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This 16 page Class Notes was uploaded by Citlalli Sauer on Tuesday October 20, 2015. The Class Notes belongs to PHYS410 at San Diego State University taught by Staff in Fall. Since its upload, it has received 22 views. For similar materials see /class/225321/phys410-san-diego-state-university in Physics 2 at San Diego State University.
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Date Created: 10/20/15
Lecture 4 Outline lt position gt Quiz 1 Expectation values 132 14 15 What is an expectation value 15 Evolution of expectation values dltxgt dt 15 Wavefunction Position Wavefunction x 15 probability density IJU t2 Norrnalisation Ilx t2 dx l Expectation value of position oo Example problem Gaussian Integration 93gt a IJ9IS2 dx Three identical wavefunctions 93gt fjofx was dx 0 The expectation value is the average of repeated measurements on an ensemble of systems 0 Step i Prepare 3 identical wavefunctions 0 Step ii and d0 3 independent measurements i 3 identical systems before measurement 9 l 2m ii 3 indep endent measurements 9 l 2m Ensembles of identical wavefunctions i many identical systems before measurement 9 2m ll Hillll H HMHH ii many independent measurements PM r P v o The expectation value is the average of repeated measurements on an ensemble of systems 93gt g 9 IJxt2 dx Evolution of part 13 c How does an expectation value Change with time dltxgt 8 2 and from last lecture we showed that 8 iii 8 8i 8amp1quot 1 2 11 315 I Saw 2m 8x 3x 3x thus we need to work out using integration by parts see aside slide Integration by parts an aside 0 Taking the product rule of two functions f as gx d d9 a ow fdx dang re arrange and integrate both sides b dg bdf bd Af dx a gdxCLfgdx bd a gdxfgb a o transfers the derivative from one function to the other Evolution of part 23 repeating from before and using integration by parts g dxx f NOW using lim 9 gt oo x t gt 0 implies that O and also we know that st3x 1 thus we get dltxgt iii 8 6111 IJ IJ dt 2m 8x 3x dx Evolution of part 33 dltxgt M 953 8 W IJ dx 3x 3x o and using another integration by parts we get d 3975 3Q 93gt 2 dx 3x dt m 0 so to calculate the velocity of the expectation value 0 but for our purposes we need momentum p mv so p ih dx Lecture 4 Outline lt momentum et a1 a Quiz 1 0 Evolution of expectation values dltxgt dt 15 o Operators and expectation values p 15 Operators in general 15 Quantum to Classical via Ehrenfest s theorem 15 Evolution of part 12 a How does an expectation value change with time 0 ie we need to work out note indep variables 8 0 8t dlta gt 8 2 271 8 8111 8111 II Il 111 d dt x8t x m dx 2mx8a 81 81 x 0 using integration by parts to transfer derivative M M We 0 Now we know that 85138513 1 and also assume that 00 0 again lim 1 gt 00 11152 gt 0 so as 0 OO dlta gt 2h 8111 dt 81 3911 d Evolution of part 22 dltxgt dt 2m 8 11 8 1 another integration by parts on either integrand term dlta gt iii 8amp1 iii 8amp1 2x1 d d dt 2m 81 x m 81 x at this stage we postulate that the velocity is given by dlta gt 23971 8111 ltUgt dz m 81 dx but for our purposes we need momentum p mu so ltpgt ih dII ih111dx The operator is sandwiched between 11 and III Smooth operators 0 For 111523 prob density Ilat2 111ast111at o All operators are sandwiched between 111 and III 0 Expectation value of position 00 00 111asta 11152 da as111at2 da 0 Expectation value of momentum ltpgtt 00 111ast ih111ast da 00 a Note that when calcing average momentum we can t do x1 ltpgt 7A z39h gym d9 471811 x11x11a dag 81 81 Wavefunction Momentum 0 Example problem Wavepacket Via Gaussian Integration Operators in general a All classical dynamical variables can be expressed a in terms of position and momentum eg T mv and Lrgtltmvrgtltp m o The rule classical gt quantum mechanical operators replace all p s with ih8 81 ltQxpgt Qx z39h 8xr dx 0 For example the average kinetic energy is given by Wavefunction Kinetic Energy 0 Example problem Wavepacket Via Gaussian Integration Ehrenfest 5 Theorem Quantum mechanical expectations obey classical laws eg velocity 21gt dlta gtdt eg momentum p m dlta gtdt and there is also the equivalent of Newetonian force lt 8V a d2 lt93gt dt2 dltpgt W l 1 l 2Xt VXt W a b r gt
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