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This 12 page Class Notes was uploaded by Jamaal McGlynn 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 28 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 5 Outline Uncertaintly a de Broglie s formula 16 o Heisenberg s uncertainty principle 16 a 1 D Schr idinger eqn stationary states 21 Quantum mechanical expectations obey Classical laws Ehrenfest 3 Theorem eg velocity 21gt dlta gtdt eg momentum p m dlta gtdt and nally there is also the equivalent of Newtonian force T 2Xt l l VXt The wave uncertainty principle 0 Where is a wave What is its wavelength 0 Hope example Griffith Figs 17 and 18 ltpgt o Piccolo vs double bass example a but Fourier analysis puts this on a rm footing o Homework read up on Fourier analysis Heisenberg s uncertainty principle a de Broglie Wavelength related to momentum h 27Th hiking and 19 A o wavenumber k length 1 and angular frequency w Hz a For a given 111 spread uncertainty in wavelength implies a momentum spread uncertainty 0x0 p 2 a We will develop this further later Wavefunction Standard Deviations 03 0 Position variance 0 512 ltxgt2 o The general form for variance A 2 o WQ W maQa V Thus momentum variance is V w ww4m V ie need 4 integrals to nd 0x and 0p in the process also gives us T p2gt 2m Wavefunction 093 Up a Example problem Wavepacket and Gaussian Integrations Timedependent Schrodinger eqn following de Broglie p hk and E flea and we also have the relations p2 h2k2 th hw so to 2m 2m 2m assuming that a particle has well de ned wavelength 114x715 Ae kx wt A ipx Eth from which we can take two derivatives 8Q iE 82111 p2 q d 92 h an 3932 712 and then we can use the E relation 8 7 22 82 39h xr x11 Z 82 71 277185132 71 Stationary States 13 Having learnt about the expectations for a given 1115 25 Time to determine 111525 itself 8111ast h282111at m 82 2m 8132 Look for pde solns using separation of variables Was I 090900 Which means that for such separable solutions 8amp1 w dgp 82E dZib 8t dt 8132 das2 VIIIat and the Schrodinger eqn can be written as h dZib 2m dx2 90wa Stationary States 23 do 712 erb EZ W VW o Dividing both sides by w 90 m V go dz 2m w dx2 0 Each side must be a constant separation constant E i Ez39 Vx E E 2m w dx2 0 Two ordinary differential equations dgp 2E 712 d2rb 7 1 90 am 2m dx2 dt h VarbErb Stationary States 33 dgp iE a 7 90 0 Since we have the form dfdg X f then z39E W 6 t 114m was wt was e iEth 0 Thus we have a stationary wavefunction 1 atl2 W we amth was e iEth weal ltQxpgt W 6W Qa ih 3M e iEth d9 0 Thus for all expectation values dltQgt dt 0 Timeindependent Schrodinger Eqn 0 is just the one ordinary differential equation a but from classical mechanics the total energy kinetic potential comes from the Hamiltonian Hap Vas o The quantum mechanical Hamiltonian operator is A 712 82 H V 2m 8132 7 o time independent Schrodinger Equation Hm Ew Hamiltonian Operator A 712 82 H V 2m 8132 7 a To calculate the total energy we evaluate H 111ast H IIat d9 wltxgt We aim
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