﬁcientlyclose to the stationary distribution,plays a central role in the theory of classical random walks.For a givengraph,then,it is natural to ask if a quantum walk can mix more quickly than its classical counterpart.(Sincea unitary process cannot be mixing,we deﬁne a stochastic process from a quantum one by performinga measurement at a given time or a distribution of times.)Several recent articles[AAKV01,ABN01,NV00]have answered this question in the afﬁrmative,showing,for example,that a quantum walk on then-cycle mixes in time O n log n,a substantial improvement over the classical random walk which requires Θn2steps to mix.Quantum walks were also deﬁned in[Wat01],and used to show that undirected graph connectivity is contained in a version of quantum LOGSPACE.These articles raise the exciting possibilitythat quantum Monte Carlo algorithms could form a new family of quantum algorithms that work morequickly than their classical counterparts.Two types of quantum walks exist in the literature.Theﬁrst,introduced by[AAKV01,ABN01,NV00],studies the behavior of a“directed particle”on the graph;we refer to these as discrete-time quantumwalks.The second,introduced in[FG98,CFG01],deﬁnes the dynamics by treating the adjacency matrixof the graph as a Hamiltonian;we refer to these as continuous-time quantum walks.The landscape isfurther complicated by the existence of two distinct notions of mixing time.The“instantaneous”notion[ABN01,NV00]focuses on particular times at which measurement induces a desired distribution,whilethe“average”notion[AAKV01],another natural way to convert a quantum process into a stochastic one,focuses on measurement times selected randomly from some interval.In this article,we analyze both the continuous-time and a discrete-time quantum walk on the hypercube.In both cases,the walk is shown to have an instantaneous mixing time atπ4n.(Of course,in the discretewalk all times are integers.)Recall that the classical walk on the hypercube mixes in timeΘn log n,sothat the quantum walk is faster by a logarithmic factor.Moreover,in the discrete-time case the walk mixesin time less than the diameter of the graph,sinceπ41;astonishingly,in the continuous-time case theprobability distribution at tπ4n is exactly uniform.Both of these things happen due to a marvelousconspiracy of destructive interference between terms of different phase.These walks show i.)a similarity between the two notions of quantum walks,and ii.)a disparity between the two notions of quantum mixing times.As mentioned above,both walks have an instantaneous mixing time at timeπ4n.On the other hand,we show that the average mixing time of the discrete-time walk isΩn32,slower than the classical walk,and that for the continuous-time walk there is no time at which the time-averaged probability distribution is close to uniform in the sense of[AAKV01].Our results suggest that for large graphs(large compared to their mixing time)the instantaneous notion of mixing time is more appropriate than the average one,since the probability distribution is close to uniform only in a narrow window of time.The analysis of the hypercubic quantum walk exhibits a number of features markedly different fromthose appearing in previously studied walks.In particular,the dimension of the relevant Hilbert space is,for the hypercube,exponential in the length of the desired walk,while in the cycle these quantities are roughly equal.This requires that interference be handled in a more delicate way than is required for the walk on the cycle;in particular,the general bound of[AAKV01]yields an exponential upper bound on the mixing time for the discrete-time walk.We begin by deﬁning quantum walks and discussing various notions of mixing time.We then analyze the two quantum walks on the hypercube in Sections2and3.(Most of the technical details for the discrete-time walk are relegated to an appendix.)1.1Quantum walks and mixing timesAny graph G V E gives rise to a familiar Markov chain by assigning probability1d to all edges leaving each vertex v of degree d.Let P t u v be the probability of visiting a vertex v at step t of the random walk on G starting at u.If G is undirected,connected,and not bipartite,then lim t∞P t u exists1and is independent of u.A variety of well-developed techniques exist for establishing bounds on the rate at which P t u achieves this limit(e.g.,[Vaz92]);if G happens to be the Cayley graph of a group(as are,for example,the cycle and the hypercube),then techniques from Fourier analysis can be applied[Dia88].Below we will use some aspects of this approach,especially the Diaconis-Shahshahani bound on the total variation distance[DS81].For simplicity,we restrict our discussion to quantum walks on Cayley graphs;more general treatments of quantum walks appear in[AAKV01,FG98].Before describing the quantum walk models we set down some notation.For a group G and a set of generatorsΓsuch thatΓΓ1,let X GΓdenote the undirected Cayley graph of G with respect toΓ.For a ﬁnite set S,we let L S f:S denote the collection of-valued functions on S with∑s S f s2 1. This is a Hilbert space under the natural inner product f g∑s S f s g s.For a Hilbert space V,a linear operator U:V V is unitary if for all v w V,v w U v U w;if U is represented as a matrix,this is equivalent to the condition that U†U1where†denotes the Hermitian conjugate.There are two natural quantum walks that one can deﬁne for such graphs,which we now describe.T HE DISCRETE-TIME WALK:This model,introduced by[AAKV01,ABN01,NV00],augments the graph with a direction space,each basis vector of which corresponds one of the generators inΓ.A step of the walk then consists of the composition of two unitary transformations;a shift operator which leaves the direction unchanged while moving the particle in its current direction,and a local transformation which op-erates on the direction while leaving the position unchanged.To be precise,the quantum walk on X GΓis deﬁned on the space L GΓL G LΓ.LetδγγΓbe the natural basis for LΓ,andδg g G the natural basis for L G.Then the shift operator is S:δgδγδgγδγ,and the local transformation isˇD1D where D is deﬁned on LΓalone and1is the identity on L G.Then one“step”of the walk corresponds to the operator UˇDV.If we measure the position of the particle,but not its direction,at time t,we observe a vertex v with probability P t v∑γΓU tψ0δvδγ2whereψ0L GΓis the initial state.T HE CONTINUOUS-TIME WALK:This model,introduced in[FG98],works directly on L G.The walk evolves by treating the adjacency matrix of the graph as a Hamiltonian and using the Schr¨o dinger equation. Speciﬁcally,if H is the adjacency matrix of X GΓ,the evolution of the system at time t is given by U t, where U t eq e iHt(here we use the matrix exponential,and U t is unitary since H is real and symmetric). Then if we measure the position of the particle at time t,we observe a vertex v with probability P t vU tψ0δv2whereψ0is the initial state.Note the analogy to classical Poisson processes:since U t e iHt 1In fact,this limit exists under more general circumstances;see e.g.[MR95].1iHt iHt22,the amplitude of making s steps is the coefﬁcient it s s!of H s,which up to normalization is Poisson-distributed with mean t.Remark.In[CFG01],the authors point out that deﬁning quantum walks in continuous time allows unitarity without having to extend the graph with a direction space and a chosen local operation.On the other hand,it is harder to see how to carry out such a walk in a generically programmable way using only local information about the graph,for instance in a model where we query a graph toﬁnd out who our neighbors are.Instead,continuous-time walks might correspond to special-purpose analog computers,where we build in interactions corresponding to the desired Hamiltonian and allow the system to evolve in continuous time.In both cases we start with an initial wave function concentrated at a single vertex corresponding to the identity u of the group.For the continuous-time walk,this corresponds to a wave functionψ0v ψ0δv(so thatψ0u1andψ0v0for all v u).For the discrete-time walk,we start with a uniform superposition over all possible directions,ψ0vγψ0δvδγ1Γif v u 0otherwise.For the hypercube,u000.In order to deﬁne a discrete-time quantum walk,one must select a local operator D on the directionspace.In principle,this introduces some arbitrariness into the deﬁnition.However,if we wish D to respectthe permutation symmetry of the n-cube,and if we wish to maximize the operator distance between D andthe identity,we show in Appendix A that we are forced to choose Grover’s diffusion operator[Gro96],which we recall below.We call the resulting walk the“symmetric discrete-time quantum walk”on then-cube.(Watrous[Wat01]also used Grover’s operator to deﬁne quantum walks on undirected graphs.)Since for large n Grover’s operator is close to the identity matrix,one might imagine that it would takeΩn12steps to even change direction,giving the quantum walk a mixing time of n32,slower than the classical random walk.However,like many intuitions about quantum mechanics,this is simply wrong.Since the evolution of the quantum walk is governed by a unitary operator rather than a stochastic one,unless P t is constant for all t,there can be no“stationary distribution”lim t∞P t.In particular,for anyε0,there are inﬁnitely many(positive,integer)times t for which U t1εso that U tψuψuεand P t is close to the initial distribution.However,there may be particular stopping times t which induce distributions close to,say,the uniform distribution,and we call these instantaneous mixing times:Deﬁnition1We say that t is anε-instantaneous mixing time for a quantum walk if P t Uε,where A B12∑v A v B v denotes total variation distance and U denotes the uniform distribution.For these walks we show:Theorem1For the symmetric discrete-time quantum walk on the n-cube,t kπ4n is anε-instantaneousmixing time withεO n76for all odd k.and,even more surprisingly,Theorem2For the continuous-time quantum walk on the n-cube,t kπ4n is a0-instantaneous mixingtime for all odd k.Thus in both cases the mixing time isΘn,as opposed toΘn log n as it is in the classical case.Aharonov et al.[AAKV01]deﬁne another natural notion of mixing time for quantum walks,in whichthe stopping time t is selected uniformly from the set0T1.They show that the time-averageddistributions¯P T1T∑T1t0P t do converge as T∞and study the rate at which this occurs.For a continuous-time random walk,we analogously deﬁne the distribution¯P T v1T T0P t v d t.Then we call a time at which¯P T is close to uniform an average mixing time:Deﬁnition2We say that T is anε-average mixing time for a quantum walk if¯P T Uε.In this paper we also calculate theε-average mixing times for the hypercube.For the discrete-time walk,it is even longer than the mixing time of the classical random walk:Theorem3For the discrete-time quantum walk on the n-cube,theε-average mixing time isΩn32ε. This is surprising given that the instantaneous mixing time is only linear in n.However,the probability distribution is close to uniform only in narrow windows around the odd multiples ofπ4n,so¯P T is far from uniform for signiﬁcantly longer times.We also observe that the general bound given in[AAKV01]yields an exponential upper bound on the average mixing time,showing that it is necessary to handle interference for walks on the hypercube more carefully than for those on the cycle.For the continuous-time walk the situation is even worse:while it possesses0-instantaneous mixing times at all odd multiples ofπ4n,the limiting distribution lim T∞¯P T is not uniform,and we show the following:Theorem4For the continuous-time quantum walk on the n-cube,there existsε0such that no time is an ε-average mixing time.Our results suggest that in both the discrete and continuous-time case,the instantaneous mixing time is a more relevant notion than the average mixing time for large,well-connected graphs.2The symmetric discrete-time walkIn this section we prove Theorem1.We treat the n-cube as the Cayley graph of n2with the regular basis vectors e i010with the1appearing in the i th place.Then the discrete-time walk takes place in the Hilbert space L n2n where n1n.Here theﬁrst component represents the position x of the particle in the hypercube,and the second component represents the particle’s current“direction”;if this is i,the shift operator willﬂip the i th bit of x.As in[AAKV01,NV00],we will not impose a group structure on the direction space,and will Fourier transform only over the position space.For this reason,we will express the wave functionψL n2n as a functionΨ:n2n,where the i th coordinate ofΨx is the projection ofψintoδxδi,i.e.the complex amplitude of the particle being at position x with direction i.The Fourier transform of such an elementΨis˜Ψ:n2n,where˜Ψk∑x1k xΨx.Then the shift operator for the hypercube is S:Ψx∑n i1πiΨx e i where e i is the i th basis vector in the n-cube,andπi is the projection operator for the i th direction.The reason for considering the Fourier transform above is that the shift operator isdiagonal in the Fourier basis:speciﬁcally it maps˜Ψk Sk˜Ψk whereS k 1k101k2...01k nFor the local transformation,we use Grover’s diffusion operator on n states,D i j2nδi j.The advantage of Grover’s operator is that,like the n-cube itself,it is permutation symmetric.We use thissymmetry to rearrange UkSkD to put the negated rows on the bottom,UkSkD2n12n2n2n12n......2n12n2n2n2n1......where the top and bottom blocks have n k and k rows respectively;here k is the Hamming weight of k.The eigenvalues of Uk then depend only on k.Speciﬁcally,Ukhas the eigenvalues1and1withmultiplicity k1and n k1respectively,plus the eigenvaluesλλwhereλ12kn2ink n k e iωkandωk0πis described bycosωk12knsinωk2nk n kIts eigenvectors with eigenvalue1span the k1-dimensional subspace consisting of vectors with support on the k“ﬂipped”directions that sum to zero,and similarly the eigenvectors with eigenvalue1span the n k1-dimensional subspace of vectors on the n k other directions that sum to zero.We call these the trivial eigenvectors.The eigenvectors ofλλe iωk arev k v k1in k 1 kWe call these the non-trivial eigenvectors for a given k.Over the space of positions and directions these eigenvectors are multiplied by the Fourier coefﬁcient1k x,so as a function of x and direction1j n the two non-trivial eigenstates of the entire system,for a given k,arev k x j1k x2n21if kj1i n k if k j0with eigenvalue e iωk,and its conjugate vkwith eigenvalue e iωk.We take for our initial wave function a particle at the origin000in an equal superposition of directions.Since its position is aδ-function in real space it is uniform in Fourier space as well as over thedirection space,giving˜Ψ0k2n2n 11This is perpendicular to all the trivial eigenvectors,so theiramplitudes are all zero.The amplitude of its component along the non-trivial eigenvector vkisa k Ψ0vk2n2kni1kn(1)and the amplitude of vk is ak.Note that ak22n2,so a particle is equally likely to appear in eithernon-trivial eigenstate with any given wave vector.At this point,we note that there are an exponential number of eigenvectors in which the initial state has a non-zero amplitude.In Section2.1,we observe that for this reason the general bound of Aharonov et al. [AAKV01]yields an exponential(upper)bound on the mixing time.In general,this bound performs poorly whenever the number of important eigenvalues is greater than the mixing time.Instead,we will use the Diaconis-Shahshahani bound on the total variation distance in terms of the Fourier coefﬁcients of the probability[Dia88].If P t x is the probability of the particle being observed at position x at time t,and U is the uniform distribution,then the total variation distance is bounded byP t U214∑k00k11˜Pt k214n1∑k1nk˜Pt k2(2)Here we exclude both the constant term and the parity term k 11;since our walk changes position at every step,we only visit vertices with odd or even parity at odd or even times respectively.Thus U here means the uniform distribution with probability 2n 1on the vertices of appropriate parity.To ﬁnd ˜Pt k ,we ﬁrst need ˜Ψt k .As Nayak and Vishwanath [NV00]did for the walk on the line,we start by calculating the t th matrix power of U k .This isU t k a 1t aa a1t c ......b 1t bc bb 1t ......wherea cos ωk t 1t n k b cos ωk t 1t k and c sin ωk t Starting with the uniform initial state,the wave function after t steps is˜Ψt k 2n 2n cos ωk t n k sin ωk t n kcos ωk t n kksin ωk t k (3)In the next two sections we will use this diagonalization to calculate the average and instantaneous mixing times,which are Ωn 32and Θn respectively.2.1Bounds on the average mixing time of the discrete-time walkIn this section,we prove Theorem 3.To do this,it’s sufﬁcient to calculate the amplitude at the origin.Fouriertransforming Equation 3back to real space at x 00gives ψt 02n 2∑k ˜Ψt k 2n n n ∑k 0n k cos ωk t cos ωk tnThe probability the particle is observed at the origin after t steps is thenP t 0ψt 022nn ∑k 0n k cos ωk t 2Let k 1x n 2.For small x ,k is near the peak of the binomial distribution,and ωk cos 1x π2x O x 3so the angle θbetween ωk for successive values of k is roughly constant,θ2n O x 2leading to constructive interference if θt 2π.Speciﬁcally,let t m be the even integer closest to πmn forinteger m .Then cos ωk t mcos 2πkm O x 3mn 1O x 6m 2n 2.By a standard Chernoff bound,2n ∑k 1x n 2n k o 1so long as x ωn 12.Let x νn n 12where νn is a function that goes to inﬁnity slowly as a function of n .We then write P t 0o 12n1x n 2∑k 1x n 2n k 1O x 6m 2n 221O νn 6m 2nwhich is 1o 1as long as m o n 12νn 3,in which case t mo n 32νn 3.For a functionψ:n2n withψ21and a set S n2,we say thatψis c-supported on S if the probability x S is at least c,i.e.∑x S d nψx d2c.The discussion above shows thatψt m is 1o1-supported on0for appropriate t mπmn.Note that ifψis c-supported on0then,as U is local,U kψmust be c1c2c1-supported on W k,the set of vertices of weight k.(The factor of1c is due to potential cancellation with portions of the wave function supported outside0.)Inparticular,at times t m k,for k n2n,ψt m k is1o1-supported on W n2x.If x x nωn,then W12δn2n o1and,evidently,the average1T∑T i P i has total variation distance1o1from the uniform distribution if T o n32.Thus we see that in the sense of[AAKV01],the discrete-time quantum walk is actually slower than the classical walk.In the next section,however,we show that its instantaneous mixing time is only linear in n.We now observe that the general bound of[AAKV01]predicts an average mixing time for the n-cube which is exponential in n.In that article it is shown that the variation distance between¯P T and the uniform distribution(or more generally,the limiting distribution lim T∞¯P T)is bounded by a sum over distinct pairs of eigenvalues,¯PT U 2T∑i j s tλiλja i2λiλj(4)where a iψ0v i is the component of the initial state along the eigenvector v i.(Since this bound includes eigenvaluesλj for which a j0,we note that it also holds when we replace a i2with a i a j,using the same reasoning as in[AAKV01].)For the quantum walk on the cycle of length n,this bound gives an average mixing time of O n log n. For the n-cube,however,there are exponentially many pairs of eigenvectors with distinct eigenvalues,all ofwhich have a non-zero component in the initial state.Speciﬁcally,for each Hamming weight k there are nk non-trivial eigenvectors each with eigenvalue e iωk and e iωk.These complex conjugates are distinct from each other for0k n,and eigenvalues with distinct k are also distinct.The number of distinct pairs is thenn1∑k1nk24n∑k k0nknkΩ4nTaking a k2n2from Equation1and the fact thatλiλj2since theλi are on the unit circle, we see that Equation4gives an upper bound on theε-average mixing time of sizeΩ2nε.In general,this bound will give a mixing time of O Mεwhenever the initial state is distributed roughly equally over M eigenvectors,and when these are roughly equally distributed overω1distinct eigenvalues.2.2The instantaneous mixing time of the discrete-time walkTo prove Theorem1we could calculateΨt x by Fourier transforming˜P t k back to real space for all x. However,this calculation turns out to be signiﬁcantly more awkward than calculating the Fourier trans-form of the probability distribution,˜P t k,which we need to apply the Diaconis-Shahshahani bound.Since P t xΨt xΨt x,and since multiplications in real space are convolutions in Fourier space,we perform a convolution over n2:˜Pt k∑k ˜Ψt k˜Ψt k kwhere the inner product is deﬁned on the direction space,u v∑n i1u i v i.We write this as a sum over j, the number of bits of overlap between k and k,and l,the number of bits of k outside the bits of k(and so overlapping with k k).Thus k has weight j l,and k k has weight k j l.Calculating the dot product˜Ψt k˜Ψt k k explicitly from Equation3as a function of these weights and overlaps gives˜P t k 12k∑j0n k∑l0kjn klcosωj l t cosωk j l t A sinωj l t sinωk j l t(5)whereA cosωk cosωj l cosωk j l sinωj l sinωk j lThe reader can check that this gives˜P t01for the trivial Fourier component where k0,and˜P t n 1t for the parity term where k n.Using the identities cos a cos b12cos a b cos a b and sin a sin b12cos a b cos a b we can re-write Equation5as˜P t k 12k∑j0n k∑l0kjn kl1A2cosωt1A2cosωt12k∑j0n k∑l0kjn klY(6)whereωωj lωk j l.The terms cosωt in Y are rapidly oscillating with a frequency that increases with t.Thus,unlike the walk on the cycle,the phase is rapidly oscillating everywhere,as a function of either l or j.This will make the dominant contribution to˜P t k exponentially small when t nπ4,giving us a small variation distance when we sum over all k.To give some intuition for the remainder of the proof,we pause here to note that if Equation6were an integral rather than a sum,we could immediately approximate the rate of oscillation of Y toﬁrst order at the peaks of the binomials,where j k2and l n k 2.One can check that dωk d k2n and hence dωd l dωd j4n.Since A1,we would then write˜Pt k O 12k∑j0n k∑l0kjn kle4i jt n e4ilt nwhich,using the binomial theorem,would give˜Pt k O 1e4it n2k1e4it n2n kcos k2tncos n k2tn(7)In this case the Diaconis-Shahshahani bound and the binomial theorem giveP t U214∑0k nnkcos k2tncos n k2tn2122cos22tnn1cos22tnn1If we could take t to be the non-integer valueπ4n,these cosines would be zero.This will,in fact,turn out to be the right answer.But since Equation6is a sum,not an integral,we have to be wary of resonances where the oscillations are such that the phase changes by a multiple of2πbetween adjacent terms,in which case these terms will interfere constructively rather than destructively.Thus to show that theﬁrst-order oscillation indeed dominates,we have a signiﬁcant amount of work left to do.The details of managing these resonances can be found in Appendix B.The process can be summarized as follows:i.)we compute the Fourier transform of the quantity Y in Equation6,since the sum of Equation6 can be calculated for a single Fourier basis function using the binomial theorem;ii.)the Fourier transform0.20.40.60.810.20.40.60.81(a)Variation distance at time t as a function of t n .(b)Log 2Probability as a function of Hamming weight.Figure 1:Graph (a)plots an exact calculation of the total variation distance after t steps of the quantum walk for hypercubes of dimension 50,100,and 200,as a function of t n .At t n π4the variation distance is small even though the walk has not had time to cross the entire graph.This happens because the distribution is roughly uniform across the equator of the n -cube where the vast majority of the points are located.Note that the window in which the variation distance is small gets narrower as n increases.Graph (b)shows the log 2probability distribution on the 200-dimensional hypercube as a function of Hamming distance from the starting point after 157π4n steps.The probability distribution has a plateau of 2199at the equator,matching the uniform distribution up to parity.of Y can be asymptotically bounded by the method of stationary phase.The dominant stationary point corresponds to the ﬁrst-order oscillation,but there are also lower-order stationary points corresponding to faster oscillations;so iii.)we use an entropy bound to show that the contribution of the other stationary points is exponentially small.To illustrate our result,we have calculated the probability distribution,and the total variation distance from the uniform distribution (up to parity),as a function of time for hypercubes of dimension 50,100,and 200.In order to do this exactly,we use the walk’s permutation symmetry to collapse its dynamics to a function only of Hamming distance.In Figure 1(a)we see that the total variation distance becomes small when t n π4,and in Figure 1(b)we see how the probability distribution is close to uniform on a “plateau”across the hypercube’s equator.Since this is where the vast majority of the points are located,the total variation distance is small even though the walk has not yet had time to cross the entire graph.3The continuous-time walkIn the case of the hypercube,the continuous-time walk turns out to be particularly easy to analyze.Theadjacency matrix,normalized by the degree,is H x y1n if x and y are adjacent,and 0otherwise.Interpreting H as the Hamiltonian treats it as the energy operator,and of course increasing the energy makes the system run faster;we normalize by the degree n in order to keep the maximum energy of the system,and so the rate at which transitions occur,constant as a function of n .The eigenvectors of H and U t are simply the Fourier basis functions:if v k x1k x then Hv k 12k n v k and U t v k e it 12k n v k where we again use k to denote the Hamming weight of k .If our initial wave vector has a particle at 0,then its initial Fourier spectrum is uniform,and at time t we have˜Ψt k 2n 2e it 12k n Again writing the probability P as the convolution of Ψwith Ψin Fourier space,。

谈论太空旅行的英语作文50词左右

谈论太空旅行的英语作文50词左右全文共6篇示例，供读者参考篇1Space Travel - A Cosmic Adventure!Hi there! My name is Sam and I'm 10 years old. I love learning about space and dreaming about traveling to other planets someday. Space travel is so fascinating to me. Let me tell you all about it!First off, what exactly is space travel? It's when humans go up into outer space using rockets and spacecraft. We've been sending astronauts and probes out of Earth's atmosphere for many decades now. Some of the most famous space missions were the Apollo moon landings in the 1960s and 1970s. Can you imagine walking on the moon? Those astronauts were so brave and cool!Nowadays, astronauts travel to and live aboard the International Space Station (ISS) for months at a time. The ISS is like a big house in space, orbiting around the Earth. Up there, they conduct all sorts of experiments and studies. Being inmicrogravity and experiencing weightlessness must feel so weird and fun!But why do we explore space? Well, there are lots of reasons. Scientists want to learn more about the universe, planets, stars, black holes, and everything out there. Maybe we'll find evidence of alien life someday! Space agencies also test new technologies that could help us live and work in space long-term.Some people even dream of sending humans to Mars one day and establishing colonies there. How awesome would it be to be one of the first kids born and raised on another planet? Just imagine going outside without a spacesuit and bouncing around on Mars' lower gravity. You could jump really high!When astronauts go outside the spacecraft to do repairs or experiments, they wear special pressurized spacesuits to protect them. But imagine if your tether broke while you were out on a spacewalk? You could float away into darkness forever! So scary.That's why we need new innovations to make space exploration safer and more sustainable. Maybe someday they'll figure out how to create artificial gravity or soils for growing food on long space missions. And how neat would it be to look out your window and see rockets launching from a space hotel?I think the greatest challenge for space travel is how to get enough fuel, supplies, and energy for really long-distance voyages. The distances between planets, moons, and stars are just mind-bogglingly huge. Our current chemical rockets would take forever to reach even nearby stars.But scientists are working on advanced propulsion systems like ion engines, nuclear thermal rockets, and even experimental warp drives. Nuclear reactors could power spacecraft for millions of miles. Or maybe we could use solar sails that get pushed along by the sun's light?If we want to travel light years across the galaxy, we'll definitely need better solutions. Some experts think we may need to build giant ships to carry everything we'd need to survive these multi-generational journeys. We're talking entire mobile space colonies housing thousands of people! That's like having an entire city in space.Just imagine being part of that crew - you'd be born, live, work and have kids all while cruising through the cosmos. Your great-grandkids would be the ones to finally reach the new planet or star system after decades of travel. It'd be hard leaving Earth forever, but you'd get to become one of humanity's great space pioneers!Those space family ships would require facilities like farms, schools, rec areas and science labs, just like a little world unto themselves. They'd have to recycle all their air, water and waste for hundreds of years in transit. So much could go wrong over those vast distances and time spans. But solving those challenges could help humans become a space-faring species.In the very far future, maybe we'll build ships that can travel close to light-speed using exotic matter or warping space-time itself. Or perhaps we'll discover some cosmic shortcut through wormholes to zip across the universe. That'd be incredible! Just think - we could visit alien worlds, see strange nebulas and supernovas up close, and uncover the mind-blowing secrets of the cosmos.Space travel isn't just vital for exploration, but it could be our survival if anything catastrophic ever happens on Earth. With space colonies and farms on other worlds, we'd ensure that human civilization lives on no matter what. All our eggs wouldn't be stuck in one planetary basket anymore. How reassuring is that?So while I probably won't be the first kid on Mars, I'm still super excited to see where space exploration goes next. MaybeI'll get to journey to the moon, an asteroid, or one of the awesome moons of Saturn or Jupiter someday. A girl can dream!I hope you've enjoyed learning about space travel from me, and that you're just as psyched about humanity's space-faring future. After all, space is the greatest cosmic frontier, just waiting for us to explore it. Who knows what mind-blowing discoveries and adventures are out there? Let's go find out!篇2Title: My Dream of Travelling to SpaceHey there! My name is Tommy, and I'm a 10-year-old kid who loves dreaming about the stars. Ever since I was a little boy, I've always been fascinated by the idea of going to space. Whenever I look up at the night sky, I can't help but imagine myself soaring through the vast expanse of the universe, exploring new worlds and making incredible discoveries.I've read countless books and watched countless movies about space travel, and each time, I'm left in awe. The thought of leaving Earth's atmosphere and venturing into the unknown is both thrilling and terrifying. But you know what? I'm not scared! In fact, I can't wait to experience the wonders of space firsthand.Imagine being one of the first people to set foot on Mars! Can you picture it? The red, dusty landscape stretching out before you, the alien sky above, and the knowledge that you're standing on a world that no human has ever walked on before. How incredible would that be?Or what about exploring the moons of Jupiter or Saturn? Imagine floating through the icy clouds of Europa or sailing across the methane seas of Titan. It's like something straight out of a science fiction movie, but it could one day be a reality!And who knows what other amazing breakthroughs await us in the future? Maybe we'll find a way to harness the energy of black holes or discover new forms of life on distant exoplanets. The possibilities are endless, and that's what makes space exploration so exciting!Now, I know what you're thinking: "Tommy, space travel is incredibly dangerous and expensive. How could a kid like you ever hope to be an astronaut?" Well, let me tell you something. I'm not just dreaming – I'm planning! I'm already studying hard in school, especially in math and science, because those are the subjects that will help me understand the complex world of space exploration.And sure, it's going to be tough. I'll have to work harder than I ever have before, and there will be plenty of challenges and setbacks along the way. But you know what? I'm not afraid of hard work. In fact, I embrace it! Because every challenge I overcome will bring me one step closer to my dream of travelling to space.Who knows, maybe one day you'll see me on the news, suiting up for a mission to Mars or boarding a spacecraft bound for the outer reaches of our solar system. And when that day comes, you can bet I'll be grinning from ear to ear, because I'll know that all my hard work and dedication has finally paid off.So, to all the other kids out there who dream of exploring the cosmos, I say this: never give up on your dreams! Study hard, work hard, and never let anyone tell you that your goals are too lofty or unrealistic. Because with enough determination and perseverance, anything is possible – even travelling to the stars.Well, that's all from me for now. I've got to get back to my homework (yes, even aspiring astronauts have to do their math and science homework!). But before I go, I'll leave you with one final thought: when you look up at the night sky tonight, take a moment to marvel at the vastness of the universe and all the wonders it holds. And who knows? Maybe one day, you'll be theone exploring those wonders firsthand. Dream big, my friends, and never stop reaching for the stars!篇3Space Travel - A Journey to the Stars!Hi there! I'm really excited to share my thoughts on space travel with you. It's something that has fascinated me ever since I was a little kid watching cartoons about rocket ships and aliens. The idea of leaving our planet and exploring the vast unknown just seems so cool to me!First off, I think space travel is incredibly important for helping us learn more about the universe we live in. There's still so much we don't know or understand about what's out there. By sending spacecraft and eventually even people to other planets, moons, and far away galaxies, we can study them up close. Imagine getting to walk on Mars or one of Jupiter's moons – how awesome would that be?!Space exploration also allows scientists to conduct experiments in a unique environment with little gravity and no atmosphere. This could lead to new discoveries in fields like physics, biology, and medicine that wouldn't be possible if we just stayed on Earth. Maybe we'll even find cures for diseases orinvent some kind of amazing new technology thanks to space research.Another reason I'm fascinated by space travel is the possibility of finding alien life somewhere out in the cosmos. While little green men might just be science fiction, there could very well be some kind of microbes or other primitive lifeforms living on planets circling distant stars. Finding evidence of extraterrestrial life, even if it's just tiny microorganisms, would be one of the most profound discoveries in human history!But despite the risks, I still dream of the day when human beings finally set foot on another planet in our solar system. Can you imagine how it would feel to take your first steps on the rusty red soil of Mars knowing you're the first person ever to walk on another world? Or to gaze up at the massive ice-crusted oceans of Jupiter's moon Europa, wondering what strangefish-like creatures might lurk in those mysterious depths?Just picturing it gives me goosebumps!There's also the tantalizing prospect of sending crewed missions to planets orbiting other stars light years from Earth if we can develop advanced propulsion technology. I'm talking about real-life interstellar travel straight out of the science fiction movies! While that seems implausible with our current level oftechnology, who's to say we won't crack the secrets of warp drives or hyper-fast ion engines or something even more amazing a few centuries from now?With an interstellar spacecraft, the mind-boggling distances between stars wouldn't be such an obstacle anymore. We could journey to the weird dimly-lit worlds of the TRAPPIST-1 system just 40 light years away. Or ride doughnut-shaped ringworlds constructed by super-advanced alien civilizations (if they exist!). The possibilities would be endless.I know interstellar travel is still the stuff of dreams right now. But I like to imagine a future where the great journey to explore strange new worlds beckons the bravest and most intrepid explorers of our time – the future Christopher Columbuses, Ferdinand Magellans, and Neil Armstrongs of interstellar space. Maybe that'll be me someday if I study real hard and become an astronaut? A kid can dream!For now though, I'll be cheering on all the exciting robotic space missions like the Mars rovers and the James Webb Space Telescope. Learning about their discoveries is the next best thing to being there myself. I feel so lucky to be living in an age where we're constantly making new inroads into understanding thefinal frontier. Visiting other planets used to be pure science fiction, but now it's becoming science fact before our very eyes.So that's why I'm so obsessed with space travel and exploration. I think it represents the curiosity, daring, and thirst for adventure that has driven human accomplishments throughout our entire history – but now on the grandest scale imaginable. Space is the next frontier, and I can't wait to see where our journey amongst the stars takes us next. Maybe sometime in the not-so-distant future, people will look up at the night sky and know that's not just stars and blackness up there, but uncharted worlds waiting to be explored. How amazing is that?Well, I've rambled on enough about my passion for space travel. I'd better stop here before my teacher thinks I'm writing a book report instead of an essay! Let's just say the unknown depths of space fill me with wonder and excitement about all the possibilities out there just waiting to be discovered. We've only scratched the surface so far...the greatest adventures still lie ahead. Isn't that thrilling to think about? Space – the final frontier. These are the voyages...篇4Space Travel: A Kid's Dream Come TrueDo you dream of traveling to space one day? I sure do! The thought of blasting off in a rocket ship and zooming through the stars gets me so excited. There's a whole universe out there waiting to be explored, and I can't wait to be one of the brave space explorers who gets to see it all!Just imagine what it would be like to leave Earth behind and journey into the inky blackness of space. You'd be surrounded by twinkling stars as far as the eye can see, almost like a billion little night lights in the sky. And planets of all different colors might come into view – maybe you篇5Space Travel: A Young Explorer's DreamHi there! My name is Sam and I'm 10 years old. Today, I want to tell you all about my biggest dream - to travel in space! Ever since I was a little kid, I've been fascinated by the stars, planets, and the vast unknown that lies beyond our Earth.I still remember the first time I looked through a telescope and saw the craters on the Moon up close. It was like beingtransported to another world! From that moment on, I became obsessed with learning everything I could about space.In school, I always raise my hand whenever the teacher talks about the solar system or space exploration. I can rattle off all the planet names in order from the Sun - Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune. I even know some of the moons like Ganymede, Europa, and our very own Moon!My favorite planet is definitely Mars though, probably because it's known as the Red Planet. I've read so much about the Mars rovers like Curiosity that have been exploring the surface. Wouldn't it be so cool to drive around on Mars yourself? Or maybe even be one of the first people to set foot there? Just imagine leaving your bootprint on an alien world. Mind-blowing!Jupiter is a close second favorite planet of mine, mainly because of its hugeness and that crazy storm called the Great Red Spot. It's a giant hurricane bigger than the entire Earth that has been raging for centuries! I'd love to get a close-up view of that monster storm from space someday.Speaking of views from space, I often dream about what it must be like to look back at our beautiful blue-and-green planet hanging in the inky blackness. Astronauts say there's nothing that compares to that incredible sight. I have a poster of theiconic "Pale Blue Dot" photo hanging above my bed to inspire my space dreams.Once in space, I'd love to try operating the robot arms to catch visiting spacecraft or perform experiments. Or maybe even go outside on a spacewalk! You know, being one of the brave few to put on a spacesuit and float around in the eerie vacuum, tethered to your spaceship home. I'd get to witness incredible views few have ever seen before with my own eyes.And who knows? If I pursued my space dreams far enough, I could perhaps even travel to other planets or moons in our solar system someday. How crazy would it be to actually set foot on Mars after studying it for so long from Earth? Or sail past the bizarre clouds of Jupiter in a篇6Space Travel - A Journey to the StarsHi there! My name is Timmy and I'm 10 years old. Today, I'm going to tell you all about space travel and why I think it's the coolest thing ever!Have you ever looked up at the night sky and wondered what's out there? I sure have! There are billions and billions ofstars twinkling away, and even more planets and moons. Space is absolutely huge – it just goes on forever and ever. Isn't that amazing?For a long time, humans could only dream about travelling to space. We had to use our imaginations and make up stories about visiting other worlds. But then, something incredible happened – we actually figured out how to leave Earth and explore the cosmos!The first person to go to space was a Russian guy named Yuri Gagarin. On April 12, 1961, he blasted off in a tiny capsule called Vostok 1. He only got to orbit the Earth once before coming back down, but it was still an unbelievable achievement. Imagine how scared yet excited he must have felt!After that, the Space Race between Russia and America really heated up. The Americans managed to send people to orbit the moon on the famous Apollo missions. And in 1969, Neil Armstrong became the first human to actually walk on the lunar surface! He said his famous words: "That's one small step for man, one giant leap for mankind."Walking on the moon must have been out of this world (pun intended!). The astronauts got to experience low gravity, bouncing around like kangaroos. They could see the whole Earthhanging in the inky blackness of space. What an incredible view that must have been.Sadly, humans haven't been back to the moon in a very long time. But that doesn't mean we've stopped exploring space! We've sent tons of robotic probes and rovers to other planets and moons. We have space telescopes floating around, peering deep into the mysteries of the universe. And we even have a whole International Space Station orbiting our planet, with astronauts living and working up there!I can't wait until we're ready to send people farther out into the solar system, maybe even to Mars one day. Wouldn't it be wild to be one of the first humans standing on another planet, looking up at Earth as just a tiny bright speck? Elon Musk and other smart people are working hard to make that happen.But oh man, exploring space and seeing things nobody has ever seen before would make it all worth it in my book. I dream of soaring among the stars, touching down on strange new worlds, and maybe even meeting alien life (but hopefully not the man-eating kind!).If I get really good at school and study hard, maybe I can grow up to be an astronaut or rocket scientist myself someday. I'll do whatever it takes to fulfill that lifelong dream of journeyingto the great unknown of outer space. There's a whole galaxy out there, waiting to be explored!Well, that's all for my essay on space travel. Don't you agree it would be the most epic adventure ever? I can't wait to see what mind-blowing new discoveries and space missions the future will hold. The universe is our oyster, filled with endless possibilities just waiting for us! Let's go explore.。

量子飞船英文作文

量子飞船英文作文Paragraph 1: Wow, have you ever heard of quantum spaceships? They are like something out of a sci-fi movie, but they actually exist! These incredible machines use the principles of quantum mechanics to travel through space at unimaginable speeds. It's mind-blowing to think about the possibilities that quantum spaceships could bring to the exploration of the universe.Paragraph 2: Picture this: you're sitting in a comfortable seat inside a quantum spaceship, ready for the adventure of a lifetime. Suddenly, the ship disappears from its current location and reappears light-years away in a matter of seconds. The sensation of traveling through space in a quantum spaceship must be absolutely exhilarating. Just imagine the rush of adrenaline as you zip through the cosmos at speeds faster than the speed of light!Paragraph 3: One of the most fascinating aspects of quantum spaceships is the concept of entanglement. Inquantum mechanics, entanglement refers to the phenomenon where particles become connected in such a way that the state of one particle affects the state of another, regardless of the distance between them. This means that quantum spaceships could potentially harness the power of entanglement to communicate instantaneously across vast distances, revolutionizing interstellar communication.Paragraph 4: Another mind-boggling feature of quantum spaceships is the ability to exist in multiple places at once. According to the principles of quantum mechanics, particles can exist in a superposition of states until they are observed or measured. Applying this concept to a spaceship, it means that a quantum spaceship could potentially occupy multiple points in space simultaneously, allowing for instantaneous travel between different locations. It's like having the ability to be in two places at once!Paragraph 5: The technology behind quantum spaceshipsis still in its infancy, but scientists and researchers are working tirelessly to make this futuristic dream a reality.They are exploring ways to overcome the immense challenges posed by quantum mechanics and engineering a spaceship that can harness the power of quantum phenomena. It's anexciting time to be alive, as we witness the birth of a new era in space exploration.Paragraph 6: In conclusion, quantum spaceships are a testament to the incredible advancements humans have made in the field of science and technology. These extraordinary machines hold the potential to revolutionize space travel and our understanding of the universe. As we continue to push the boundaries of what is possible, it's only a matter of time before quantum spaceships become a common sight in our exploration of the cosmos. The future is trulylimitless!。

Quantum Walk on the Line (Extended Abstract)

1
Introduction
Random walks on graphs have found many applications in computer science, including randomised algorithms for 2-Satisﬁability, Graph Connectivity and probability ampliﬁcation (see, e.g., [14]). Recently, Sch¨ oning [19] discovered a random walk based algorithm similar to that of Papadimitriou [17] that gives an elegant (and the most eﬃcient known) solution to 3-Satisﬁability. In general, Markov chain simulation has emerged as a powerful algorithmic tool and has had a profound impact on random sampling and approximate counting [10]. Notable among its numerous applications are estimating the volume of convex bodies [6]1 and approximating the permanent [9]. A few months ago, Jerrum, Sinclair and Vigoda [11] used this approach to solve the long standing open problem of approximating the permanent for general non-negative matrices. In the spirit of developing similar techniques for quantum algorithms, we consider quantum walk on graphs. To date, few general techniques are known for developing and analysing quantum algorithms: Fourier sampling, which is typiﬁed by the seminal work of Simon [21] and Shor [20], and amplitude ampliﬁcation, which originated in the seminal work of Grover [8]. Barring applications of these techniques, the search for

今天发生的新鲜事英语作文100字左右

全文分为作者个人简介和正文两个部分：作者个人简介：Hello everyone, I am an author dedicated to creating and sharing high-quality document templates. In this era of information overload, accurate and efficient communication has become especially important. I firmly believe that good communication can build bridges between people, playing an indispensable role in academia, career, and daily life. Therefore, I decided to invest my knowledge and skills into creating valuable documents to help people find inspiration and direction when needed.正文：今天发生的新鲜事英语作文100字左右全文共3篇示例，供读者参考篇1New Experience at School TodayThis morning, something really cool happened at school that I've never done before. We had a guest speaker come to our science class who worked for NASA! He talked all about the Marsrovers and showed us real pictures and videos from the planet. It was amazing to see those red rocks and dusty landscapes that are actually on Mars. He even brought in a model of one of the rovers for us to pass around. I've always dreamed of becoming an astronaut, so getting to hear directly from someone involved in space exploration was super inspiring. I can't wait until we get to go on a field trip to the planetarium next month!And here is an expanded version around 2000 words:New Experience at School TodayYou'll never believe what happened at school today! We had the most amazing guest speaker come talk to us during science class. I'm still trying to process how cool the whole thing was.It started off just like any other Friday. I dragged myself out of bed, got ready, and headed to my first few periods feeling pretty tired and zoning out like usual. But then right before fourth period, my science teacher Mr. Matthews made an announcement that we were having a special visitor. He said a real NASA engineer was going to give a presentation to our class!At first, I thought I must have misheard him. Why would someone who works for NASA want to come speak at our littlehigh school in the middle of nowhere? I looked around and most of my classmates had the same confused expressions on their faces. But sure enough, when the late bell rang, a man in his 40s or 50s strolled in dressed in a NASA polo shirt and apparel from the Jet Propulsion Laboratory. My mind was blown.Mr. Matthews introduced him as Dr. Bryan Jackson, an engineer who has spent over 20 years working on the Mars rover projects sending robotic rovers to explore the surface of the red planet. As soon as he started speaking, I was absolutely captivated. In a friendly, down-to-earth style, Dr. Jackson walked us through the complete history of the Mars exploration rovers from the early concepts to the finally launches and missions still happening today.It was amazing to get a behind-the-scenes look at what went into designing, testing, and piloting these incredibly advanced rovers from millions of miles away. Who knew so many pivotal decisions had to be made, like what scientific instruments to equip the rovers with based on weight constraints? Or how they had to construct the rovers to be driven remotely by scientists on Earth because it takes 20 minutes for a signal to reach Mars? Dr. Jackson really brought the whole process to life.Of course, the real showstopper was all the incredible photos and video footage he had gathered over his decades working on the Mars missions. We stared in awe at these crystal-clear images of the rusty orange Martian landscape stretching out as far as the eye could see. Seeing those iconic shots of the rovers parked among the rocky alien terrain with mountains or craters in the background was unreal. It looked like something straight out of a movie, except it was 100% real.Dr. Jackson sprinkled in all sorts of fascinating facts too, like how the rovers have operated for over 15 years when they were only designed for a 90-day mission on Mars. Or how in order to land them on Mars, they had to use a "sky crane" maneuver to lower the rovers down from a hovering rocket stage because the planet's atmosphere is too thin for regular parachute landings. Mind-blowing stuff.But probably the coolest part was when Dr. Jackson brought out a scale model of the Perseverance rover that had been 3D printed at JPL. It was amazingly detailed and accurate, from the six aluminum wheels to the robotic arm used to collect rock samples. We all passed it around the classroom, geeking out over how intricate and sturdy yet portable the design was. Forme, holding a model of an actual vehicle driving around on another planet was an incredibly humbling experience.Hearing Dr. Jackson speak, you could really tell how passionate he was about space exploration and the quest to find signs of ancient microbial life on Mars. He beamed with enthusiasm as he talked about analyzing the rover's findings or fixing problems from millions of miles away. You can't get that sort of dedication or expertise from reading about it in books or watching videos. Having him visit in person was an irreplaceable opportunity.At the end, Dr. Jackson opened it up for questions from the class. My hand shot right up to ask him what he would say to a high school student who dreams about working at NASA or JPL like he does. He smiled and told me, "Work hard, study hard, and never stop pursuing the things you're passionate about." He talked about all the STEM fields you can get into that lead to jobs like his, whether it's engineering, physics, chemistry, computer science or countless other paths. Just hearing that validation and encouragement from someone actually in my dream career field gave me such a motivational boost.As Dr. Jackson was packing up his materials, I caught up with him to thank him again for visiting. I told him how inspiring itwas and that I really hoped I could work in space exploration or rover design someday too. He was so friendly and said he's always happy to help get students interested in STEM fields. Before he left, Dr. Jackson signed a small printout of the Perseverance rover for me. You'd better believe I'll be hanging that up above my desk!I can honestly say this was one of the most engaging and eye-opening experiences I've ever had at school. Having that kind of real-world connection to the cutting-edge work being done in STEM fields really brings it all into perspective. It was incredible to learn directly from someone involved in interplanetary travel and see actual images from another world with my own eyes. Getting to interact with Dr. Jackson definitely reinforced my interest in pursuing a career that lets me be part of that kind of amazing scientific exploration and discovery.I'm feeling really motivated now to double down and focus as I start applying to colleges and picking a STEM major. Physics and engineering were already at the top of my list, but this just crystallized how perfect those paths could be. Just thinking about designing robotic systems that could investigate other planets or even galaxies gives me chills. The future of space travel is so ripe with possibilities.Maybe someday I'll even get a chance to work on a mission that lands a rover on Mars just like the ones Dr. Jackson told us about. How cool would it be to have a career where you're helping plan the intricate details and maneuvers to transport a high-tech robot millions of miles away to study the surface of an alien world? You could be part of paradigm-shifting new discoveries or find evidence that life once existed elsewhere in our solar system. The prospects are absolutely mind-blowing.At the very least, experiences like today have really opened my eyes to the amazing work actually being done in STEM careers. A few years ago, I don't think I fully grasped just how cutting-edge and future-focused fields like aerospace engineering or deep space exploration really are. Dr. Jackson and his photos and videos brought it all to vivid life in a way textbooks or documentaries never could.I'll never forget holding that realistic 3D model of the Perseverance rover that's currently trundling across Mars as we speak. It was almost spiritual, being able to inspect and feel the same design that's resided on another planet. Knowing humans' innate curiosity and innovative spirit allowed us to construct that machine and pilot it so flawlessly 200 million miles away is such an epic perspective.Experiences like this have motivated me to work harder than ever on my STEM education so I can hopefully contribute to making similar breakthroughs and discoveries in my career someday. Traveling to other planets was just science fiction until recent decades. Now it's cutting-edge reality, and so much more is on the horizon if our drive for exploration continues.Whether it's rovers uncovering signs of life on Mars or one day sending human crews there, I want to play a role in that next chapter of cosmic discovery. Getting inspired like I did today is paramount for keeping scientific curiosity and innovation alive for generations to come. I'm incredibly grateful my school could arrange a visit from Dr. Jackson that brought these vital fields to life. Interacting with his first-hand experiences and expertise is sure to stick with me and influence my studies and career path for years to come. Here's to reaching for the stars!篇2Today was just a regular school day, or so I thought. Little did I know the excitement that was in store! During morning break, a squirrel somehow got into the building and chaos ensued as teachers tried to catch it. We all gathered around watching and cheering them on. After an epic chase through the halls, the squirrel was finally captured and released outside.What a way to liven up an otherwise boring Monday! I'll never forget the hilarious sight of our principal wielding a butterfly net while giving chase.Today started off just like any other Monday. I dragged myself out of bed, threw on my uniform, and headed out the door for another week of classes. Little did I know, however, that this supposedly ordinary day had a fun surprise in store that would provide endless entertainment.I made my way through the all too familiar routine - homeroom, first period math, second period English. An oppressive cloud of boredom and sleep deprivation hung over me as I mindlessly copied down equations and grammar rules. My drooping eyelids threatened to fully close as the teacher's monotone droned on and on.But just as I was about to fully succumb to the overwhelming desire to zonk out, the morning bell rang for break time, pulling me back from the brink of oblivion. I gathered my things and prepared for the short reprieve of freedom and fresh air. However, just as I stepped into the hallway, I was met with a scene of total pandemonium.Students were running in every direction, shrieking and pointing wildly. Teachers rushed out of their classrooms,concerned looks on their faces as they tried to make sense of the commotion. That's when I saw it - a tiny ball of fur scurrying frantically down the hall, weaving between the forest of legs with remarkable agility.At first, I thought my sleep-deprived mind was playing tricks on me. But there was no mistaking the furry culprit - a squirrel had somehow found its way inside the school! The audacious little guy seemed utterly unfazed by his extremely urbanized surroundings as he sped past, tiny legs spinning wildly.The chase was on as teachers began barking orders, determined to catch the unruly intruder before any serious damage occurred. They brandished makeshift tools of capture like brooms, boxes, and even the iconic butterfly net wielded by our fearless principal Mr. Johnson.What ensued was a scene of pure comedic gold that had the entire student body doubled over in laughter. The poor squirrel, desperate for an escape route, pulled off a series of Matrix-style maneuvers as he narrowly evaded his pursuers at every turn. He bounded over stray backpacks and dove under benches, all while the teachers grew increasingly flustered.At one point, the squirrel made a beeline for the stairwell, leading to a dramatic chase up and down multiple floors. I lostcount of how many times Mr. Johnson came skidding around a corner, net poised for attack, only to be mockingly greeted by the squirrel's twitching tail as he scampered off in the opposite direction.Despite their valiant efforts, the teachers simply could not outwit nature's tiny ninja. Just when they thought they had him cornered, the slippery squirrel would locate an impossibly small gap in a door frame or ventilation grate and disappear once more.The chase raged on for what felt like an eternity until finally, through sheer dumb luck, the squirrel found himself trapped inside a empty classroom. With no other options left, he froze in place as the teachers closed in from all sides. Mr. Johnson skillfully dropped the butterfly net over him and - after a few tense moments of struggling - emerged victorious with the captured critter.The hallways erupted with cheers and applause as the squirrel was safely carried outside and released back into the world where he belonged. As relieved as I was that the madness had concluded, I couldn't help but feel a pang of sadness watching him scamper off into the bushes. Our lives may havereturned to normal, but that squirrel had injected a badly needed dose of excitement into our mundane routines.I'll never forget the image of Mr. Johnson frantically swinging that butterfly net while his toupee threatened to dislodge with each wild swing. Or the sight of Ms. Wilkins attempting to coax the squirrel over by littering a trail of nuts behind her. It was a welcome detour from the usualsoul-crushing boredom that came with just another Monday.As I returned to class in a bit of a daze, I couldn't help but feel a new sense of appreciation for life's spontaneous joys. There's something to be said for an unplanned adventure amidst the repetition - a reminder that any day could be the day you find yourself chasing a squirrel with a butterfly net. I realized that if I kept my eyes open, I'd be rewarded with many more such occurrences to shake up the same old, same old. From now on, I vowed to expect the unexpected and accept each day's little surprises as gifts to be savored to the fullest.篇3Today Was a Wild Ride! (100 words)You won't believe what went down at school today! During second period, the fire alarm randomly started blaring. We allevacuated to the football field, confused but kinda stoked to miss class. After like an hour, the firemen finally showed up...but it was just a prank! Some idiot pulled the alarm as a joke. We were so heated to waste that whole morning. By lunchtime, though, people were redan laughing about it. I just hope that prankster gets busted. What a crazy day!Expanded Version (around 2000 words):Today was definitely one for the books – a morning I'll never forget. It started off as a pretty typical Monday. I groggily dragged myself out of bed at 6:30 am, threw on my usual jeans and t-shirt, scarfed down a bowl of cereal, and headed out the door for another dreadful day at Westbrook High.First period English Lit was its usual bore. Mrs. Steinberg droned on and on about symbolism in Of Mice and Men while I dozed off, scoring some much-needed sleep after staying up way too late last night binging episodes of The Last of Us. I had gotten through like three episodes before realizing I had barely made a dent in my huge English paper that's due on Wednesday. So much for my weekend productivity!The real madness began halfway through second period Chemistry. We were in the middle of a riveting lecture on chemical bonding (cue the sarcasm) when suddenly, the piercingwail of the fire alarm exploded through the hallways. My teacher, Mr. Davis, abruptly stopped mid-sentence with a confused look on his face."Is this a drill?" someone asked. Mr. Davis shrugged his shoulders and told us to swiftly head outside and line up on the football field like we practiced.As a herd of confused students spilled out of every classroom, filling the hallways with a sea of chaos, the reality dawned on me: this was no drill. There must be a legitimate emergency going on. Maybe a small fire had broken out in the science lab? Or some sort of gas leak? My adrenaline began pumping as all sorts of worst-case scenarios started racing through my mind.Once we made it outside to the football field, a blast of icy mid-January air danced across my cheeks, making me instantly regret not throwing on a jacket over my t-shirt this morning. Teachers attempted to corral their students into neat lines, while the rest of us stood around in befuddled clusters, the din of hundreds of gossiping voices echoing across the empty field."Yo, did you see any smoke or anything?" I asked Jason and Emma, my two best friends who were huddled up near me."Not a clue, man. This is crazy!" Jason responded, seeming unusually enthused by the dramatic turn of events."I heard it might have been a fire in the chem lab," Emma added, her teeth chattering from the bitter cold. "But who knows if that's true?"We idled around on the field for what felt like forever, basically the entire student body and faculty abandoning ship from the building. After about an hour with no sign of any fire trucks or update on the situation, I noticed antsy side conversations breaking out here and there, with people growing increasingly skeptical that this was an actual emergency.Then, finally, a couple fire trucks came screaming onto campus, their deafening sirens slicing through the crisp winter air. Here we go, I thought to myself. At least we'll get some answers now.But nothing could have prepared me for what happened next. The fire chief strode up to the school principal, a tall barrel-chested man named Mr. Reynolds. A few words were exchanged, and then the most unexpected thing happened: Mr. Reynolds violently flung his arms down in a rare display of outrage, letting out a primal yell that sliced through the nervous murmuring of the crowd like a hot knife.You could have heard a pin drop in that moment. We all stood there in stunned silence as it became evident that something had gone horribly awry."Everybody listen up!" Mr. Reynolds' voice boomed through the bullhorn. "This was a prank! A prank that has disrupted valuable classroom time while wasting the resources and putting the lives of our first responders at risk."A deafening collective gasp arose from the crowd as we all began looking around at each other in disbelief. Prank? Someone had pulled the fire alarm as a joke? Who in their right mind would do something so stupid and irresponsible?"We are going to conduct a rigid investigation to find out who is responsible for this foolish prank," Mr. Reynolds went on, the tension thickening with every word. "And when we find out who it is, you will be suspended, face potential expulsion, and quite possibly criminal charges!"A roll of nervous laughter rippled through the crowd at the prospect of some legend potentially catching criminal charges over a measly prank. In that moment though, I couldn't help but feel a tinge of anger and resentment toward this anonymous prankster. What an incredibly idiotic, selfish, and immature thingfor someone to do – ruining an entire morning for hundreds of students and teachers over what, a cheap laugh?It took a while, but eventually order was restored and we were allowed back inside to resume our day. The vibe for the remainder of the day was anything but normal though. An unmistakable tension hung thick in the air, with teachers clearly in foul moods and roughly half the students riled up over the prospect of an expelled prankster in our midst."Can you believe someone actually did that?" I vented to Jason as we grabbed a quick bite in the cafeteria during lunch period. "Like, I get pranks are funny and all, but there's a line you don't cross. Disrupting an entire day of school and dispatching fire crews for no reason? That's just crazy.""I know, man, what a dick move," Jason grumbled through a mouth full of lukewarm chicken nuggets. "Though you gotta respect the sheer cojones on that madlad, right? Imagine the rush of pulling off something so next-level?""Easy for you to say," I shot back. "Your parents don't have to worry about being slapped with a huge bill if we have to cover the fire department's costs.""Yeesh, chill bro," Jason replied, throwing his hands up defensively. "I'm just saying, you gotta admit the sheer audacity of it is pretty legendary. Like, that kicks serious ass on hiding a whoopee cushion under a teacher's chair."I just shook my head and let the subject drop, resisting the urge to point out the paradox Jason was missing – that the more "legendary" and outrageous a prank is, the higher likelihood it steps over an ethical line. There's a valid reason that pranks on the scale of faking a fire alarm are grounds for suspension or even expulsion these days. Schools and communities have been cracking down harder than ever after too many instances of pranks going horrifically awry or ending in legitimate danger.Still, deep down, I recognized that tiny imp of immaturity we all have residing in the back of our minds, the one that can't help but be at least slightly awed by sheer unfiltered rebellion and anarchy, consequences be damned. It was what made the prankster's identity take on an almost mythical aura throughout the hallways that day, with dozens of names being breathlessly thrown out as the suspected culprit, Second period Chem appearing to harbor the most blabbermouths gleefully gossiping about witnessing so-and-so carrying out the dastardly deed.Far from settling the growing hysteria, the final two periods of the day only seemed to stoke it further. By the time the last bell rang at 2:45, you could cut the tension with a knife. Students packed the hallways in frenzied clusters, loudly swapping rumors and trying to piece together which mythological hero in our midst had committed the ultimate act of rebellion."Hey, did you hear it was Greg Sampson from your Calc class?" Emma rushed over to me and Jason as we exited our English Lit class, barely able to contain her giddiness."No way, everyone's saying it was actually Liam Becker," I countered. "Apparently he left second period Chem right before the alarm went off.""This is so fucking legendary!" Jason bellowed, looking like a kid on Christmas morning. "Whoever it was will go down in Westbrook history as an absolute madlad!"I was about to retort when I noticed a few teachers further down the hallway, clustered in a tight circle while engaged in a very intense discussion.。

光子上旋下旋的英语

光子上旋下旋的英语Photon upspin and downspin,also known as photon polarization,refer to the intrinsic angular momentum of a photon along its direction of motion.Polarization can be understood as the orientation of the electromagnetic field of a photon.When a photon is in an upspin state,its angular momentum points in the same direction as its momentum.This means that the electromagnetic field oscillates vertically,with the electric and magnetic fields perpendicular to each other.In contrast,when a photon is in a downspin state,its angular momentum points in the opposite direction to its momentum, causing the electromagnetic field to oscillate horizontally.Photon polarization plays a crucial role in many quantum phenomena,such as the transmission of information in quantum communication and the generation of entangled photon pairs in quantum optics experiments.By manipulating the polarization of photons,researchers can control their behavior and harness their unique properties for various applications.In conclusion,understanding the concepts of photon upspin and downspin is essential for exploring thefascinating world of quantum mechanics and its wide range of potential applications in technology and communication.。

光学专业英语

Iris – aperture stop虹膜孔俓光珊retina视网膜[ˈrɛtnə]Color Blind 色盲weak color 色弱Myopia – near-sighted 近视（Myopia[maɪˈopiə]）Sensitivity to Light感光灵敏度（Sensitivity [ˌsɛnsɪˈtɪvɪti]）boost推进[bust]lag behind落后于Hyperopic – far-sighted 远视Dynamic Range 动态范围（Dynamic[daiˈnæmik]）critical fusion frequency 临界融合频率（critical[ˈkrɪtɪkəl]）fusionˈfjuʒən]CFF临界闪变频率visual sensation视觉Chromaticity Diagram色度图 Chromaticity[ˌkroməˈtɪsɪti]Color Temperature色温HSV Model色彩模型(hue色度[hju]saturation饱和度value纯度CIE Model 相干红外能量模式Complementary Colors补色Bar Pattern条状图形Heat body 热稠化approximate近似violet紫罗兰Body Curve人体曲线Color Gamut色阶adjacent邻近的normal illumination法线照明Primary colors红黄蓝三原色Color saturation色饱和度Color Triangle颜色三角Color Notation颜色数标法Color Difference色差TV Signal Processing电视信号处理Gamma Correction图像灰度校正Conversion Tables换算表out of balance失衡wobble摇晃back and forth前后clear (white) panel白光板vibrant震动fuzzy失真quantum leap量子越迁SVGA (800x600)derive from起源自culprit犯人render呈递inhibit抑制,约束stride大幅前进blemish污点obstruction障碍物scratch刮伤substance物质实质主旨residue杂质criteria标准parameter参数adjacent邻近的接近的asynchrony异步cluster串群mutually互助得algorithm运算法则Chromatic Aberrations色差Fovea小凹Visual Acuity视觉灵敏度Contrast Sensitivity对比灵敏度Temporal (time) Response反应时间rendition表演,翻译animation活泼又生气ghost重影Parallax视差deficient缺乏的不足的Display panel显示板NG.( Narrow Gauge)窄轨距dichroic mirror二色性的双色性的Brewster Angle布鲁斯特角Polarized Light极化光Internal reflection内反射Birefringence 双折射Extinction Ratio 消光系数Misalignment 未对准Quarter Waveplates四分之一波片blemish污点瑕疵Geometric几何学的ripple波纹capacitor电容器parallel平行的他tantalum钽(金属元素)exsiccate使干燥exsiccate油管,软膏furnace炉子镕炉electrolytic电解的,由电解产生的module模数analog类似物out of the way不恰当pincushion针垫拉lateral侧面得rectangle长方形fixture固定设备control kit工具箱DVIconnector DVI数局线Vertical垂直的horizontal 水平的interlace隔行扫描mullion竖框直楞sawtooth锯齿[ˈsɔtuθ]toggle套索钉keypad数字按键键盘tangential切线diagnostic tool诊断工具sagittal direction径向的sagittal[ˈsædʒɪtl]cursor position光标位置3Yw'/#p3`ray aberration光线相差weighting factor权种因子variables变量for now暂时,目前.眼下check box复选框Airy disk艾里斑exit pupil出[射光]瞳optical path difference光称差with respect to关于diffraction limited衍射极限wavefront aberration波阵面相差spherical aberration球面象差paraxial focus傍轴焦点chromatic aberration象差local coordinate system局部坐标系统coordinate system坐标系orthogonal直角得,正交的conic sections圆锥截面account for解决,得分parabolic reflector拋物面反射镜radius of curvature曲率半径spherical mirror球面镜geometrical aberration几何相差incident radiation入射辐射global coordinate总体坐标in terms of根据按照reflected beam反射束FYI=for your information供参考Constructive interference相长干涉phase difference相差achromatic singlet消色差透镜Interferometer干涉仪boundary constraint边界约束,池壁效radii半径Zoom lenses变焦透镜Beam splitters分束器discrete不连续的,分离的objective/eye lens物镜/目镜mainframe主机rudimentary根本的,未发展的photographic照相得摄影得taxing繁重的,费力得algebra代数学trigonometry三角学geometry几何学calculus微积分学philosophy哲学lagrange invariant拉格朗日不变量spherical球的field information场信息Standard Lens标准透镜Refracting Surface折射面astigmatism散光HDTV高清晰度电视DLV ( Digital Light Valve)数码光路真空管，简称数字光阀diffraction grating衍射光珊field angle张角paraxial ray trace equations近轴光线轨迹方称back focal length后焦距principal plane主平面vertex顶点,最高点astigmatism散光,因偏差而造成的曲解或错判medial中间的,平均的variance不一致conic圆锥的,二次曲线field of view视野collimator瞄准仪convolution回旋.盘旋,卷积fuzzy失真,模糊aberrated异常的[ˈæbəˌretɪd]asymmetry不对称得[eˈsɪmɪtri]indicative可表示得[ɪnˈdɪkətɪv]parabolic拋物线得[ˌpærəˈbɑlɪk]suffice足够,使满足specification规格,说明书[ˌspɛsəfɪˈkeʃən]straightforward易懂的,直接了当的[stretˈfɔrwəd],solidify凝固,巩固.Constraints 约束,限制metrology度量衡field coverage视场,视野dictate口述, 口授, 使听写, 指令, 指示, 命令, 规定irradiance发光, 光辉,辐照度aerial空气得,空中得halide卤化物的monochromatic单色的,单频的polychromatic多色的aspherical非球面的spherical球面的alignment列队,结盟power(透镜)放大率equiconvergence 同等收敛EFL(effective focal length)有效焦距workhorse广为应用的设备biconvex两面凸的global optimization整体最优化concave凹得,凹面得cylindrical圆柱得solid model实体模型Modulation Transfer Function调制传递函数in the heat of在最激烈的时候protocol协议,规定triplet三重态sanity心智健全zinc锌,涂锌的selenide 硒化物,硒醚miscellaneous各色各样混在一起, 混杂的, 多才多艺的versus与...相对polynomial多项式的coefficient系数explicit function显函数" wYgi%distinct清楚的,截然不同的emanate散发, 发出, 发源rudimentary根本的,未发展的intersection角差点PRTE=paraxial ray trace equation旁轴光线轨迹方程achromats 消色差透镜cardinal points基本方位separations分色dashed虚线blow up放大overlay覆盖,覆盖图multiplayer 多层的humidity 湿度float glass浮法玻璃square one 出发点,端点square up to 准备开打,坚决地面对reflecting telescope 反射式望远镜diagnostic tools诊断工具Layout plots规划图Modulation transfer function调制转换功能FFT快速傅里叶变换Point spread function点传播功能wavelength波长angle角度absorption吸收system aperture系统孔径lens units透镜单位wavelength range波长范围singlet lens单业透镜spectrum光谱diffraction grating衍射光栅asphere半球的LDE=Lens data editor Surface radius of curvature表面曲率半径surface thickness表面厚度material type材料种类semi-diameter半径focal length焦距aperture type孔径类型aperture value孔径值field of view视场microns微米F, d, and C= blue hydrogen, yellow helium, red hydrogen lines, primary wavelength主波长sequential mode连续模式object surface物表面The front surface of the lens透镜的前表面stop光阑The back surface of the lens透镜的后表面The image surface像表面symmetric相对称的biconvex两面凸的The curvature is positive if the center of curvature of the surface is to the right of the vertex. It is negative if the center of curvature is to the left of the vertex.如果曲率中心在最高点的右边,曲率值为正,如果曲率中心在最高点的左边,则曲率为负image plane像平面Ray Aberration光线相差tangential direction切线方向sagittal direction径向paraxial focus旁轴的Marginal边缘的spherical aberration球面像差Optimization Setup最优化调整variable变量mathematical sense数学角度MFE= Merit Function Editor, Adding constraints增加约束focal length焦矩长度operand操作数the effective focal length有效焦矩primary wavelength主波长initiate开始spot diagram位图表Airy disk艾里斑axial chromatic aberration轴向色差with respect to关于至于exit pupil出射光瞳OPD=optical path difference光学路径差diffraction limited衍射极限chromatic aberration色差chromatic focal shift色焦距变换paraxial focus傍轴焦点axial spherical aberration轴向球差(longitudinal spherical aberration 纵向球差:沿光轴方向度量的球差) lateral spherical aberration垂轴球差(在过近轴光线像点A‵的垂轴平面内度量的球差)coma、comatic aberration彗差meridional coma子午彗差sagittal coma弧矢彗差astigmatism像散local coordinate system本地坐标系统meridional curvature of field子午场曲sagittal curvature of field弧矢场曲decentered lens偏轴透镜orthogonal直角的垂直的conic section圆锥截面account for说明,占有,得分stigmatic optical system无散光的光学系统Newtonian telescope牛顿望远镜parabolic reflector抛物面镜foci焦距chromatic aberration,色差superpose重迭parabola抛物线spherical mirror球面镜RMS=Root Mean Square均方根wavefront波阵面spot size光点直径Gaussian quadrature高斯积分rectangular array矩阵列grid size磨粒度PSF=Point Spread Function点扩散函数FFT=Fast Fourier Transform Algorithm快速傅里叶变换Cross Section横截面Obscurations昏暗local coordinates局部坐标系统vignette把…印为虚光照Arrow key键盘上的箭头键refractive折射reflective反射in phase同相的协调的Ray tracing光线追迹diffraction principles衍射原理order effect式样提出的顺序效果energy distribution能量分配Constructive interference相长干涉dispersive色散的Binary optics二元光学phase advance相位提前achromatic single消色差单透镜diffractive parameter衍射参数Zoom lenses变焦透镜Athermalized lenses绝热透镜Interferometers干涉计Beam splitter分束器Switchable component systems可开关组件系统common application通用symmetry对称boundary constraint边界约束multi-configuration (MC) MC Editor (MCE) perturbation动乱,动摇index accuracy折射率准确性index homogeneity折射率同种性index distribution折射率分配abbe number离差数Residual剩余的Establishing tolerances建立容差figure of merit质量因子tolerance criteria公差标准Modulation Transfer Function (MTF)调制传递函数boresight视轴，瞄准线Monte Carlo蒙特卡洛Tolerance operands误差操作数conic constant ]MC1"{_qT 圆锥常数astigmatic aberration像散误差Mechanical tilt机械倾斜，机械倾角Tolerance Data Editor (TDE)公差资料编辑器compensator补偿棱镜estimated system performance预估了的系统性能iteratively反复的，重迭的statistical dependence统计相关性sequential ray trace model连续光线追迹模型imbed埋葬，埋入multiple多样的，多重的，若干的Non-Sequential Components不连续的组件Corner cube角隅棱镜，三面直角透镜Sensitivity Analysis灵敏度分析Faceted reflector有小面的反射镜emit发射，发出nest嵌套overlap交迭outer lens外透镜brute force强力seidel像差系数aspect ratio长宽比MRA边缘光线角MRH边缘光线高度asynchronous不同时的,异步Apodization factor变迹因子hexapolar六角形dithered高频脉冲衍射调制传递函数（DMTF），衍射实部传递函数（DRTF），衍射虚部传递函数（DITF），衍射相位传递函数（DPTF），方波传递函数（DSWM）logarithmic对数的parity奇偶% Uc,I e longitudinal aberrations 纵向像差赛得系数:球差（SPHA，SI）彗差（COMA，S2），像散（ASTI，S3），场曲（FCUR，S4），畸变（DIST，S5），轴向色差（CLA，CL）和横向色差（CTR，CT）.横向像差系数:横向球差（TSPH），横向弧矢彗差（TSCO），横向子午彗差（TTCO），横向弧矢场曲（TSFC），横向子午场曲（TTFC），横向畸变（TDIS）横向轴上色差（TLAC）。

宇航员登月英语作文初一

宇航员登月英语作文初一As the world watched in awe, the first human stepped onto the surface of the moon. It was a moment that would go down in history as one of the greatest achievements of mankind. The Apollo 11 mission had been a success, and the astronauts had accomplished what many had thought was impossible. But what was it like to be an astronaut on that historic mission? Let's take a closer look.The journey to the moon began on July 16, 1969, whenthe Apollo 11 spacecraft was launched from the Kennedy Space Center in Florida. The crew consisted of three astronauts: Neil Armstrong, Buzz Aldrin, and Michael Collins. Armstrong and Aldrin would be the first humans to walk on the moon, while Collins would remain in orbitaround the moon.The journey to the moon took three days, during which time the astronauts had to endure cramped living conditions, weightlessness, and the constant noise and vibration of thespacecraft's engines. They also had to perform a number of tasks to keep the spacecraft running smoothly, such as checking the systems, monitoring the fuel levels, and making course corrections.Finally, on July 20, 1969, the lunar module, named Eagle, separated from the command module and began its descent to the moon's surface. Armstrong and Aldrin would be the first humans to walk on the moon, but before they could do that, they had to land the lunar module safely.The landing was a tense and nerve-wracking experience. The lunar module had never been tested in the actual conditions of the moon's surface, and there was a real possibility that something could go wrong. Armstrong had to manually guide the spacecraft to a safe landing site, using only his eyes and his experience as a test pilot to judge the distance and speed of the spacecraft.Finally, after what seemed like an eternity, the lunar module touched down on the moon's surface. Armstrong's famous words, "That's one small step for man, one giantleap for mankind," echoed around the world as he stepped onto the lunar surface. Aldrin followed shortly after, and the two astronauts spent several hours exploring the moon's surface, collecting samples, and conducting experiments.Meanwhile, Michael Collins orbited the moon, waitingfor his fellow astronauts to return to the lunar module. The journey back to Earth was no less challenging than the journey to the moon, with the astronauts having to endure the stresses of re-entry into Earth's atmosphere and the risk of a malfunction during the descent.But despite the challenges and the risks, the Apollo 11 mission was a success. It proved that humans could travel to the moon and back, and it inspired generations of people to dream big and aim for the stars. The legacy of that mission lives on today, as we continue to explore the universe and push the boundaries of what is possible.。

宇航员在哪里工作英语作文

Astronauts are individuals who are trained by space agencies to travel into outer space and perform various tasks,such as conducting experiments,repairing equipment,or simply exploring the cosmos.Their work environment is primarily in space,but their preparation and training take place on Earth.Here is an English composition about where astronauts work:Title:The Working Environment of AstronautsAstronauts,the modernday explorers of the universe,have a unique and challenging work environment.Their primary workplace is not a traditional office or a factory,but the vast expanse of outer space.However,the journey to space and the return to Earth involve several stages of work that take place both on the ground and in orbit.On Earth:Training and PreparationBefore an astronaut ever leaves the planet,they undergo rigorous training at facilities like the Johnson Space Center in Houston,Texas,or the Yuri Gagarin Cosmonaut Training Center in Star City,Russia.Here,they learn about spacecraft systems,spacewalk procedures,and how to handle emergencies.They also train in simulators that mimic the conditions of space,such as the Neutral Buoyancy Lab where astronauts practice spacewalks underwater.In Transit:Launch and ReturnThe journey to space begins with a launch,typically from a site like the Kennedy Space Center in Florida or the Baikonur Cosmodrome in Kazakhstan.Astronauts work closely with ground control teams to ensure a successful launch.Upon return,they must work with the spacecrafts reentry systems to safely land back on Earth,often in remote areas where recovery teams are waiting.In Space:The International Space Station ISSThe most common workplace for astronauts today is the International Space Station,a collaborative project involving multiple countries.Here,astronauts live and work in a microgravity environment,conducting scientific research,maintaining the stations systems,and preparing for future missions.The ISS is a hub of international cooperation, with astronauts from different countries working together on a daily basis.Spacewalks and Extravehicular Activities EVAsAstronauts may also work outside the spacecraft during spacewalks,which are essential for tasks such as repairing the ISS or deploying satellites.These activities require careful planning and coordination,as well as the use of specialized equipment like the Extravehicular Mobility Unit EMU,which provides life support and mobility in the vacuum of space.Back on Earth:PostMission AnalysisAfter returning from a mission,astronauts continue to work,often participating in debriefings and analyzing the data collected during their time in space.This work is crucial for improving future missions and for advancing our understanding of space and its effects on the human body.In conclusion,the work of an astronaut is not confined to a single location.It spans from the training facilities on Earth to the orbital laboratories in space,and back to the analysis and reporting on Earth.It is a dynamic and multifaceted career that requires a combination of physical prowess,intellectual curiosity,and the ability to work effectively in a team.。

Continuous-time Quantum Walks on a Cycle Graph

We begin with formulating the basic equations of our model. The Hamiltonian of an electron placed in the QD-cycle is 1 N −1 H cycle = ∑ c† c j + c† (1) j c j +1 , 4 j =0 j +1
Hale Waihona Puke ()where c† j ( c j ) are creation (annihilation) operators for an electron on site j ; N is the number of QDs in the cycle, and cN ≡ c0 . We renormalize the time for convenience, so that it becomes dimensionless, and all the amplitudes further on are given in terms of hopping amplitude between neighboring QDs. The point contact, placed next to each QD, consists of two reservoirs of electrons: source and drain that are coupled through the potential barrier shaped by PC gates, see Fig. 1. The Hamiltonian of j-th PC can be written as H PC , j = ∑ El , j al+, j al , j + ∑ Er , j ar+, j ar , j + ∑ Ωlr , j al+, j ar , j + ar+, j al , j , (2)

量子纠缠双缝干涉英语范例

量子纠缠双缝干涉英语范例Engaging with the perplexing world of quantum entanglement and the double-slit interference phenomenon in the realm of English provides a fascinating journey into the depths of physics and language. Let's embark on this exploration, delving into these intricate concepts without the crutchesof conventional transition words.Quantum entanglement, a phenomenon Albert Einstein famously referred to as "spooky action at a distance," challengesour conventional understanding of reality. At its core, it entails the entwining of particles in such a way that the state of one particle instantaneously influences the stateof another, regardless of the distance separating them.This peculiar connection, seemingly defying the constraints of space and time, forms the bedrock of quantum mechanics.Moving onto the enigmatic realm of double-slit interference, we encounter another perplexing aspect of quantum physics. Imagine a scenario where particles, such as photons or electrons, are fired one by one towards a barrier with twonarrow slits. Classical intuition would suggest that each particle would pass through one of the slits and create a pattern on the screen behind the barrier corresponding tothe two slits. However, the reality is far more bewildering.When observed, particles behave as discrete entities, creating a pattern on the screen that aligns with the positions of the slits. However, when left unobserved, they exhibit wave-like behavior, producing an interferencepattern consistent with waves passing through both slits simultaneously. This duality of particle and wave behavior perplexed physicists for decades and remains a cornerstoneof quantum mechanics.Now, let's intertwine these concepts with the intricate fabric of the English language. Just as particles become entangled in the quantum realm, words and phrases entwineto convey meaning and evoke understanding. The delicate dance of syntax and semantics mirrors the interconnectedness observed in quantum systems.Consider the act of communication itself. When wearticulate thoughts and ideas, we send linguistic particles into the ether, where they interact with the minds of others, shaping perceptions and influencing understanding. In this linguistic entanglement, the state of one mind can indeed influence the state of another, echoing the eerie connectivity of entangled particles.Furthermore, language, like quantum particles, exhibits a duality of behavior. It can serve as a discrete tool for conveying specific information, much like particles behaving as individual entities when observed. Yet, it also possesses a wave-like quality, capable of conveying nuanced emotions, cultural nuances, and abstract concepts that transcend mere words on a page.Consider the phrase "I love you." In its discrete form, it conveys a specific sentiment, a declaration of affection towards another individual. However, its wave-like nature allows it to resonate with profound emotional depth, evoking a myriad of feelings and memories unique to each recipient.In a similar vein, the act of reading mirrors the double-slit experiment in its ability to collapse linguistic wave functions into discrete meanings. When we read a text, we observe its words and phrases, collapsing the wave of potential interpretations into a singular understanding based on our individual perceptions and experiences.Yet, just as the act of observation alters the behavior of quantum particles, our interpretation of language is inherently subjective, influenced by our cultural background, personal biases, and cognitive predispositions. Thus, the same text can elicit vastly different interpretations from different readers, much like the varied outcomes observed in the double-slit experiment.In conclusion, the parallels between quantum entanglement, double-slit interference, and the intricacies of the English language highlight the profound interconnectedness of the physical and linguistic worlds. Just as physicists grapple with the mysteries of the quantum realm, linguists navigate the complexities of communication, both realmsoffering endless opportunities for exploration and discovery.。

量子力学入门英语

Until now , we know that both quantum mechanics and relativity theory are absolutely right . But there are some paradoxes between them ，which may be the origin of revolution theory . 迄今为止，我们知道量子力学与相对论都是非常正确的理论。但是，它们之间还是存在一些不兼容的地方。这些地方有可能成为未来革命性理论的起点。
Hello , everyone! My name is Zheng Zerui , from School of Materials Science and Engineering. My teammates are Zhang Yuhan , Wang Yabo , Wu Qihang , Li Wei, Zhang Wei, Li Jiang , Chen Penghui. 大家好！我是材料学院的郑泽锐，我的队友有张宇寒、王亚博、吴起航、李伟、张维、李江、陈鹏辉。
schrödinger
For a general quantum system, Schrödinger equation is
ℏ2 ∂ iℏ Ψ (r , t ) = [− ∇ 2µ ∂t
2
+ V ( r )] Ψ ( r , t )
•
In conclusion , there are two points: A . Quantum mechanics is founded by the hypothesis that energy is discrete . B . The core work of quantum mechanics is calculating the Schrödinger equation. So quantum mechanics is very easy. 总而言之，这儿有两点比较重要： A . 量子力学是以能量是不连续为假设建立的。 B . 量子力学的核心工作就是计算薛定谔方程。所以说，量子力学是很简单的。

宇宙科学潮汐锁定的英语范文

宇宙科学潮汐锁定的英语范文Here's a sample English text on the topic of tidal locking in astronomy, written in an informal and conversational tone while maintaining paragraph independence and diversity in language expression:You know, in the vastness of space, there's a fascinating phenomenon called tidal locking. It's like when two celestial bodies dance together, and one gets locked in place, always showing the same side to its partner. Kind of like two dancers who've been dancing so long, they're perfectly synced.Imagine a moon orbiting a planet. Over millions of years, the moon's gravity pulls on the planet, and the planet's gravity pulls back, causing the moon to gradually settle into a rhythm. Eventually, the moon stops rotating on its own axis and just revolves around the planet, always showing the same face. It's a cosmic romance, if you will.And it's not just moons that get tidally locked. Even some planets in our solar system's extreme outer regions have been found to be tidally locked to their suns. It's a cosmic dance that goes on for ages, with each partner perfectly aligned in their own orbit.One cool thing about tidal locking is that it can give us a clue about the age of a celestial body. If a moon is tidally locked to its planet, it means they've been dancing together for a long time. It's a bit like seeing an old married couple, still going strong after decades of companionship.。

太空宇宙飞船介绍作文英语

As a high school student with a keen interest in astronomy and space exploration, Ive always been fascinated by the concept of spacecraft. The idea of venturing into the vast expanse of the cosmos, exploring the unknown, and pushing the boundaries of human knowledge is both thrilling and humbling. In this essay, I will delve into the world of spacecraft, discussing their history, types, and the impact theyve had on our understanding of the universe.Spacecraft, in their most basic form, are vehicles designed to travel through outer space. The history of spacecraft can be traced back to the early 20th century when the concept of space travel was still a mere fantasy. However, the launch of Sputnik 1 by the Soviet Union in 1957 marked the beginning of the space age, proving that it was indeed possible to send a manmade object into space.Since then, there have been numerous types of spacecraft developed for various purposes. One of the most wellknown types is the satellite. Satellites orbit the Earth and are used for a wide range of applications, from communication and navigation to weather monitoring and scientific research. They have become an integral part of our daily lives, enabling us to watch television, make phone calls, and access the internet.Another type of spacecraft is the probe. Probes are unmanned spacecraft designed to explore other planets and celestial bodies in our solar system. Some of the most famous probes include Voyager 1 and 2, which were launched in 1977 and have since traveled beyond the heliosphere, the bubblelike region of space dominated by the solar wind. These probeshave provided us with invaluable information about the planets theyve visited and have helped us to understand the composition and structure of our solar system.Manned spacecraft, such as the Apollo missions and the Space Shuttle, have allowed humans to travel into space and conduct experiments that would be impossible on Earth. The Apollo 11 mission, which landed on the moon in 1969, was a monumental achievement that demonstrated the incredible capabilities of human ingenuity and technology.Spacecraft have also played a crucial role in our understanding of the universe beyond our solar system. The Hubble Space Telescope, launched in 1990, has provided us with stunning images of distant galaxies, nebulae, and stars, allowing us to explore the vastness of the cosmos and gain insights into the origins and evolution of the universe.As we continue to develop new technologies and explore the cosmos, spacecraft will undoubtedly play an even more significant role in our quest for knowledge. The recent success of the Mars rovers, which have been exploring the red planet for years, is a testament to the potential of spacecraft to unlock the mysteries of our universe.In conclusion, spacecraft have revolutionized our understanding of space and have opened up a world of possibilities for future exploration. From satellites that connect us to the world to probes that venture into the far reaches of our solar system, spacecraft have become an essential part of our lives and our pursuit of knowledge. As we continue to push theboundaries of space travel, I am excited to see what new discoveries and advancements await us in the future.。

Quantum Mechanics

Quantum MechanicsQuantum mechanics, a cornerstone of modern physics, delves into the perplexing behavior of matter and energy at the atomic and subatomic levels. This realm, where classical physics fails to provide adequate explanations, is governed by principles that often defy our intuitive understanding of the world. At the heart of quantum mechanics lies the concept of quantization, which postulates that energy, like matter, exists in discrete packets known as quanta. Thisrevolutionary idea, first proposed by Max Planck in 1900, challenged the long-held belief that energy flowed continuously. Another fundamental tenet of quantum mechanics is wave-particle duality. This principle asserts that particles, such as electrons and photons, can exhibit both wave-like and particle-like properties. The famous double-slit experiment, where electrons passing through two slits create an interference pattern, exemplifies this paradoxical nature. The wave function, a mathematical construct, describes the probability of finding aparticle at a specific location and time, blurring the lines between deterministic classical physics and the probabilistic nature of the quantum world. The Heisenberg uncertainty principle, formulated by Werner Heisenberg in 1927, further complicates our understanding of the quantum realm. This principle states that it is impossible to simultaneously determine with perfect accuracy both the position and momentum of a particle. The more precisely we know one quantity, the less precisely we can know the other. This inherent uncertainty arises not from limitations in our measurement tools but from the fundamental nature of quantum systems. Quantum entanglement, a phenomenon that puzzled even Einstein, occurs when two or more particles become interconnected, their fates intertwined regardless of the distance separating them. Measuring the state of one entangled particle instantaneously determines the state of the other, seemingly defying the limitations imposed by the speed of light. This "spooky action at a distance," as Einstein called it, has profound implications for our understanding of reality and the nature of causality. The bizarre and counterintuitive principles of quantum mechanics have led to groundbreaking technologies that shape our modern world. Lasers, transistors, and magnetic resonance imaging (MRI) are just a few examples of innovations that rely on the principles of quantum mechanics. Moreover, quantumcomputing, an emerging field with the potential to revolutionize fields like medicine, materials science, and artificial intelligence, exploits the unique properties of quantum systems to perform calculations beyond the capabilities of classical computers. As we delve deeper into the quantum realm, we continue to unravel its mysteries and explore its implications. Quantum mechanics challenges our classical intuitions and compels us to rethink our understanding of reality. This ongoing exploration promises not only to advance our technological capabilities but also to reshape our fundamental understanding of the universe and our place within it.。

the definition of time

The Definition of timeAuthor’s email:Abstract:Define time as a mapping of the whole universe transformations.In this definition about the whole transformations,the concepts of relative time and local time don’t exist.And this definition is consistent with the most physical phenomena.The physicists of last century excluded the possibility of the existence of Electromagnetic wave conducting medium according to a series of experiments’result,in which,speed repeating effect relative to the light speed was not observed.But this deduction overlooked the other possibility that the matters with masses and their movements are the wave on the Electromagnetic wave conducting medium as well.Nothing can really go through the space,light speed is the maximum conducting speed in the space,so it doesn’t exist speed repeating over the speed of light.I assume that the universe is composed of the space elementary quantum,which is the Electromagnetic wave conducting medium. Keywords:the definition of time;Space elementary quantum(SEQ);Another explanation on double-slit experiment,and uncertainty relationIntroduction:Traditional physics theories haven’t defined‘time’clearly,but‘time’is the important physical quantity used in various physics equations.So whatever it is Academic authoritative or informal,a clear definition of time could be needed at first.Starting from discussing the universe’s fundamental composition,using basic algebra thinking,understanding time as a transformation system and deducing some interesting suppositions,such as the deduction like‘time’s evolution is limited but not eternal’,this article is based on the Law of conservation of energy and Law of entropy increase,and other Max Karl Ernst Ludwig Planck’achievements(April23,1858-October4,1947),and is illuminated by Theory of relativity’s concept of warping of space-time and The Quantum Theory’s concept of field quantum. Some preparatory assumptions or basic sets:1the universe is in expansion.2the universe runs with the Law of conservation of energy and Law of entropy increase3the universe is composed of Space elementary quantum(SEQ)4various field quantum and all various micro particle are just different Energy excited states of Space elementary quantum(SEQ)5Space elementary quantum(SEQ)has a ground state with ground energy(as ground state spin’s form)this can be one possible explanation on the phenomenon of parity nonconservation. The course of defining time:A I assume that the universe is composed of Space elementary quantum(SEQ).various field quantum and all various micro particle are just different Energy excited states of Space elementary quantum(SEQ).Space elementary quantum(SEQ)is the Electromagnetic wave conducting medium.The matters with masses and their movements are the wave on the Electromagnetic wave conducting medium as well,and in this assumption framework,Nothing can really go through the space, light speed is the maximum conducting speed in the space,so it doesn’t exist speed repeatingover the speed of light.Everything is just some energy state of Space elementary quantum(SEQ).B The composition’s features of the universe:according to the Law of conservation of energy and Planck’s finding about energy quantization,I assume that the number of Space elementary quantum(SEQ)is limited,further more,I assume the number of Space elementary quantum in the universe is N,every Space elementary quantum(SEQ)has M energy states,so the universecan generates to the Nth power of M’s transformation.And these M energy state could be understood as a algebra system with two effects like a ring including Gravitational and electromagnetic effects.With the the Law of conservation of energy and Law of entropy increase Constraint,thepossible transformations’number is much less than the Nth power of M’s.C The definition of time:C1Supposing the possible number of the universe transformations is J,J is much less than the Nth power of M’sC2The time interval between two adjacent transformations is Planck time,it’s easy to be understood to the minimum time interval.C3The direction of time’s evolution is entropy increase.C4The possible transformations can be mapped to possible entropy set,but not bijection, because multiple transformations could be mapped to the one entropy value.I suppose there are k entropy values in the entropy set,and the J transformations can be divided into K.The transformations with the same entropy value are called parallel transformations,but it can only occur one transformation in reality from the parallel transformations with the same entropy. These set including K entropy values can be mapped to the time as bijection.C5according to what mentioned above,we can deduce that the transformations are not unlimited so that the time is discrete and limited.Another explanation on some Existing phenomena:1uncertainty relation.As long as it is considered that Electromagnetic wave is conducting wave on the Space elementary quantum(SEQ),it is easy to understand uncertainty relation.In this assumption,the speed is conducting speed,and the location is equivalent to conducting on certain Space elementary quantum(SEQ).When the location is determined,how could the conducting speed be determined,vice versa.And I think that it is impossible for now to get the accurate location,when the galaxies is moving and the universe is expanding constantly.2Double-slit experiment.Based on Space elementary quantum(SEQ)assumption,electron’s conducting in the space conforms some probability function,and really causes the space multi-path oscillating,and these oscillating can be accumulated.Electron excitation caused space oscillating both of the two slits at the sides of excitation source,it is easy to understand that interference Fringe could be observed.Even in case of one-slit experiment,when the accumulated Wavelets cross the slit,slit’s Unsmooth edges can make different reflections and generate interference that could be observed if the sensor is sensitive enough.The definition mentioned above in this article is just a kind of understanding on time,I still hope more people can put forward their own definition of time.The defining time is an inescapable job if we want to get a step further in physics.Pay my respects to Max Karl Ernst Ludwig Planck and the other physicists!。

中考英语太空探索的历史与未来单选题40题

中考英语太空探索的历史与未来单选题40题1. Neil Armstrong is famous for being the first person to walk on the _____.A.moonB.sunC.earthD.mars答案：A。

本题考查太空探索的常识。

Neil Armstrong 是第一个登上月球的人。

B 选项太阳不能行走；C 选项地球不是Neil Armstrong 第一个行走的星球；D 选项火星目前还没有人行走过。

2. The first satellite launched into space was _____.A.Sputnik 1B.Apollo 11C.ChallengerD.Columbia答案：A。

本题考查太空探索历史中的重要事件。

Sputnik 1 是第一颗发射到太空的人造卫星。

B 选项Apollo 11 是载人登月任务；C 选项Challenger 和D 选项Columbia 都是航天飞机。

3. Who was the first American woman in space?A.Sally RideB.Judy ResnikC.Kathryn SullivanD.Mae Jemison答案：A。

本题考查太空探索中的人物。

Sally Ride 是第一位进入太空的美国女性。

B 选项Judy Resnik 也是宇航员但不是第一位美国女性宇航员；C 选项Kathryn Sullivan 也不是第一位；D 选项Mae Jemison 是第一位黑人女宇航员但不是第一位美国女性宇航员。

4. The International Space Station is a joint project of several countries. How many countries are involved?A.5B.15C.20D.25答案：B。

本题考查国际空间站的相关知识。

漫步太空插上科学的翅膀作文七百字

漫步太空插上科学的翅膀作文七百字英文回答：Walking in space with the wings of science is an exhilarating experience that opens up a whole new world of possibilities. As an astronaut, I have had the privilege of exploring the vastness of the universe and witnessing its wonders firsthand.One of the most remarkable aspects of space exploration is the ability to conduct scientific experiments in a zero-gravity environment. This unique setting allows us to study the effects of microgravity on various materials and biological systems. For example, we can observe how flames behave differently in space compared to on Earth, which has led to advancements in fire safety technology.In addition to conducting experiments, spacewalks also provide an opportunity for us to repair and maintain spacecraft and satellites. Just like a mechanic working ona car, we use specialized tools and equipment to fix any issues that may arise. This hands-on approach ensures that our space vehicles are in optimal condition, allowing us to continue our exploration of the cosmos.Furthermore, being in space gives us a different perspective on our planet and its place in the universe. Looking down at Earth from space, I am struck by its beauty and fragility. It reminds me of the importance of taking care of our planet and preserving its resources for future generations.中文回答：漫步太空，借助科学的翅膀，是一种令人振奋的体验，它打开了一个全新的可能性世界。

科学文献

_^? q @@ ? ? ? ? I@ @@ k I @@ @ p2
? p1 ?
? ? p1 ? ?? k J (q) ?? !? t ^_^_^_^_^@ q @ @@ I@ @ @ p2
Figure 1: Tree-level Feynman diagrams for e+e? ! hadrons at O( s); in which the lepton side is represented by an arbitrary current, J ; and the bosonquark coupling by ! . that are parallel to p1;2 but with energy Q=2; the energy of the partons in the parton model. These allow each matrix element to be written as the product of two parts, one corresponding to parton-model production, the other to the emission of the gluon without any knowledge of J or ! . The nal result is then o d n F d 3 = C2 s d?2 (1 ? dx1dx2 x ) 2 x2jM2(r1; q ? r1)j2 + x2jM2(q ? r2; r2)j2 ; 1 2 x1)(1 ? 2 (1) where xi 2 pi q = q q are the energy fractions in the cmf of q; d?2 is an element of the parton-model phase-space, and is the azimuthal angle of the gluon. Both d?2 and have di erent interpretations for the two parts of the matrix element: in the rst they refer to the phase-space to produce a quark with momentum r1 in the parton model, and the azimuthal angle around r1; while in the second they refer to an antiquark with momentum r2. With the exception of neglecting quark masses, this expression is exact for arbitrary currents of vector or axialvector type, so applies equally well to the W decays in WW pair production for example. It is also valid at the helicity level, so if polarisation e ects are retained in the lowest-order matrix element, they are correctly treated at O( s). As a Monte Carlo prescription, (1) has a simple interpretation: rst partonmodel events are generated according to the exact matrix element, then gluon emission is generated according to 1 d = CF s x2 + x2 ; 1 2 (2) 2 (1 ? x1)(1 ? x2) 0 dx1 dx2 and nally parton 1 or 2 is chosen with relative probability x2 and x2 to retain 1 2 its parton-model direction, with the hadron plane being rotated uniformly in 3