量子力学-薛定谔方程

Lecture 10-11Schrödinger(薛定谔)equationsPrior to 1925 quantum physics was a “hodgepodge” of hypotheses, principles, theorems and recipes . It was not a logically consistent theory.Once we know this wavefunction we know “everything” about the system!Part 1Dynamic EquationsIf we know the forces acting upon the particle than,according to classical physics , we know everything about a particle at any moment in the future.22,(),()d r F ma F U r U r m dt ==-∇-∇=r r r r r r r r22221()()0E r E r c t∂∇-=∂r r r r A differential equation by itself does not fully determine theunknown function ()(,)r t or E r t r r r .Part 2Dynamic Equation of Wave function---- Schrödinger equations用21()sin cos 2x kx kx ψ⎡⎤=+⎢⎥⎣⎦描述的粒子，只能有5种动量取值，分别是0,2,2,,k k k k -- ，对应的几率分别是11111,,,,28888，这些几率总和应该为1。

()()1111111()0220,2888811111()128888ki i i k i i p P p p k k k k P p ====⋅+⋅+-⋅+⋅+-⋅==++++=∑∑h h h h 1212(),()1(),()1(),()1k ki i i i i x P x x P x x p x xd x x p x dx x x xd x x d ψψ=======⇒=∑∑⎰⎰⎰⎰¡¡¡¡Do we have the same recipe for calculation of average momentum by using wave function in position representation? Yes, of course, we have!To find the expectation (average) value of p , we first need to represent p in terms of x and t . Consider the derivative of the wave function of a free particle with respect to x:0001(,)exp ()p i x t p x E t ⎡⎤ψ=-⎢⎥⎣⎦h We find that0000000000**000(,)(,)(,)(,)(,)()(,)(,)(,)(0)δ∂ψ=ψ⇒ψ=ψ∂∂==ψ-ψ=ψψ=∂∂-∂⎰⎰p p p p p p p p i x t p x t x t p x t x p p x t i x t dx p x t x t dx i xp x　h h h This suggests we define the momentum operator asThe expectation value of the momentum is20022220221()sin cos 2()()()()()=()()()()()ψϕϕϕϕϕ------⎡⎤=+⎢⎥⎣⎦=--++--++k k k k k k k k k k k k x kx kx C x C x C x C x C x x x x x x h h h h h h h h h h h h1()px i p x e ϕ=h ()()111111()022028888===⋅+⋅+-⋅+⋅+-⋅=∑ki i i p P p p k k k k h h h h So,we can not have definite values for the dynamical variables, such as the momentum, when the state of a particle is determined by the wave function with respect to x. We have to find the other way to describe thedynamical variables in Quantum Mechanics.For every dynamical variable or any observable thereis a corresponding Quantum Mechanical OperatorPhysical Quantities →OperatorsOperators are important in quantum mechanics. All observables have corresponding operators.Operators ↔Symbols for mathematical operation✧ The position x is its own operator ˆxx =. Done. Other operators are simpler and just involve multiplication 22x x x x ∧==⋅. ✧ The potential energy operator is just multiplication by V(x).✧ The momentum operator is defined as ˆp i x ∂=-∂h00000002211ˆˆˆˆ1()()[11ˆ(()](())()222ˆ()()())ˆ()()1ˆ()2pp xipp x p xp xix x piix px p pp xix p pp p ppx eipip x i e p eexp x p xx xpT x p p x p p x xpxx eT x T xϕϕϕϕμμμϕϕϕϕϕϕϕμϕ=====⎛⎫∂-=-==⎪∂⎝⎭===hh hhhhh()px xϕ=h000000001(,)exp()1ˆ(,)exp()(,)ˆ(,)(,)pp pp pix t p x E ti iEE x t i p x E t i x ttE x t E x t⎡⎤ψ=-⎢⎥⎣⎦∂⎧⎫⎡⎤⎛⎫ψ=-=ψ-⎬ ⎪⎢⎥∂⎝⎭⎣⎦⎭ψ=ψhh hh h Eigenvalue equation of an operatoreigenvalueDeriving the Schrödinger Equation using operators: This was a plausibility argument, not a derivation. We believe the Schrödinger equation not because of this argument, but because its predictions agree withexperiments.Schrödinger EquationNotes:The Schrödinger Equation is THE fundamental equation of Quantum Mechanics.There are limits to its validity. In this form it applies only to a single, non-relativistic particle (i.e. one withnon-zero rest mass and speed much less than c.)●On the left hand picture 13 velocity vectors of an individual fly are shown; the chain●On the right hand picture the same 13velocity vectors are assigned to 1 fly each todemonstrate that the ensemble average yields the same result, i.e. <v e> = 0,provided that each and every fly does the same thing on average.●i.e. time average = ensemble average. The new subscripts "e" and "r" denote ensemble and space, respectively. This is a simple version of a very far reaching concept in stochastic physics known under the catch word "ergodic hypothesis".●As long as every fly does - on average - the same thing, the vector average overtime of the ensemble is identical to that of an individual fly - if we sum up a fewthousand vectors for one fly, or a few million for lots of flies does not make anydifference. However, we also may obtain this average in a different way:●We do not average one fly in time obtaining <v i>t , but at any given time all flies inspace.●This means, we just add up the velocity vectors of all flies at some moment in timeand obtain <v e>r , the ensemble average. It is evident (but not easy to prove for general cases) thata) Schrödinger equation is a linear homogeneous partialdifferential equation.b) The Schrödinger equation contains the complex number i.Therefore its solutions are essentially complex (unlikeclassical waves, where the use of complex numbers isjust a mathematical convenience.)c) The wave equation has infinite number of solutions,someof which do not correspond to any physical or chemical reality.1. For an electron bound to an atom/molecule, the wavefunction must be everywhere finite, and it must vanish in the boundaries2. Single valued3. Continuous4. Gradient (dψ/dr) must be continuous5. ψψ*dτ is finite, so that ψ can be normalizedd) Solutions that do not satisfy these properties(above)DONOT generally correspond to physicallyrealizable circumstances.e) Conditions on the wave function(波函数的三个基本条件——有限、单值、连续)1. In order to avoid infinite probabilities, the wave functionmust be finite everywhere.2. The wave function must be single valued.3. The wave function must be twice differentiable. Thismeans that it and its derivative must be continuous. (An exception to this rule occurs when V is infinite.)4. In order to normalize a wave function, it must approachzero as x approaches infinity.f) Only the physically measurable quantities must be real.These include the probability, momentum and energy.Can think of the LHS of the Schrödinger equation as a differentialoperator that represents the energy of the particle ?This operator is called the Hamiltonian of the particle , and usually given the symbolˆH.Hamiltonian is a linear differential operator .222ˆ(,)2d V x t H m dx ⎡⎤-+ψ≡ψ⎢⎥⎣⎦h Kineticenergy Potential energyHence there is an alternative (shorthand) form for thetime-dependent Schrödinger equation:Part 3Time-independent Schrödinger equation (TISE), i.e.stationary state(定态)Schrödinger equationSuppose potential is independent of time(),()U x t U x =Look for a separated solution, substitute (,)()()x t x T t ψψ=into• This only tells us that T(t) depends on the energy E . It doesn’t tell us what the energy actually is. For that we have to solve the space part.• T(t) does not depend explicitly on the potential U(x). But there is an implicit dependencebecause the potential affectsthe possible values for the energy E .This is the time-independent Schrödinger equation (TISE) or so-called stationary state Schrödinger equation.Solution to full TDSE isEven though the potential is independent of time the wavefunction still oscillates in time . But probability distribution is static()()2*//2*,,()()()()()iEt iEt P x t x t x e x e x x x ψψψψψ+-=ψ===h hFor this reason a solution of the TISE is known as a StationaryState(定态)Stationary state Schrödinger Equation Notes:• In one-dimension space, the TISE is an ordinary differential equation (not a partial differential equation)• The TISE is an eigenvalue equation for the Hamiltonianoperator:ˆ()()Hx E x ψψ=Part 4 Probability current density and continuity equation Definition of probability current densityIn non-relativistic quantum mechanics, the probability current of the wave function Ψ is defined asin the position basis and satisfies the quantum mechanical continuity equationwith the probability density defined as.If one were to integrate both sides of the continuity equation with respect to volume, so thatthen the divergence theorem implies the continuity equation is equivalent to the integral equationwhere the V is any volume and S is the boundary of V. This is the conservation law for probability in quantum mechanics.In particular, if is a wavefunction describing a single particle, theintegral in the first term of the preceding equation (without the time derivative) is the probability of obtaining a value within V when the position of the particle is measured. The second term is then the rate at which probability is flowing out of the volume V. Altogether the equation states that the time derivative of the change of the probability of the particle being measured in V is equal to the rate at which probability flows into V. Derivation of continuity equationThe continuity equation is derived from the definition of probability current and the basic principles of quantum mechanics. Suppose is the wavefunction for a single particle in the positionbasis (i.e. is a function of x, y, and z). Thenis the probability that a measurement of the particle's position will yield a value within V. The time derivative of this iswhere the last equality follows from the product rule and the fact that the shape of V is presumed to be independent of time (i.e. the time derivative can be moved through the integral). In order to simplify this further, consider the time dependent Schrödinger equationand use it to solve for the time derivative of :When substituted back into the preceding equation for this gives.Now from the product rule for the divergence operatorand since the first and third terms cancel:If we now recall the expression for P and note that the argumentof the divergence operator is justthis becomeswhich is the integral form of the continuity equation .The differential form follows from the fact that the preceding equation holds for all V, and as the integrand is a continuousfunction of space, it must vanish everywhere:For all whole space we have()()2lim lim 0lim lim 0V V V V V S V S dV j dV t j dV j ds →∞→∞→→∞→∞⎛⎫∂ψ ⎪=-∇⋅= ⎪∂⎝⎭∇⋅=⋅=⎰⎰⎰⎰⎰⎰⎰⎰⎰⎰⎰r r r r rwhich meansthat must be continuousat any positionin the whole space.So the wavefunction and its derivative must be continuous.(An exception to this rule occurs when V is infinite.)One more, if the(,)()Et i x t x e ϕ-ψ=hand ()x ϕis real, the probability current*Im ()()0j x x m ϕϕ⎡⎤=∇≡⎣⎦r r h over the whole 1D space which means j r is always continuous whatever the wavefunction ()x ϕand its derivative ()x ϕ'are continuous or not. However, ()x ϕhas to be continuous for an acceptable physical solution for that the probability density is uniquely defined(唯一确定). As to ()x ϕ', it may not be continuous especially at the point where the potential energy is infinite.It is easy to prove that ()x ϕ' has to be continuous at the point 0x where the potential energy just has a limited high step.Have a fun!。