Technical Note 21 Hermite Polynomials and Their Use for Integration
纳米管制作皮肤感应器 翻译 中英
最后译文：纳米管弹性制作出皮肤般的感应器美国斯坦福大学的研究者发现了一种富有弹性且透明的导电性能非常好的薄膜，这种薄膜由极易感触的碳纳米管组成，可被作为电极材料用在轻微触压和拉伸方面的传感器上。
“这种装置也许有一天可以被用在被截肢者、受伤的士兵、烧伤方面接触和压迫的敏感性的恢复上，也可以被应用于机器人和触屏电脑方面”，这个小组如是说。
鲍哲南和他的同事们在他们的弹透薄膜的顶部和底部喷上一种碳纳米管的溶液形成平坦的硅板，覆盖之后，研究人员拉伸这个胶片，当胶片被放松后，纳米管很自然地形成波浪般的结构，这种结构作为电极可以精准的检测出作用在这个材料上的力量总数。
事实上，这种装配行为上很像一个电容器，用硅树脂层来存储电荷，像一个电池一样，当压力被作用到这个感应器上的时候，硅树脂层就收紧，并且不会改变它所储存的电荷总量。
这个电荷是被位于顶部和底部的硅树脂上的纳米碳管测量到的。
当这个复合膜被再次拉伸的时候，纳米管会自动理顺被拉伸的方向。
薄膜的导电性不会改变只要材料没有超出最初的拉伸量。
事实上，这种薄膜可以被拉伸到它原始长度的2.5倍，并且无论哪种方向不会使它受到损害的拉伸它都会重新回到原始的尺寸，甚至在多次被拉伸之后。
当被充分的拉伸后，它的导电性喂2200S/cm，能检测50KPA的压力，类似于一个“坚定的手指捏”的力度，研究者说。
“我们所制作的这个纳米管很可能是首次可被拉伸的，透明的，肤质般感应的，有或者没有碳的纳米管”小组成员之一Darren Lipomi.说。
这种薄膜也可在很多领域得到应用，包括移动设备的屏幕可以感应到一定范围的压力而不仅限于触摸；可拉伸和折叠的几乎不会毁坏的触屏感应器；太阳能电池的透明电极；可包裹而不会起皱的车辆或建筑物的曲面；机器人感应装置和人工智能系统。
其他应用程序“其他系统也可以从中受益—例如那种需要生物反馈的—举个例子，智能方向盘可以感应到，如果司机睡着了，”Lipomi补充说。
高分子化学术语中英文对照
高分子macromolecule, polymer 又称“大分子”。
2 超高分子supra polymer3 天然高分子natural polymer4 无机高分子inorganic polymer5 有机高分子organic polymer6 无机-有机高分子inorganic organic polymer7 金属有机聚合物organometallic polymer8 元素高分子element polymer9 高聚物high polymer10 聚合物polymer11 低聚物oligomer 曾用名“齐聚物”。
12 二聚体dimer13 三聚体trimer14 调聚物telomer15 预聚物prepolymer16 均聚物homopolymer17 无规聚合物random polymer18 无规卷曲聚合物random coiling polymer19 头-头聚合物head-to-head polymer20 头-尾聚合物head-to-tail polymer21 尾-尾聚合物tail-to-tail polymer22 反式有规聚合物transtactic polymer23 顺式有规聚合物cistactic polymer24 规整聚合物regular polymer25 非规整聚合物irregular polymer26 无规立构聚合物atactic polymer27 全同立构聚合物isotactic polymer 又称“等规聚合物”。
28 间同立构聚合物syndiotactic polymer 又称“间规聚合物”。
29 杂同立构聚合物heterotactic polymer 又称“异规聚合物”。
30 有规立构聚合物stereoregular polymer, tactic polymer 又称“有规聚合物”。
31 苏型双全同立构聚合物threo-diisotactic polymer32 苏型双间同立构聚合物threo-disyndiotactic polymer33 赤型双全同立构聚合物erythro-diisotactic polymer34 赤型双间同立构聚合物erythro-disyndiotactic polymer35 全同间同等量聚合物equitactic polymer36 共聚物copolymer37 二元共聚物binary copolymer38 三元共聚物terpolymer39 多元聚合物multipolymer40 序列共聚物sequential copolymer41 多层共聚物multilayer copolymer42 多相聚合物multiphase polymer43 统计[结构]共聚物statistical copolymer44 无规共聚物random copolymer46 周期共聚物periodic copolymer47 梯度共聚物gradient copolymer48 嵌段共聚物block copolymer 又称“嵌段聚合物(block polymer)” 。
色谱专用术语英语翻译
不知各位是否用的着色谱图 chromatogram 色谱峰 chromatographic peak峰底peak base峰高 h，peak height峰宽 W，peak width半高峰宽 Wh/2，peak width at half height峰面积 A，peak area拖尾峰 tailing area前伸峰 leading area假峰ghost peak畸峰 distorted peak反峰 negative peak拐点 inflection point原点 origin斑点 spot区带 zone复班 multiple spot区带脱尾 zone tailing基线 base line基线漂移 baseline drift基线噪声 N，baseline noise统计矩 moment一阶原点矩γ1，first origin moment二阶中心矩μ2，second central moment三阶中心矩μ3，third central moment液相色谱法 liquid chromatography，LC液液色谱法 liquid liquid chromatography，LLC液固色谱法 liquid solid chromatography，LSC正相液相色谱法 normal phase liquidchromatography反相液相色谱法 reversed phase liquidchromatography，RPLC柱液相色谱法 liquid column chromatography高效液相色谱法 high performance liquidchromatography，HPLC尺寸排除色谱法 size exclusion chromatography，SEC凝胶过滤色谱法gel filtration chromatography凝胶渗透色谱法 gel permeation chromatography，GPC亲和色谱法 affinity chromatography离子交换色谱法 ion exchange chromatography，IEC离子色谱法 ion chromatography离子抑制色谱法 ion suppression chromatography离子对色谱法 ion pair chromatography疏水作用色谱法 hydrophobic interactionchromatography制备液相色谱法 preparative liquid chromatography平面色谱法 planar chromatography纸色谱法 paper chromatography薄层色谱法 thin layer chromatography，TLC高效薄层色谱法high performance thin layerchromatography，HPTLC浸渍薄层色谱法 impregnated thin layerchromatography凝胶薄层色谱法 gel thin layer chromatography离子交换薄层色谱法 ion exchange thin layerchromatography制备薄层色谱法 preparative thin layerchromatography薄层棒色谱法 thin layer rod chromatography液相色谱仪liquid chromatograph制备液相色谱仪 preparative liquid chromatograph凝胶渗透色谱仪 gel permeation chromatograph涂布器 spreader点样器 sample applicator色谱柱 chromatographic column棒状色谱柱monolith column monolith column微粒柱 microparticle column填充毛细管柱 packed capillary column空心柱open tubular column微径柱 microbore column混合柱mixed column组合柱 coupled column预柱 precolumn保护柱 guard column预饱和柱 presaturation column浓缩柱 concentrating column抑制柱 suppression column薄层板 thin layer plate浓缩区薄层板 concentratingthin layer plate荧光薄层板 fluorescence thin layer plate反相薄层板 reversed phase thin layer plate梯度薄层板 gradient thin layer plate烧结板 sintered plate展开室 development chamber 往复泵 reciprocating pump注射泵 syringe pump气动泵 pneumatic pump蠕动泵peristaltic pump检测器 detector微分检测器differential detector积分检测器 integral detector总体性能检测器 bulk property detector溶质性能检测器solute property detector(示差)折光率检测器[differential] refractive indexdetector荧光检测器fluorescence detector紫外可见光检测器 ultraviolet visible detector电化学检测器 electrochemical detector蒸发(激光)光散射检测器 [laser] light scatteringdetector光密度计 densitometer薄层扫描仪thin layer scanner柱后反应器 post-column reactor体积标记器 volume marker记录器 recorder积分仪integrator馏分收集器 fraction collector工作站 work station固定相 stationary phase固定液 stationary liquid载体 support柱填充剂 column packing化学键合相填充剂 chemically bonded phasepacking薄壳型填充剂pellicular packing多孔型填充剂 porous packing吸附剂 adsorbent离子交换剂 ion exchanger基体 matrix载板 support plate粘合剂 binder流动相 mobile phase洗脱(淋洗)剂 eluant，eluent展开剂 developer等水容剂isohydric solvent改性剂 modifier显色剂 color [developing] agent死时间 t0，dead time保留时间 tR，retention time调整保留时间 t'R，adjusted retention time死体积 V0，dead volume保留体积 vR，retention volume调整保留体积 v'R，adjusted retention volume柱外体积 Vext，extra-column volune粒间体积 V0，interstitial volume(多孔填充剂的)孔体积 VP，pore volume of porouspacking液相总体积 Vtol，total liquid volume洗脱体积 ve，elution volume流体力学体积 vh，hydrodynamic volume相对保留值 ri.s，relative retention value分离因子α，separation factor流动相迁移距离 dm，mobile phase migrationdistance流动相前沿 mobile phase front溶质迁移距离 ds，solute migration distance比移值 Rf，Rf value高比移值hRf，high Rf value相对比移值 Ri.s，relative Rf value保留常数值 Rm，Rm value板效能 plateefficiency折合板高 hr，reduced plate height分离度R，resolution液相载荷量 liquid phase loading离子交换容量 ion exchange capacity负载容量 loading capacity渗透极限 permeability limit排除极限 Vh，max，exclusion limit拖尾因子 T，tailing factor柱外效应 extra-column effect管壁效应 wall effect间隔臂效应 spacer arm effect边缘效应 edge effect斑点定位法 localization of spot放射自显影法 autoradiography 原位定量 in situ quantitation生物自显影法bioautography归一法 normalization method内标法internal standard method外标法 external standard method叠加法 addition method普适校准(曲线、函数) calibration function or curve 谱带扩展(加宽) band broadening(分离作用的)校准函数或校准曲线 universalcalibration function or curve [of separation]加宽校正 broadening correction加宽校正因子 broadening correction factor溶剂强度参数ε0，solvent strength parameter洗脱序列 eluotropic series洗脱(淋洗) elution等度洗脱 gradient elution 梯度洗脱 gradient elution(再)循环洗脱 recycling elution线性溶剂强度洗脱 linear solvent strength gradient程序溶剂 programmed solvent程序压力programmed pressure程序流速 programmed flow展开development上行展开 ascending development下行展开descending development双向展开 two dimensional development环形展开 circular development离心展开centrifugal development向心展开 centripetal development径向展开 radial development多次展开multiple development分步展开 stepwise development 连续展开 continuous development梯度展开 gradient development匀浆填充 slurry packing停流进样 stop-flow injection阀进样 valve injection柱上富集 on-column enrichment流出液 eluate柱上检测 on-column detection柱寿命 column life柱流失 column bleeding 显谱 visualization活化 activation反冲 back flushing脱气 degassing沟流 channeling过载overloading。
一维谐振子波函数
一维谐振子波函数摘要：一、一维谐振子的基本概念二、一维谐振子的波函数1.波函数的实值与复值2.波函数的时间依赖性三、一维谐振子的能量本征函数四、应用与结论正文：一、一维谐振子的基本概念一维谐振子是一种物理模型，用于描述在一维空间中运动的粒子受到弹性势能的影响而发生振动的现象。
在这个模型中，粒子被限制在一个有限的空间范围内，如一个线性的势阱。
一维谐振子的研究有助于理解简谐振动在其他领域的应用，如机械振动、电磁波等。
二、一维谐振子的波函数1.波函数的实值与复值在研究一维谐振子时，我们需要关心波函数。
波函数是描述粒子在空间中位置的函数，通常表示为Ψ(x)。
在一维谐振子问题中，波函数可以分为实部和虚部，即Ψ(x) = A * cos(kx - ωt) + Bi * sin(kx - ωt)，其中A和B为实数，k 为波数，ω为角频率，t为时间。
2.波函数的时间依赖性由于波函数中含有时间变量t，因此我们需要了解波函数随时间的变化规律。
从薛定谔方程可以看出，波函数的时间偏导数含有虚数单位i，所以一般情况下波函数为复值函数。
而波函数的模平方不随时间变化，表示粒子在某一位置的概率密度。
三、一维谐振子的能量本征函数在一维谐振子问题中，能量本征函数是描述粒子能量的函数。
对于简谐振子，能量本征函数可以表示为Ψ(x) = C * exp(-x/2) * Hermite polynomials(x)，其中C为归一化常数，Hermite polynomials(x)为赫尔墨特多项式。
这些本征函数满足薛定谔方程，并具有归一化和正交性质。
四、应用与结论一维谐振子的研究在物理学、力学等领域具有广泛的应用。
通过对一维谐振子的研究，我们可以更好地理解简谐振动的特点，为实际问题的解决提供理论依据。
在实际应用中，一维谐振子模型可以扩展到更高维度的谐振子模型，从而为多维系统的分析提供方法。
DearEditor,DearR...
Dear Editor, Dear Reviewer,I deeply appreciate the time and effort you’ve spent on reviewing my manuscript. Your comments are really thoughtful and in-depth and I do honestly agree with most of them. Before I address the comments individually, please allow me to explain several difficulties I encountered, which eventually resulted in the limitations of the manuscript, which were also pointed out in your comments.The primary purpose of writing this manuscript is to borrow the data assimilation framework for understanding the impact of the uncertainties in the physical models and their parameters, which has been actively developed and successfully applied in other disciplines, into the seismic modeling and inversion community. This adoption process is not trivial and to make it tractable and also more readable I made simplifications, which lead to limitations of the derived formulation. In addition to the Gaussian assumptions for the stochastic noise processes, which will be addressed in detail later, I also simplified the form of the wave equation, which does not include rotation of the Earth, self-gravitation and other important effects such as poroelasticity. The stochastic noise processes in the dynamic model and its boundary and initial conditions are treated as additive, which might not be suitable for all situations. In the revised manuscript, I pointed out the limitations of my formulation more clearly both in the introduction section and also through out the text.I think the justification for introducing stochastic noises into the wave equation and its initial and boundary conditions is two-folded. First, our deterministic model is not perfect and it is difficult, if possible at all, to fully eliminate all its deficiencies. Second, the impact of the uncertainties in the dynamic model, in particular, the impact on the estimation of model parameters, needs to be evaluated, especially when the procedures of seismological inversions are becoming more and more precise under the full-wave framework. This manuscript is an attempt to address some of the issues under the full-wave framework. Its scope and depth are inherently limited by my own background and capability. But I do hope that it could become useful at some point during the development and application of the full-wave methodology.Responses to major remarks and questions:1.The Gaussian assumption for the stochastic noise processes indeed brings muchconvenience into the derivation. And a direct benefit in terms of readability is that the resulting equations have similarities with classical formulations based on least-squares. The quadratic-form misfit functional and quadratic-form model regularization, which are often employed in classical formulations, correspond to Gaussian likelihood and Gaussian priors used in this manuscript. The Bayesian framework itself is not limited to Gaussian statistics. An example of using exponential and truncated exponential distributions with the Bayesian framework and a grid-search optimization algorithm for centroid moment tensor inversions is given in a separate manuscript, “Rapid Centroid Moment Tensor (CMT) Inversion in a Three-Dimensional Earth Structure Model for Earthquakes inSouthern California (Lee, Chen & Jordan, GJI)”, which is currently under review.For probability densities that are non-Gaussian, the formulation still has applicability if a Gaussian distribution provides a sufficiently good approximation to the actual distribution from a practical point of view. An optimal Gaussian approximation can be found from the first and second moments of the actual distribution. For nonlinear systems such as the one used in solving seismological inverse problems, the system is linearized around the current mean and the covariance is propagated using the linearized dynamics. For limited propagation ranges, a Gaussian distribution could indeed provide a sufficiently good approximation locally. When nonlinearities are high and uncertainties need to be evolved over long ranges, the Gaussian assumption may no longer be valid. A promising new development to account for non-Gaussian probabilities is the generalized polynomial chaos (gPC) theory, which uses a polynomial based stochastic space to represent and propagate uncertainties. The completeness of the space warrants accurate representations of any probability densities and certain bases can be selected to represent particular types of probability densities with the fewest number of terms. In gPC, a perfect Gaussian distribution can be represented using 2 Hermite polynomials and a uniform distribution can be represented using 2 Legendre polynomials. Polynomial math can therefore be employed to make the interactions among various probability densities tractable and the results can be calculated in the polynomial space, which has favorable properties in terms of continuity and differentiability. I am still working on the formulation based on the gPC theory. If successful, it will be documented in a future publication. In the introduction section of the revised manuscript, I’ve added a paragraph to indicate the limitations and applicability of the Gaussian assumption.2.The origins of the uncertainties are sometimes difficult to classify and explain andbecause of such unknown and/or unexplained origins, we often work toward reducing them to stochastic processes and try to quantify their statistical properties using direct and/or indirect methods. I adopted the Bayesian framework in this study, which is essentially a subjective interpretation of probability. Some types of uncertainties depend on chance (i.e. aleatory or statistical) and others are due to the lack of knowledge (i.e. epistemic or systematic). It is difficult to separate different types of uncertainties in the derivation, therefore individual stochastic noises introduced in the derivation do not correspond to a particular type of uncertainty and the noises in the wave equation and its boundary/initial conditions could have both aleatory and epistemic origins. Some common origins of uncertainties, in addition to the uncertainties in the model parameters, include but are not limited to, the errors in the mathematical model, the numerical method used for solving the mathematical model, and errors in the initial and boundary conditions. For epistemic uncertainties, efforts need to be made to better understand the system and sometimes model errors can be evaluated by using improved observations. But in general, to evaluate epistemic uncertainties requires more effort and usually involves discovering new physics or mechanisms, which could be nonlinear. Treating those stochastic noises as additive is certainly a limitation of my formulation and I pointed that out in therevised manuscript. And I also included more explanations about possible origins of the uncertainties that I am aware of in the revised manuscript. But the primary purpose of introducing those stochastic noise processes is to account for uncertainties due to unknown/unexplained origins. The recent development of Dempster-Shafer theory provides a systematic framework for representing epistemic plausibility and has been used in machine learning. A short essay about this new development can be found at /assets/downloads/articles/article48.pdf, but to adopt it in this study is too involved and might not be necessary at the current development stage of full-wave seismological inversions. In terms of nomenclature, I realized that “model error” might be misleading and I changed that to “model residual” in the revised manuscript.3.Possible reasons that could cause deviations from the traction-free boundarycondition might include lithosphere-atmosphere coupling, deviation from the continuum model for materials in the near-surface environment, numerical errors caused by, for instance, errors in the numerical representation of the actual topography, etc. The Earth is constantly in motion. The quiescent-past initial condition for one seismic event could be violated in practice if we consider motions caused by, for instance, other seismic events, the Earth’s ambient noise field and the constant hum of the Earth caused by atmosphere-ocean-seafloor coupling. I’ve added those possible explanations and some references into the revised manuscript. These are some possible causes that I am aware of. It is certainly not complete. There might be also unknown mechanisms or noises that can cause deviations from the theoretical initial and boundary conditions.4.It is true that the errors in the elastic parameters are not independent. I’veremoved the sentence from the revised manuscript. In practice, we only need to introduce 21 independent distributions. To account for equality constraints among elastic parameters, the delta distribution can be introduced to represent the corresponding conditional probabilities. A delta distribution can be treated as the limit of a Gaussian distribution with its variance approaching zero. Going through the same steps in the derivation, the equations for updating the elastic parameters will then include a number of equations that repeat the symmetry conditions among all elastic parameters. I’ve added several sentences in the revised manuscript to clarify this point. It is also true that the Gaussian distribution is only an approximation to the actual distribution, since the elastic parameters need to satisfy stability requirements. The Gaussian approximation is only valid locally when the current mean (i.e. the reference elastic tensor for the current iteration) satisfy the stability requirements and the variance is not too large. I added several sentences in the revised manuscript to clarify this point. Some of the positivity constraints can be removed through a change of variable, but I did not explore in that direction. The quadratic model regularization term that is often used in the objective functions in the classical formulations also imply a Gaussian distribution for structural parameters. But I do fully agree that one should not adopt Gaussian distributions for elastic parameters too easily just for mathematical convenience.Responses to minor remarks and questions:1.I do realize that using the term “full-physics” actually contradicts with theprimary goal of the formulation, which is actually to account for inadequacies in the physical model. On the other hand, I am also concerned that the use of “full waveform” might cause the misunderstanding that I am inverting the completed seismograms from first arrival to coda point by point. I replaced “full-physics” in the title and throughout the text with the term “full-wave”. I hope that is an acceptable term.2.I have re-worded this sentence and added the two references of Bamberger et al.,provided by the reviewer. Many thanks for correcting me on this.3.I do agree that adding the source index indeed complicates the formulation evenfurther. However it might make the discussion on computational costs and the distinctions between scattering-integral and adjoint formulations more clear. I agree that at this point I did sacrifice readability for some degree of clarity. I do apologize for the inconvenience caused by this notation.4.I thank the reviewer for raising this issue. Yes, I did consider this notation.However I was concerned that readers who are not familiar with the methodology might mistaken it as the global optimal. I used the iteration index γ in the iterative Euler-Lagrange equations to indicate the optimal models for each iteration. I hope this is acceptable.5.The statement on source inversion is indeed over simplified. I was trying tomotivate the discussion on separating phase and amplitude information in the complete waveforms. I’ve re-worded the sentence and added a paragraph in section 4.2 to discuss finite-source inversion results from Fichtner & Tkalčić in the revised manuscript.6.I fully acknowledge the importance of the Born approximation in tomography. Ihave re-worded the sentence in the revised manuscript to avoid misleading the readers. The limitation of the Born approximation that I am referring to only applies to direct waveform inversions using waveform differences as data functionals. I’ve re-worded some sentences to emphasize this point. The Born approximation can be used for obtaining the exact sensitivity kernels of other types of data functionals such as cross-correlation travel-time. It actually plays a fundamental role in seismic tomography.7.Yes, it should be correlogram. Many thanks for correcting this mistake in mymanuscript.8.Yes, I agree that the example that I used is not full-physics. I changed “full-physics” to “full-wave” and re-worded a few sentences in the paragraph to emphasize that.9.I’ve added the reference for Fichtner et al. (2010b) and a more extended discussabout the improvements both in resolution and in resolving anisotropy.10.I fully agree. The number of simulation counts used in this manuscript is just togive a general guideline for estimating computational costs. The design of line search is flexible and can change the number of simulations. I added a sentence in the revised manuscript to emphasize that.11.Yes, I have added a few sentences to clarify that this method only works for non-dissipative media. The PML absorbing boundary conditions need to be handledwith care. One possibility that seems to work is to store the wave-field going through the absorbing boundaries and play it back during the simulation with the negative time step. But in this case, additional storage as well as IO costs, which could be substantial depending on the size of the mesh, is needed.12.I have corrected and updated the references. Many thanks for checking andpointing out the errors. I really appreciate it.I hope the responses above address your comments and answer your questions satisfactorily. Thanks very much for your review and I truly appreciate your comments. Best regards,Po Chen。
量子力学索引英汉对照
21-centimeter line, 21厘米线AAbsorption, 吸收Addition of angular momenta, 角动量叠加Adiabatic approximation, 绝热近似Adiabatic process, 绝热过程Adjoint, 自伴的Agnostic position, 不可知论立场Aharonov-Bohm effect, 阿哈罗诺夫-玻姆效应Airy equation, 艾里方程；Airy function, 艾里函数Allowed energy, 允许能量Allowed transition, 允许跃迁Alpha decay, 衰变；Alpha particle, 粒子Angular equation, 角向方程Angular momentum, 角动量Anomalous magnetic moment, 反常磁矩Antibonding, 反键Anti-hermitian operator, 反厄米算符Associated Laguerre polynomial, 连带拉盖尔多项式Associated Legendre function, 连带勒让德多项式Atoms, 原子Average value, 平均值Azimuthal angle, 方位角Azimuthal quantum number, 角量子数BBalmer series, 巴尔末线系Band structure, 能带结构Baryon, 重子Berry's phase, 贝利相位Bessel functions, 贝塞尔函数Binding energy, 束缚能Binomial coefficient, 二项式系数Biot-Savart law, 毕奥-沙法尔定律Blackbody spectrum, 黑体谱Bloch's theorem, 布洛赫定理Bohr energies, 玻尔能量；Bohr magneton, 玻尔磁子；Bohr radius, 玻尔半径Boltzmann constant, 玻尔兹曼常数Bond, 化学键Born approximation, 玻恩近似Born's statistical interpretation, 玻恩统计诠释Bose condensation, 玻色凝聚Bose-Einstein distribution, 玻色-爱因斯坦分布Boson, 玻色子Bound state, 束缚态Boundary conditions, 边界条件Bra, 左矢Bulk modulus, 体积模量CCanonical commutation relations, 正则对易关系Canonical momentum, 正则动量Cauchy's integral formula, 柯西积分公式Centrifugal term, 离心项Chandrasekhar limit, 钱德拉赛卡极限Chemical potential, 化学势Classical electron radius, 经典电子半径Clebsch-Gordan coefficients, 克-高系数Coherent States, 相干态Collapse of wave function, 波函数塌缩Commutator, 对易子Compatible observables, 对易的可观测量Complete inner product space, 完备内积空间Completeness, 完备性Conductor, 导体Configuration, 位形Connection formulas, 连接公式Conservation, 守恒Conservative systems, 保守系Continuity equation, 连续性方程Continuous spectrum, 连续谱Continuous variables, 连续变量Contour integral, 围道积分Copenhagen interpretation, 哥本哈根诠释Coulomb barrier, 库仑势垒Coulomb potential, 库仑势Covalent bond, 共价键Critical temperature, 临界温度Cross-section, 截面Crystal, 晶体Cubic symmetry, 立方对称性Cyclotron motion, 螺旋运动DDarwin term, 达尔文项de Broglie formula, 德布罗意公式de Broglie wavelength, 德布罗意波长Decay mode, 衰变模式Degeneracy, 简并度Degeneracy pressure, 简并压Degenerate perturbation theory, 简并微扰论Degenerate states, 简并态Degrees of freedom, 自由度Delta-function barrier, 势垒Delta-function well, 势阱Derivative operator, 求导算符Determinant, 行列式Determinate state, 确定的态Deuterium, 氘Deuteron, 氘核Diagonal matrix, 对角矩阵Diagonalizable matrix, 对角化Differential cross-section, 微分截面Dipole moment, 偶极矩Dirac delta function, 狄拉克函数Dirac equation, 狄拉克方程Dirac notation, 狄拉克记号Dirac orthonormality, 狄拉克正交归一性Direct integral, 直接积分Discrete spectrum, 分立谱Discrete variable, 离散变量Dispersion relation, 色散关系Displacement operator, 位移算符Distinguishable particles, 可分辨粒子Distribution, 分布Doping, 掺杂Double well, 双势阱Dual space, 对偶空间Dynamic phase, 动力学相位EEffective nuclear charge, 有效核电荷Effective potential, 有效势Ehrenfest's theorem, 厄伦费斯特定理Eigenfunction, 本征函数Eigenvalue, 本征值Eigenvector, 本征矢Einstein's A and B coefficients, 爱因斯坦A,B系数；Einstein's mass-energy formula, 爱因斯坦质能公式Electric dipole, 电偶极Electric dipole moment, 电偶极矩Electric dipole radiation, 电偶极辐射Electric dipole transition, 电偶极跃迁Electric quadrupole transition, 电四极跃迁Electric field, 电场Electromagnetic wave, 电磁波Electron, 电子Emission, 发射Energy, 能量Energy-time uncertainty principle, 能量-时间不确定性关系Ensemble, 系综Equilibrium, 平衡Equipartition theorem, 配分函数Euler's formula, 欧拉公式Even function, 偶函数Exchange force, 交换力Exchange integral, 交换积分Exchange operator, 交换算符Excited state, 激发态Exclusion principle, 不相容原理Expectation value, 期待值FFermi-Dirac distribution, 费米-狄拉克分布Fermi energy, 费米能Fermi surface, 费米面Fermi temperature, 费米温度Fermi's golden rule, 费米黄金规则Fermion, 费米子Feynman diagram, 费曼图Feynman-Hellman theorem, 费曼-海尔曼定理Fine structure, 精细结构Fine structure constant, 精细结构常数Finite square well, 有限深方势阱First-order correction, 一级修正Flux quantization, 磁通量子化Forbidden transition, 禁戒跃迁Foucault pendulum, 傅科摆Fourier series, 傅里叶级数Fourier transform, 傅里叶变换Free electron, 自由电子Free electron density, 自由电子密度Free electron gas, 自由电子气Free particle, 自由粒子Function space, 函数空间Fusion, 聚变Gg-factor, g-因子Gamma function, 函数Gap, 能隙Gauge invariance, 规范不变性Gauge transformation, 规范变换Gaussian wave packet, 高斯波包Generalized function, 广义函数Generating function, 生成函数Generator, 生成元Geometric phase, 几何相位Geometric series, 几何级数Golden rule, 黄金规则"Good" quantum number, "好"量子数"Good" states, "好"的态Gradient, 梯度Gram-Schmidt orthogonalization, 格莱姆-施密特正交化法Graphical solution, 图解法Green's function, 格林函数Ground state, 基态Group theory, 群论Group velocity, 群速Gyromagnetic railo, 回转磁比值HHalf-integer angular momentum, 半整数角动量Half-life, 半衰期Hamiltonian, 哈密顿量Hankel functions, 汉克尔函数Hannay's angle, 哈内角Hard-sphere scattering, 硬球散射Harmonic oscillator, 谐振子Heisenberg picture, 海森堡绘景Heisenberg uncertainty principle, 海森堡不确定性关系Helium, 氦Helmholtz equation, 亥姆霍兹方程Hermite polynomials, 厄米多项式Hermitian conjugate, 厄米共轭Hermitian matrix, 厄米矩阵Hidden variables, 隐变量Hilbert space, 希尔伯特空间Hole, 空穴Hooke's law, 胡克定律Hund's rules, 洪特规则Hydrogen atom, 氢原子Hydrogen ion, 氢离子Hydrogen molecule, 氢分子Hydrogen molecule ion, 氢分子离子Hydrogenic atom, 类氢原子Hyperfine splitting, 超精细分裂IIdea gas, 理想气体Idempotent operaror, 幂等算符Identical particles, 全同粒子Identity operator, 恒等算符Impact parameter, 碰撞参数Impulse approximation, 脉冲近似Incident wave, 入射波Incoherent perturbation, 非相干微扰Incompatible observables, 不对易的可观测量Incompleteness, 不完备性Indeterminacy, 非确定性Indistinguishable particles, 不可分辨粒子Infinite spherical well, 无限深球势阱Infinite square well, 无限深方势阱Inner product, 内积Insulator, 绝缘体Integration by parts, 分部积分Intrinsic angular momentum, 内禀角动量Inverse beta decay, 逆衰变Inverse Fourier transform, 傅里叶逆变换KKet, 右矢Kinetic energy, 动能Kramers' relation, 克莱默斯关系Kronecker delta, 克劳尼克LLCAO technique, 原子轨道线性组合法Ladder operators, 阶梯算符Lagrange multiplier, 拉格朗日乘子Laguerre polynomial, 拉盖尔多项式Lamb shift, 兰姆移动Lande g-factor, 朗德g-因子Laplacian, 拉普拉斯的Larmor formula, 拉摩公式Larmor frequency, 拉摩频率Larmor precession, 拉摩进动Laser, 激光Legendre polynomial, 勒让德多项式Levi-Civita symbol, 列维-西维塔符号Lifetime, 寿命Linear algebra, 线性代数Linear combination, 线性组合Linear combination of atomic orbitals, 原子轨道的线性组合Linear operator, 线性算符Linear transformation, 线性变换Lorentz force law, 洛伦兹力定律Lowering operator, 下降算符Luminoscity, 照度Lyman series, 赖曼线系MMagnetic dipole, 磁偶极Magnetic dipole moment, 磁偶极矩Magnetic dipole transition, 磁偶极跃迁Magnetic field, 磁场Magnetic flux, 磁通量Magnetic quantum number, 磁量子数Magnetic resonance, 磁共振Many worlds interpretation, 多世界诠释Matrix, 矩阵；Matrix element, 矩阵元Maxwell-Boltzmann distribution, 麦克斯韦-玻尔兹曼分布Maxwell's equations, 麦克斯韦方程Mean value, 平均值Measurement, 测量Median value, 中位值Meson, 介子Metastable state, 亚稳态Minimum-uncertainty wave packet, 最小不确定度波包Molecule, 分子Momentum, 动量Momentum operator, 动量算符Momentum space wave function, 动量空间波函数Momentum transfer, 动量转移Most probable value, 最可几值Muon, 子Muon-catalysed fusion, 子催化的聚变Muonic hydrogen, 原子Muonium, 子素NNeumann function, 纽曼函数Neutrino oscillations, 中微子振荡Neutron star, 中子星Node, 节点Nomenclature, 术语Nondegenerate perturbationtheory, 非简并微扰论Non-normalizable function, 不可归一化的函数Normalization, 归一化Nuclear lifetime, 核寿命Nuclear magnetic resonance, 核磁共振Null vector, 零矢量OObservable, 可观测量Observer, 观测者Occupation number, 占有数Odd function, 奇函数Operator, 算符Optical theorem, 光学定理Orbital, 轨道的Orbital angular momentum, 轨道角动量Orthodox position, 正统立场Orthogonality, 正交性Orthogonalization, 正交化Orthohelium, 正氦Orthonormality, 正交归一性Orthorhombic symmetry, 斜方对称Overlap integral, 交叠积分PParahelium, 仲氦Partial wave amplitude, 分波幅Partial wave analysis, 分波法Paschen series, 帕邢线系Pauli exclusion principle, 泡利不相容原理Pauli spin matrices, 泡利自旋矩阵Periodic table, 周期表Perturbation theory, 微扰论Phase, 相位Phase shift, 相移Phase velocity, 相速Photon, 光子Planck's blackbody formula, 普朗克黑体辐射公式Planck's constant, 普朗克常数Polar angle, 极角Polarization, 极化Population inversion, 粒子数反转Position, 位置；Position operator, 位置算符Position-momentum uncertainty principles, 位置-动量不确定性关系Position space wave function, 坐标空间波函数Positronium, 电子偶素Potential energy, 势能Potential well, 势阱Power law potential, 幂律势Power series expansion, 幂级数展开Principal quantum number, 主量子数Probability, 几率Probability current, 几率流Probability density, 几率密度Projection operator, 投影算符Propagator, 传播子Proton, 质子QQuantum dynamics, 量子动力学Quantum electrodynamics, 量子电动力学Quantum number, 量子数Quantum statics, 量子统计Quantum statistical mechanics, 量子统计力学Quark, 夸克RRabi flopping frequency, 拉比翻转频率Radial equation, 径向方程Radial wave function, 径向波函数Radiation, 辐射Radius, 半径Raising operator, 上升算符Rayleigh's formula, 瑞利公式Realist position, 实在论立场Recursion formula, 递推公式Reduced mass, 约化质量Reflected wave, 反射波Reflection coefficient, 反射系数Relativistic correction, 相对论修正Rigid rotor, 刚性转子Rodrigues formula, 罗德里格斯公式Rotating wave approximation, 旋转波近似Rutherford scattering, 卢瑟福散射Rydberg constant, 里德堡常数Rydberg formula, 里德堡公式SScalar potential, 标势Scattering, 散射Scattering amplitude, 散射幅Scattering angle, 散射角Scattering matrix, 散射矩阵Scattering state, 散射态Schrodinger equation, 薛定谔方程Schrodinger picture, 薛定谔绘景Schwarz inequality, 施瓦兹不等式Screening, 屏蔽Second-order correction, 二级修正Selection rules, 选择定则Semiconductor, 半导体Separable solutions, 分离变量解Separation of variables, 变量分离Shell, 壳Simple harmonic oscillator, 简谐振子Simultaneous diagonalization, 同时对角化Singlet state, 单态Slater determinant, 斯拉特行列式Soft-sphere scattering, 软球散射Solenoid, 螺线管Solids, 固体Spectral decomposition, 谱分解Spectrum, 谱Spherical Bessel functions, 球贝塞尔函数Spherical coordinates, 球坐标Spherical Hankel functions, 球汉克尔函数Spherical harmonics, 球谐函数Spherical Neumann functions, 球纽曼函数Spin, 自旋Spin matrices, 自旋矩阵Spin-orbit coupling, 自旋-轨道耦合Spin-orbit interaction, 自旋-轨道相互作用Spinor, 旋量Spin-spin coupling, 自旋-自旋耦合Spontaneous emission, 自发辐射Square-integrable function, 平方可积函数Square well, 方势阱Standard deviation, 标准偏差Stark effect, 斯塔克效应Stationary state, 定态Statistical interpretation, 统计诠释Statistical mechanics, 统计力学Stefan-Boltzmann law, 斯特番-玻尔兹曼定律Step function, 阶跃函数Stem-Gerlach experiment, 斯特恩-盖拉赫实验Stimulated emission, 受激辐射Stirling's approximation, 斯特林近似Superconductor, 超导体Symmetrization, 对称化Symmetry, 对称TTaylor series, 泰勒级数Temperature, 温度Tetragonal symmetry, 正方对称Thermal equilibrium, 热平衡Thomas precession, 托马斯进动Time-dependent perturbation theory, 含时微扰论Time-dependent Schrodinger equation, 含时薛定谔方程Time-independent perturbation theory, 定态微扰论Time-independent Schrodinger equation, 定态薛定谔方程Total cross-section, 总截面Transfer matrix, 转移矩阵Transformation, 变换Transition, 跃迁；Transition probability, 跃迁几率Transition rate, 跃迁速率Translation,平移Transmission coefficient, 透射系数Transmitted wave, 透射波Trial wave function, 试探波函数Triplet state, 三重态Tunneling, 隧穿Turning points, 回转点Two-fold degeneracy , 二重简并Two-level systems, 二能级体系UUncertainty principle, 不确定性关系Unstable particles, 不稳定粒子VValence electron, 价电子Van der Waals interaction, 范德瓦尔斯相互作用Variables, 变量Variance, 方差Variational principle, 变分原理Vector, 矢量Vector potential, 矢势Velocity, 速度Vertex factor, 顶角因子Virial theorem, 维里定理WWave function, 波函数Wavelength, 波长Wave number, 波数Wave packet, 波包Wave vector, 波矢White dwarf, 白矮星Wien's displacement law, 维恩位移定律YYukawa potential, 汤川势ZZeeman effect, 塞曼效应。
量子力学中的特殊函数求解
量子力学中的特殊函数求解在量子力学中，特殊函数扮演着重要的角色。
这些函数具有独特的性质和特殊的求解方法，对于描述微观世界中的物理现象起到至关重要的作用。
本文将介绍几个在量子力学中常见的特殊函数及其求解方法。
一、拉盖尔多项式(Laguerre Polynomials)拉盖尔多项式是具有广泛应用的一类特殊函数，其定义为：\[L_n(x) = e^x \left(\frac{d}{dx}\right)^n (e^{-x} x^n)\]这里，n为非负整数，x为实数。
拉盖尔多项式在量子力学中常用于描述粒子在势场中的行为，特别是在氢原子的能级结构中起到重要作用。
拉盖尔多项式的求解方法有多种，其中一种常用的方法是借助微分方程的性质。
通过拉盖尔方程的微分形式以及递推关系，可以得到拉盖尔多项式的具体表达式。
由于拉盖尔多项式的求解相对复杂，算法和数值方法在实际应用中也被广泛采用。
二、勒让德多项式(Legendre Polynomials)勒让德多项式是一类重要的特殊函数，广泛应用于球面坐标系下的问题求解和角动量相关的物理量计算。
勒让德多项式的定义如下：\[P_\ell(x) = \frac{1}{2^\ell \ell!} \frac{d^\ell}{dx^\ell} (x^2 - 1)^\ell\]这里，\(\ell\)为非负整数，x为实数。
勒让德多项式可以通过递推关系或者其微分方程的求解来得到。
勒让德多项式的正交性质使其在量子力学中的角动量计算和球对称问题的求解中非常有用。
三、厄密多项式(Hermite Polynomials)厄密多项式是量子力学中应用最广泛的特殊函数之一，常用于描述谐振子的能量本征态和波函数。
厄密多项式的定义如下：\[H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2}\]这里，n为非负整数，x为实数。
厄密多项式的求解方法可以通过其递推关系或者其微分方程的性质来实现。
激光原理(英文)-文档资料
Special cases
Types of resonators
10
Classical mechanics analog
Fabry-Perot boundary conditions
Multi-mode laser
Multiple resonant frequencies
4
Single longitudinal mode lasers
• Insert etalon into cavity • Use low reflectivity etalon
Laser basics
Optics, Eugene Hecht, Chpt. 13; Optical resonator tutorial:
/tutorials/lasers/leot/
1
Laser oscillation
Laser is oscillator • Like servo with positive feedback • Greater than unity gain
Multi-longitudinal Multi-transverse&long. Single mode
7
Gaussian beams
• Zero order mode is Gaussian
• •
Intensity profile:
I
I e2r2/w2 0
beam waist: w0
2
w w0
9
Resonator options
• Best known -- planar, concentric, confocal • Confocal unique
– mirror alignment not critical – position is critical – transverse mode frequencies identical
高分子材料工程专业英语第二版课文翻译(基本全了
A 高分子化学和高分子物理UNIT 1 What are Polymer?第一单元什么是高聚物？What are polymers? For one thing, they are complex and giant molecules and are different from low molecular weight compounds like, say, common salt. To contrast the difference, the molecular weight of common salt is only 58.5, while that of a polymer can be as high as several hundred thousand, even more than thousand thousands. These big molecules or ‘macro-molecules’ are made up of much smaller molecules, can be of one or more chemical compounds. To illustrate, imagine that a set of rings has the same size and is made of the same material. When these things are interlinked, the chain formed can be considered as representing a polymer from molecules of the same compound. Alternatively, individual rings could be of different sizes and materials, and interlinked to represent a polymer from molecules of different compounds.什么是高聚物？首先，他们是合成物和大分子，而且不同于低分子化合物，譬如说普通的盐。
开启片剂完整性的窗户(中英文对照)
开启片剂完整性的窗户日本东芝公司，剑桥大学摘要：由日本东芝公司和剑桥大学合作成立的公司向《医药技术》解释了FDA支持的技术如何在不损坏片剂的情况下测定其完整性。
太赫脉冲成像的一个应用是检查肠溶制剂的完整性，以确保它们在到达肠溶之前不会溶解。
关键词：片剂完整性，太赫脉冲成像。
能够检测片剂的结构完整性和化学成分而无需将它们打碎的一种技术，已经通过了概念验证阶段，正在进行法规申请。
由英国私募Teraview公司研发并且以太赫光（介于无线电波和光波之间）为基础。
该成像技术为配方研发和质量控制中的湿溶出试验提供了一个更好的选择。
该技术还可以缩短新产品的研发时间，并且根据厂商的情况，随时间推移甚至可能发展成为一个用于制药生产线的实时片剂检测系统。
TPI技术通过发射太赫射线绘制出片剂和涂层厚度的三维差异图谱，在有结构或化学变化时太赫射线被反射回。
反射脉冲的时间延迟累加成该片剂的三维图像。
该系统使用太赫发射极，采用一个机器臂捡起片剂并且使其通过太赫光束，用一个扫描仪收集反射光并且建成三维图像(见图)。
技术研发太赫技术发源于二十世纪九十年代中期13本东芝公司位于英国的东芝欧洲研究中心，该中心与剑桥大学的物理学系有着密切的联系。
日本东芝公司当时正在研究新一代的半导体，研究的副产品是发现了这些半导体实际上是太赫光非常好的发射源和检测器。
二十世纪九十年代后期，日本东芝公司授权研究小组寻求该技术可能的应用，包括成像和化学传感光谱学，并与葛兰素史克和辉瑞以及其它公司建立了关系，以探讨其在制药业的应用。
虽然早期的结果表明该技术有前景，但日本东芝公司却不愿深入研究下去，原因是此应用与日本东芝公司在消费电子行业的任何业务兴趣都没有交叉。
这一决定的结果是研究中心的首席执行官DonArnone和剑桥桥大学物理学系的教授Michael Pepper先生于2001年成立了Teraview公司一作为研究中心的子公司。
TPI imaga 2000是第一个商品化太赫成像系统，该系统经优化用于成品片剂及其核心完整性和性能的无破坏检测。
(生物科技行业)密歇根大学生物系实验室的常用试剂配方
Table of ContentsLB Medium (1)NZ Medium (2)SM Buffer (3)SET Buffer (4)6X Prehyb Soln (5)10 X TBE (6)10 X TAE (7)20 X SSC (8)1% SDS, 0.2 M NaOH (9)14% PEG (8000), 2M NaCl, 10 mM MgSO4 (10)20% SDS (11)1.0 M Tris, pH 8.0, 1.5 M NaCl (12)10mM Tris-HCl, pH 7.5, 10mM MgSO4 (13)10 mM Tris, 50 mM EDTA, pH 7.5 (14)10 mM Tris-HCl, 1 mM EDTA, pH 7.5 (15)3 M Sodium Acetate, pH 4.8 (16)Electrophoresis dye (17)Labelling Stop dye (18)Sequencing gel dye (19)5% Acrylamide (20)6% Acrylamide in TBE, 50% Urea (21)40% Acrylamide (22)LB Medium (1 Liter)10g Bacto-tryptone5g Bacto-yeast extract10g NaClFor forty plates add 1% agar--1g. Autoclave media. When cool, add ampicillin and pour plates. For 1L of media, add 1.8 mL amp.NZ Medium (500 mL)5 g Bacto-tryptone2.5 g Bacto-yeast extract2.5 g NaCl1.25 g MgSO4For 20 plates add 1.2% agar--6g. Autoclave and pour plates at 50o CSM Buffer (1L)5.8 g NaCl1.2 g MgSo450 mL 1M Tris-HCl, pH 7.50.1 g Gelatin (doesn't dissolve)AutoclaveUsed for phage dilution and storage.SET Buffer50 mM Tris-HCl, pH 8.0, 50 mM EDTA, 20% w/v Sucroseto make 200mL:40 g Sucrose10 mL of 1M Tris20 mL of 0.5 M EDTA, disodium saltbring to 200 mL with H206X Prehybridization Solutionto make 500 mL300 mL ddH20150 mL 20X SSC50 mL 50X Denhardt's solution1 mL 0.5 M EDTA (disodium salt)2.5 mL 20% SDS6X refers to the concentration of SSC10X TBE Buffer (for polyacrylamide gels) to make one liter:60.75 g Tris3.7 g EDTA (tetrasodium salt)30 g Boric acid10X TAE Buffer (For agarose gels)to make one liter:48.20 g Tris6.75 g NaAce3.75 g EDTA (disodium salt)Adjust pH to 7.6 with acetic acid. (Approx. 20 mL)20X SSCto make one liter:175.3 g NaCl88.2 g NaCitrateadd water to bring volume to one liter.adjust to pH 7.0 with HCl.1% SDS, 0.2 M NaOHto make 100 mL:93 mL ddH205 mL 20% SDS2 mL 10 M NaOH14% PEG (8000), 2M NaCl, 10 mM MgSO4 to make one liter:140 g PEG117 g NaCl2.46 g MgSO4For use in phage DNA preparation.20% SDSto male 250 mL:50 g of SDS in a beakerAdd stir bar and H20 last.This solution will have to be heated for the SDS to dissolve.1.0 M Tris, pH 8.0, 1.5 M NaClto make one liter:121.1 g Trizma87.6 g NaClin a volume of water less than 1L. Adjust pH with HCl, then bring to 1L with H2010 mM Tris-HCl, pH 7.5, 10 mM MgSO4to make one liter:10 mL 1 M Tris-HCl2.46 g MgSO4for use in phage DNA preparation10 mM Tris, 50 mM EDTA, pH 7.5to make 200 mL:2 mL 1 M Tris20 mL 0.5 M EDTA (tetrasodium salt)178 mL ddH20adjust pH with HCl.10 mM Tris-HCl, 1 mM EDTA, pH 7.5to make 200 mL:2.0 mL 1 M Tris-HCl, pH 7.50.4 mL 0.5 M EDTA197.6 mL ddH203 M Sodium Acetate, pH 4.8to make one liter:408.1 g NaAce (trihydrate; gets cold in soln)about 700 mL H20adjust pH with glacial acetic acid (takes a lot)Measure tru pH by dilution with water; range will be between 4.8 and 5.5.Electrophoresis Dyeto make 4 mL:3 mL 50 mM EDTA, 10 mM Tris-HCl, pH 8.01 mL glycerol20 μL BPB10 μL Xylene cyanolStop dye for labelled probe1 mL 50 mM EDTA, 10 mM Tris, pH 7.5-8.5about 200 μl glyceroladd a few grains of blue dextran (8000)Sequencing gel dyefor approx 1 mL:1 mL formamide10 μL xylene cyanol10 μl BPB3 μL 10 M NaOH5% acrylamideto make 200 mL:20 mL 10X TBE25 mL 40% acrylamide155 mL H206% Acrylamide in TBE, 50% Ureato make 500 mL:50 mL 10X TBE75 mL 40% acrylamide250 g Ureabring to 500 mL with H2O40% Acrylamide (38:2 acrylamide:bis acrylamide) to make 200 mL:76 g acrylamide4 g bis acrylamidebring to 200 mL with H2O。
线性代数课程专业词汇表
线性代数课程专业词汇表英文单词或词组中文翻译书中出现页码Linear equation 线性方程 1Linear system(s) 线性方程组 1Consistent 有解 2Inconsistent 无解 2Solution set of linear system 线性方程组的解集合 2Equivalent systems 等价的线性方程组 3Row operations 行变换 5Strict triangularform 严格三角形式 5Back substitution 回代法 6Equivalent systems 等价的线性方程组 6Coefficientmatrix 系数矩阵 7Coefficientmatrix 系数矩阵 7Augmented matrix 增广矩阵 8Pivot 主元 8Free variables 自由未知量 14Lead variables 前变量 14Gaussian elimination 高斯消元法 15Overdetermined linear system 方程个数超过未知数个数的方程组15 Row echelon form 行阶梯形 15Underdetermined linear system 方程个数低于未知数个数的方程组17 Gauss-Jordan reduction 高斯 - 若当归纳法 18Reduced row echelon form 减少的行阶梯形 18Homogeneous linear system 齐次线性方程组 22Homogeneous system 齐次线性方程组 22nontrivial solution 非零解 22Trivialsolution 平凡解，全零解 22Matrix algebra 矩阵代数 30Scalars 常数 30Column vector(s) 列向量 31Euclidean n-space 欧几里得空间 31Row vector(s) 行向量 31Vector(s ）向量 31Addition of matrices 矩阵加法 32Addition of matrices 矩阵加法 32Equality of matrices 矩阵相等 32Scalar multiplication for matrices 矩阵的数乘 32Scalar multiplication of matrices 矩阵的数乘 32Zero matrix 零矩阵 33Scalar product 内积 34Linear combination 线性组合 36Consistency Theorem 解的存在性定理37Multiplication of matrices 矩阵乘法 38Identity matrix 单位矩阵 47Inverse matrix 逆矩阵 48Invertible matrix 可逆矩阵 48Nonsingular matrix 非奇异矩阵 48Singular matrix 奇异矩阵 49Transpose of a matrix 矩阵的转置 49Transpose of matrix 矩阵的转置 49Symmetric matrix 对称矩阵 51Symmetric matrix 对称矩阵 51Adjacency matrix 邻接矩阵 52Graph(s) 图 52Angle between vectors 向量的夹角 56Markov chain(s ） Markov 链 57Elementary matrix 初等矩阵 62Row equivalent 行等价 64Row equivalent matrices 行等价矩阵 64Diagonal matrix 对角矩阵 67Lower triangular 下三角 67Triangular factorization 三角分解 67Triangular matrix 三角形矩阵 67Triangular matrix 三角形矩阵 67Upper triangular 上三角 67Upper triangular matrix 上三角矩阵 67LU factorization LU 分解 68Matrixfactorizations 矩阵分解 68Partitioned matrices 分块矩阵 72Vandermonde matrix 范德蒙矩阵 72Block multiplication 分块乘法 74Inner product 内积 78Determinant(s) 行列式 90Cofactor 代数余子式 93Minor 余子式 93Cofactor expansion 代数余子式展开 94Determinant of matrix 矩阵的行列式 95Skew symmetric 反对称 105Adjoint of a matrix 伴随矩阵 106Cramer’s rule 克莱姆法则 107Cryptography 密码学 108Addition of vectors 向量的加法 119Closure properties 封闭性 119Vector space 向量空间 119Zero vector 零向量 119C[a,b] 区间 [a,b] 上的连续函数 120Isomorphism between vector spaces 向量空间的同构123Subspace(s) 子空间 123Zero subsapce 零空间 125Nullspace 零化空间 127Nullspace of matrix 矩阵的零化空间 127Span 张成 128Spanning set 生成集 129Linearly dependent 线性相关 136Linearly independent 线性无关 136Basis 基 145Dimension 维数 147Finite dimensional 有限维 147Infinite dimensional 无限维 147Standard basis 标准基 150change of basis 基的变换 151Coordinate vector 坐标向量 152Transitionmatrix 过渡矩阵 155Coordinates 坐标 157Column space 列空间 162Column space of matrix矩阵的列空间 162Rank of a matrix 矩阵的秩 162Rank of matrix 矩阵的秩 162Row space 行空间 162Row space of matrix 矩阵的行空间 162Nullity 零化度 164Rank-Nullity Theorem Rank-Nullity 定理 164 Leftinverse 左可逆 170Right inverse 右逆 170Full rank 满秩 171Linear transformation(s) 线性变换 175 Linear operator 线性算子 176Image 象 181Kernel 核 181Contraction 收缩 192Dilation 扩张 192Similarity 相似性 199Similar matrices 相似矩阵 202Trace 迹 206Angle between vectors 向量的夹角 211 Euclidean length 欧几里得长度 211Distance in 2-space 2 维空间的距离 212 Cauchy-Schwarz inequality 柯西 - 施瓦兹不等式 213 Orthogonality 正交性 213Scalar projection 数量投影 214equation of plane 平面方程 215Nonmal vector 正规向量 215Angle between vectors 向量的夹角 216Pythagorean Law Pythagorean 定理 216Correlations相关 219Correlation matrix 相关矩阵 221Covariance 协方差 222Covariance matrix 协方差矩阵 222Factor analysis 因子分析 222Fundamental subspaces 基本子空间 227Range of a matrix 矩阵的值域 227Direct sum 直和 229Least squares problem(s) 最小二乘法问题 234Projection onto column space 列空间上的投射 236Normal equations 正规方程 237Inner product space 内积空间 245Length in inner product spaces 内积空间中的长度 246 Orthogonal set(s) 正交集合 255Orthonormal set(s) 标准正交集 255orthonormal basis 标准正交基 256Orthonormal basis 标准正交基 256Orthogonal matrices 正交矩阵 258Orthogonal matrix 正交矩阵 258Approximation of functions 函数的逼近 264Fouriercoefficients 傅里叶系数 266Fourier matrix 傅里叶矩阵 269Gram-Schmidt process Gram-Schmidt 过程 274Dimension Theorem 维数定理 283Orthogonal polynomials 正交多项式 283Hermite polynomials Hermite 多项式 287Jacobi polynomials Jacobi 多项式 287Lagrange’s interpolating formula Lagrange 插值公式 288 Gaussian quadrature 高斯求积 289Characteristic value(s) 特征值 301Characteristic vector 特征向量 301Eigenvalue 特征值 301Eigenvector 特征向量 301Characteristic equation 特征方程 302Characteristic polynomial 特征多项式 302Eigenspace 特征空间 302Nilpotent 幂零的 311Companion matrix 友矩阵 313Linear differential equations 线性微分方程 313Initial valueproblems 初值问题 314Diagonalizable matrix 可对角化的矩阵 326Distance in n-space n 维空间的距离 332Complex matrix 复矩阵 346Hermite matrix Hermite矩阵 346Unitary matrix 酉矩阵 347Unitary matrix 酉矩阵 347Normal matrices 正规矩阵 351Singular values 奇异值 356Conic sections 二次曲线部分 371Quadratic equation in n variables n 个变量的二次方程 376Quadratic form in n variables n 个变量的二次型 376 Definite quadraticform 定二次型 378Indefinite quadratic form 不定二次型 378Negative definite matrix 负定矩阵 378Negative definite quadratic form 负定二次型 378Negative semidefinite matrix 半负定矩阵 378Negative semidefinite quadratic form 半负定二次型 378 Positive definitematrix 正定矩阵 378Positive definite quadratic form 正定二次型 378Positive semidefinite matrix 半正定矩阵 378Positive semidefinite quadratic form 半正定二次型 378 Local maximum 极大值 382Local minimum 极小值 382Positive definitematrix 正定矩阵 384Leading principal submatrix 顺序主子矩阵 385Nonnegative matrix 非负矩阵 392Nonnegative vector 非负向量 392Positive matrix 正矩阵 392Positive matrix 正矩阵 392Reducible matrix 可约矩阵 394Frobenius theorem Frobenius 定理 395Absolute error 绝对误差 411Relative error 相对误差 411Back substitution 回代法 419QR factorization QR 分解 448。
计算方法-插值法(二)
x0 f (x0)
x1 f (x1) x2 f (x2)
P0,1(x) P1,2(x)
P0,1,2 ( x)
(x x0) (x x0)
x3 f (x3) P2,3(x) P1,2,3(x) P0,1,2,3(x)
(x x0)
x4 f (x4)
P3,4(x) P2,3,4(x)
P1,2,3,4 ( x)
S1( x), x [ x0 ,x2 ]
Sn ( x), x [ xn1,xn ]
(1) S(x)在每个小区间[xk , xk1]上都是三次多项式 (2) S(x)满足 S(x j ) y j , j 0,1,, n (3) S(x)都在区间[a,b]上连续，导数值未知
高次插值的病态性质(德国Runge 20世纪初)
设函数
f
(x)
1 1 x2
,
x [5,5]，将[5,5]n等分取n
1个节点xi
5
ih,
h 10 ，i 0,1,,n，试就n 2,4,6,8,10作f (x)的n次Lagrange插值多项式。
n
解:
yi
f
(xi )
1 1 xi2
作n次Lagrange插值多项式
注：同样是三次多项式,三次样条插值与分段 Hermite 插值的根本区 别在于S(x)自身光滑,不需要知道 f 的导数值（除了在2个端点可能需 要）；而Hermite插值依赖于f 在所有插值点的导数值。
15
三次样条插值数学定义：
a ≤ x0, x1, …, xn ≤ b为区间[a, b]的一个分割，如果
Ln (x)
n j0
y jl j
n
1
j0
1
数值分析学习课件
所以有
π π 1 3 π = 0.258768616 H3 ( ) = + − 12 48 4 96
sin （12 ） =0.258819045
方法2 方法2： 直接用待定系数法求解： 直接用待定系数法求解：
π
由 f (0) = 0，f ( ) =
6
π
1 2
，可有 y = L1 ( x ) =
3
3
H 3 ( x) =
2
f ( 4 ) (ξ x ) R3 ( x ) = f ( x ) − P3 ( x ) = K ( x )( x − x0 )( x − x1 ) ( x − x 2 ), K ( x ) = 4!
可解。 又: h1’(x1) = 1 ⇒ C1 可解。
一般地， 一般地，已知 x0 , …, xn 处有 y0 , …, yn 和 y0’ , …, yn’ ，求 H2n+1(x) 满足 H2n+1(xi) = yi ， H’2n+1(xi) = yi’。 。 解：设 H2n+1( x ) = Σ yi hi ( x ) + Σ yi’ h i ( x )
∧
(x − xj ) ( xi − x j )
一 可解A 由余下条件 hi(xi) = 1 和 hi’(xi) = 0 可解 i 和 Bi ⇒
hi ( x ) = [1 − 2l i′( xi )( x − xi )] l i2 ( x )
∧ ∧
hi (x) 有根 x0 , …, xn, 除了xi 外都是 重根 ⇒ hi( x) = Ci ( x − xi ) li2(x) 外都是2重根
一致
易证： 记 h = max | xi +1 − xi | ，易证：当 h → 0 时，P1h ( x ) → f ( x ) 失去了原函数的光滑性。 失去了原函数的光滑性。 缺点：I(x)连续，但不光滑，精度较低, 缺点：I(x)连续，但不光滑，精度较低,仅在 连续
厄米多项式推导 -回复
厄米多项式（Hermite Polynomials）是物理学和数学中的一类正交多项式，尤其是在量子力学和统计学领域起到重要作用。
通常用H_n(x)表示厄米多项式，其满足以下递推关系：H_0(x) = 1H_1(x) = 2xH_n(x) = 2x H_(n-1)(x) - 2(n-1) H_(n-2)(x)厄米多项式的推导属于数学分析领域，有多种方法可以推导出这些多项式。
一种常用的方法是通过生成函数的方法。
假设厄米多项式的生成函数是：G(x, t) = ∑[H_n(x) * t^n / n!], 其中n 从0 到+∞我们可以利用厄米多项式满足的微分方程来求解生成函数G(x, t)。
厄米多项式满足以下的微分方程：H'_n(x) = 2n * H_(n-1)(x)代入生成函数，得到：G'(x, t) = ∑[H'_n(x) * t^n / n!]根据厄米多项式满足的微分方程，可以写成：G'(x, t) = ∑[2n H_(n-1)(x) t^n / n!]考虑到n = 0 的项是0（因为H_-1(x) 项不存在），所以我们可以将求和从n=1 开始：G'(x, t) = ∑[2n H_(n-1)(x) t^n / n!], 其中n 从1 到+∞可以将t^n/n! 替换成t^(n-1)/(n-1)!, 然后令m = n-1, 就可以将求和从m=0 开始：G'(x, t) = 2 ∑[(m+1) H_m(x) * t^m / m!], 其中m 从0 到+∞这个式子现在和生成函数G(x, t) 的形式一样，只是多了系数2和(m+1)：G'(x, t) = 2 (t d/dt) * G(x, t)将生成函数表示成G(x, t) = exp(2xt - t^2)，然后用标准的解析方法求解该微分方程。
最后可以得到：H_n(x) 是exp(-x^2) (d^n/dx^n) exp(x^2) 的前n项多项式部分(n阶多项式)通过这种方法，可以推导出任意阶的厄米多项式。
3.5 线性谐振子
那么,微分方程变为 超几何微分函数.
d 2 1 d κ 1 y 2 + ( y) + ( ) = 0 dy 2 dy 2 4
(9)
应用级数法,可以把φ展开成y的函数.该级数必须只含有限项, 才能在y→±∞时使φ(y)有限;其一般解为.
1/ 2 1 1 3 κ 1 ( y ) = A1F1 ( a; ; y ) + By1 F1 (a + ; ; y ), a = ( ) 2 2 2 2 4
∧ 1 (ξ + ) = a ξ 2
∧+
(48)
从这两个关系, 我们可以估算ψn的相邻函数ψn-1和ψn+1. 为了简便 起见,我们做如下替代
1 (ξ ) = a ξ 2
∧+
(49)
(48)式变为
aψ n = nψ n 1 ,
κ
(14)
3 E n = ω ( 2 n + ) 2
(15)
E 应用前面我们做的替换 κ = ω
得到能量值
1 En = (2n + )ω 2 3 1 En = (2n + )ω = [( 2n + 1) + )ω 2 2
结合这两个结果, 我们得到分离的能谱
1 En = ( n + )ω , n = 0,1,2,... 2
H n ( λ x ),
1 E n = ( n + ) ω 2
(18)
H n (ξ ) = ( 1) e
n
ξ
2
d ξ 2 e n dξ
n
(19) (20)
dH n = 2nH n 1 (ξ ) dξ
H n +1 (ξ ) 2ξH (ξ ) + 2nH n 1 (ξ ) = 0
薛定谔方程及提出背景
薛定谔方程在一维空间里，一个单独粒子运动于位势中的含时薛定谔方程为；(1)其中，是质量，是位置，是相依于时间的波函数，是约化普朗克常数，是位势。
类似地，在三维空间里，一个单独粒子运动于位势中的含时薛定谔方程为。
(2)假若，系统内有个粒子，则波函数是定义于-位形空间，所有可能的粒子位置空间。
用方程表达，。
其中，波函数的第个参数是第个粒子的位置。
所以，第个粒子的位置是。
不含时薛定谔方程不含时薛定谔方程不相依于时间，又称为本征能量薛定谔方程，或定态薛定谔方程。
顾名思义，本征能量薛定谔方程，可以用来计算粒子的本征能量与其它相关的量子性质。
应用分离变量法，猜想的函数形式为；其中，是分离常数，是对应于的函数．稍回儿，我们会察觉就是能量．代入这猜想解，经过一番运算，含时薛定谔方程 (1) 会变为不含时薛定谔方程：。
类似地，方程 (2) 变为。
历史背景与发展爱因斯坦诠释普朗克的量子为光子，光波的粒子；也就是说，光波具有粒子的性质，一种很奇奥的波粒二象性。
他建议光子的能量与频率成正比。
在相对论里，能量与动量之间的关系跟频率与波数之间的关系相同，所以，连带地，光子的动量与波数成正比。
1924年，路易·德布罗意提出一个惊人的假设，每一种粒子都具有波粒二象性。
电子也有这种性质。
电子是一种波动，是电子波。
电子的能量与动量决定了它的物质波的频率与波数。
1927年，克林顿·戴维孙和雷斯特·革末将缓慢移动的电子射击于镍晶体标靶。
然后，测量反射的强度，侦测结果与X射线根据布拉格定律 (Bragg's law) 计算的衍射图案相同。
戴维森-革末实验彻底的证明了德布罗意假说。
薛定谔夜以继日地思考这些先进理论，既然粒子具有波粒二象性，应该会有一个反应这特性的波动方程，能够正确地描述粒子的量子行为。
于是，薛定谔试着寻找一个波动方程。
哈密顿先前的研究引导著薛定谔的思路，在牛顿力学与光学之间，有一种类比，隐蔽地暗藏于一个察觉里。
电镜样品制备方法中英文(常用
电镜样品制备方法中英文（常用倒掉固定液，用0、IM, pII7、0的磷酸缓冲液漂洗样品三次，每次15min；用1%的餓酸溶液固定样品l-2h；倒掉固定液，用0、IM, pII7、0的磷酸缓冲液漂洗样品三次，每次15min；用梯度浓度（包括30%,50%, 70%, 80%, 90%和95%五种浓度）的乙醇溶液对样品进行脱水处理，每种浓度处理15min,再用100% 的乙醇处理两次，每次20 mine用乙醇与醋酸异戊酯的混合液（V/V二1/1）处理样品30min, 再用纯醋酸异戊酯处理样品l-2h o临界点干燥。
镀膜，观察。
处理好的样品在Hitachi TM-1000型扫描电镜中观察。
1 Double fixation： The specimen was first fixed with2、 5% glutaraldehyde in phosphate buffer （pII7、 0）for more than4hours； washed three times in the phosphate buffer； then postfixed withl% 0s04 in phosphate buffer （pII7、0）forlhour and washed three times in the phosphate buffer、 2、 Dehydration： The specimen was first dehydrated by a graded series of ethanol（30%,50%,70%,80%,90%,95% and100%）for about 15 to20 minutes at each step, transferred to the mixture of alcohol and iso-amyl acetate （v：v=l:1）for about30 minutes, then transferred to pure isoamyl acetate for aboutlhourIn the end, the specimen was dehydrated in Ilitachi Model HCP-2 critical point dryerwith liquid C02、 3、 Coating and observation： The dehydrated specimen was coated with gold-palladium and observed in Philips Model TM-1000 SEM、 No、 2 Negative staining of bacteriumThe bacterium suspension was stained byl to2%solution of phosphotungstic acid （PTA）in a pH range of6、 5 to7、 0 forl5 to30 seconds. Then, the bacterium was observed in TEM of Model JEM1230、No、3透射电镜样品制备方法样品在2、5%的戊二醛溶液中4工固定过夜，然后按下列步骤处理样品：倒掉固定液，用0、IM, pII7、0的磷酸缓冲液漂洗样品三次，每次15min；用1%的餓酸溶液固定样品l-2h；倒掉固定液，用0、IM, pH7、0的磷酸缓冲液漂洗样品三次，每次15min；用梯度浓度（包括30%,50%, 70%, 80%, 90%和95%五种浓度）的乙醇溶液对样品进行脱水处理，每种浓度处理15min,再用100% 的乙醇处理一次，每次20min；最后过度到纯丙酮处理20min o 用包埋剂与丙酮的混合液（V/V二1/1）处理样品lh；用包埋剂与丙酮的混合液（V/V=3/l）处理样品3h；纯包埋剂处理样品过夜；将经过渗透处理的样品包埋起来，7(rc加热过夜，即得到包埋好的样品。
邻乙基苯胺
邻乙基苯胺- 性质
浅黄色液体，露置于日光下逐渐变深。
熔点- 43℃。
沸点210～216℃，112～116℃ (3.333kPa)，相对密度0. 982 (22℃)。
折射率1.5584 (22℃)。
闪点91℃。
溶于水、乙醇。
邻乙基苯胺- 制法
由邻硝基乙苯还原制得。
1.铁粉还原法利用该方法每生产It产品原料消耗为：邻硝基乙苯(100%)1. 66t，氯化铵（工
业品）350kg，烧碱 160kg，铁粉（工业品）1.65t，精盐（工业品）1. 4t。
2.硫化钠还原法硫化钠和邻硝基乙苯在120～130℃下反应，将反应混合物分离后得产品，
产品收率95%。
邻乙基苯胺- 用途
本品是合成苗前选择性除草剂杀草胺、灭草胺及杀虫剂乙基杀虫脒的原料；也用于制造硫化耐晒蓝染料等。
邻乙基苯胺- 安全性
∙本品LD50为1260mg/kg。
对眼睛有强烈刺激作用。
对黏膜、上呼吸道有刺激性。
吸入体内可引起高铁血红蛋白血症，出现紫绀。
防护措施见2，6-_甲基苯胺。
∙包装采用小开口钢桶，螺纹口玻璃瓶、铁盖压口玻璃瓶、塑料瓶或金属桶（罐）外木板箱。
贮存于阴凉、通风仓间内。
远离火种、热源。
贮存期不可太长，规定三个月轮换一次。
保持容器密封。
应与氧化剂、酸类、食品化学品分开存放。
搬运时要轻装轻卸，防止包装及容器损坏。
英文特生物技术（北京）有限公司-植物质膜蛋白提取试剂盒说明书
Minute TM 植物质膜蛋白提取试剂盒目录号：SM-005-P描述：植物膜蛋白占植物细胞总蛋白的很小一部分，但是在植物生理学中起着非常重要的作用。
传统的植物膜组分分离纯化方法是蔗糖密度梯度离心法和双液相法。
这些方法虽然比较有效，但是需要超高速离心和大量的起始原材料，操作过十分繁琐和费时。
为了克服植物膜组分提取中的缺点，我们特别开发了此款植物膜组分提取试剂盒。
植物组织首先通过缓冲液A中致敏，匀浆，然后通过一个特殊的离心管柱，在此过程中匀浆的组织通过柱子特有的Z字形通路后细胞膜被切割成大小相等的碎片，后续无需超高速离心，通过差速离心法和密度梯度离心法将天然的质膜组分从未破裂的细胞，细胞核，细胞浆和细胞器的混合物中分离出来。
在每次实验中仅需使用相同量的起始材料，离心力和离心时间，即可高度富集膜组分，并保证一致性良好。
整个操作过程大约1小时可以完成。
应用：试剂盒用于快速从植物组织中分离天然膜组分，可应用于SDS-PAGE，immunoblottings，ELISA，IP，膜蛋白质结构分析，2-D，酶活性测定及其他应用。
试剂盒组份（50次）：1. 25ml Buffer A2. 10ml Buffer B3. 50个离心管柱4. 50个收集管5. 2根塑料研磨棒6. 组织分离粉储存：Buffer A和Buffer B 需在-20℃储存所需附加材料1XPBS涡旋震荡仪台式离心机重要产品信息1.仔细阅读整个操作说明。
将缓冲液A和缓冲液B完全解冻后摇匀，放置于冰上。
将离心管柱和接收管套管放置于冰上预冷。
2.离心机请调整成RCF/Xg模式，按照离心力设置离心机，所有离心步骤都需要在4℃室温下或者低温离心机中进行。
3.研究蛋白磷酸化，磷酸酶抑制剂应在使用前加入缓冲液A中。
蛋白酶抑制剂可以选择添加或不添加，如添加在使用前加入缓冲液A中（请按照蛋白酶或磷酸酶抑制剂母液比例，例如母液是100x，添加时按照1：100添加，1ml缓冲液A添加10ul抑制剂）。
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Technical Note No.21*
Options,Futures,and Other Derivatives,Ninth Edition
John Hull
Hermite Polynomials and Their Use for Integration
As explained in Section25.10,the Gaussian copula model requires functions to be integrated over a normal distribution between−∞and+∞.Gaussian quadrature ap-proximates the integral as
∞−∞
1
√
2π
e−F2/2g(F)dF≈
M
k=1
w k g(F k)(1)
The approximation gets better as M increases.It has the property that it is exact when g(F)is a polynomial of order M.
The determination the w k and F k involves Hermite polynomials.If you want to avoid getting into the details of this,values of w k and F k for different values of M can be downloaded from a spread sheet on the author’s web site.
Thefirst few Hermite polynomials are
H0(x)=1
H1(x)=2x
H2(x)=4x2−2
H3(x)=8x3−12x
H4(x)=16x4−48x2+12
A recurrence relationship for calculating higher order polynomials is
H n+1(x)=2xH n(x)−2nH n−1(x)
and an equation for the derivative with respect to x is
H n(x)=2nH n−1(x)
Define x k(1≤k≤n)as the n roots of H n(x)(that is,the n values of x for which H n(x)=0)and
w∗k=
2n−1n!
√
πn[H n−1(x k)]
A key result is
∞−∞f(x)dx≈
n
k=1
w∗k e x2k f(x k)(2)
*c Copyright John Hull.All Rights Reserved.This note may be reproduced for use in conjunction with Options,Futures,and Other Derivatives by John C.Hull.
1
Setting x=F/√
2and
f(x)=
1
√
π
e−x2g(
√
2x)
equation(2)gives
∞−∞
1
√
2π
e−F2/2g(F)dF≈
n
k=1
1
π
w∗k g(F k)
or alternatively
∞−∞
1
√
2π
e−F2/2g(F)dy≈
n
k=1
w k g(F k)
where
w k=w∗
k
√
π
F k=
√
2x k
This is the result in equation(1),with n=M.
This leaves the problem of calculating the n roots of a Hermite polynomial.A program for doing this is‘gauher’in“Numerical Recipes for C:The Art of Scientific Computing”by Press,Flanery,Teukolsky,and Vetterling,Cambridge University Press.
2。