Popular in Course
Popular in Physics 2
This 21 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 17 views. For similar materials see /class/225321/phys410-san-diego-state-university in Physics 2 at San Diego State University.
Reviews for PHYS410
Report this Material
What is Karma?
Karma is the currency of StudySoup.
Date Created: 10/20/15
Lecture 6 Outline Uncertaintly Solvay talk Ehrenfest Quantum to Classical Via Ehrenfest s theorem 15 de Broglie s formula 16 Heisenberg s uncertainty principle 16 1 D Schrodinger eqn stationary states 21 Ehrenfest s Theorem Quantum mechanical expectations obey classical laws eg velocity 11gt dltxgtdt eg momentum p m dltxgtdt and nally force ltVgtdltpgtmd2ltxgt l Ky l 2m W VXt The wave uncertainty principle 0 Where is a wave What is its wavelength 0 Hope example Grif th Figs 17 and 18 ltpgt o Piccolo vs double bass example 0 but Fourier analysis puts this on a rm footing o Homework read up on Fourier analysis Heisenberg s uncertainty principle 0 ole Broglie Wavelength is related to the momentum h 27Th p A A 39 o For a given 11 spread uncertainty in wavelength implies a momentum spread uncertainty h 0 We Will prove all of this later Timeindependent Schr dinger eqn 0 Remembering the timedependent Schredinger eqn m qlcrnf h23211xt 3t 2m 3x2 0 NOW that we ve learnt how to work out different VIJU t expectations for a given x t 0 Time to determine x 15 itself Stationary States 13 311xt h23211xt 3t 2m 3x2 0 Look for pde solns using separation of variables was t 009005 0 Which means that for such separable solutions aw den 311 on 5 E and ag 290 ih VIJU t o and the Schrodinger eqn can be written as dgp E2 12 Wa WWVW Stationary States 23 dgp 752 d2 WE 0m WW 0 Dividing both sides by w 90 V go dt 2m w dx2 0 Each side must be a constant separation constant E i E 39 V Z 90 dt 2m w dx2 0 Two ordinary differential equations d 39E E2 d2 L and V E dt h 2m dx2 Stationary States 33 dgp iE E 0 Since we have the form dfdg oc f then iE t 6 t MawwmwmwmwWV 0 Thus we have a stationary wavefunction www vwwwwwmwmawmwww ltQxpgt W 61E 7 Qa ih e EW d9 0 Thus for all expectation values dltQgt dt O Timeindependent Schrodinger Eqn is just the one ordinary differential equation but from classical mechanics the total energy kinetic potential comes from the Hamiltonian Hx p p 2Vx 7 2m The quantum mechanical Hamiltonian operator is E2 62 H v 27713932x7 time independent Schrodinger Equation 13 Ew Hamiltonian Operator 752 82 WW 0 To calculate the total energy we evaluate H IJxt H Mm d9 f W H was da f W E was dx E was da E Lecture 6 Outline Keep em Separated o Homework review it s looking 0k a 1D Schr dinger eqn stationary states 21 o and let s get cracking 0n the in nite well 22 Stationary States recall a 1 D timedependent Schrodinger eqn 8111ast 712 821145132 x11 271 at 2m 8x2 V 5132 0 Assumed separation of variables 1115 25 Was 9023 0 1 D timemdepemlent Schrodinger eqn d 39E 712 d2 d f gp and wVawEw o Stationary state solns 93715 W93 901 W93 WW where I 1 937152 b 2 o and used Hamiltonian operator to nd total energy ltHgt 111amf H 111515 d9 fwx H mag d9 Wm E was da Ef wan d9 E Hamiltonian operator new bits Hw 7 Vagww a We know that E but what about 12 m Hum HltEwgt E w Ea a so we can rework 12 111ami H2 11152 da since ltH2gt wx iEth 12 77blj 2 E15ii dx we H2 was dx E2 wxl2 dx E2 0 The variance of the Hamiltonian expectation value 0 I ltH2gt ltHgt2 I E2 E2 I 0 0 ie every measurement of H upon a separable solution lla t 9025 will result in E Combination Fried Rice 0 The timeindependent Schrodinger eqn has an in nite number of solns lms w2a w3a a each with a corresponding E1 E2 E3 1111ast 1amp1 8 iE1th 1112ast w2a 8 iE2th 0 Linear combination of complex functions fnz HZ Ci f1Z C2 f2Z C3 f3Z 0 is also a soln to a linear diffeqn Stationary gt timedependent The general soln 111523 to the time dependent SE is a linear combination of in nite set of separable solns Ilnat from timeindependent eqn I1437715 Z qulnx7 t Z e ZEnth n21 n1 At this point alarm bells should be ringing since for each individual stationary state I I nm tl2 1515 81E h 1M9 e zE Lth lawn which give all expectation values dltQgt dt 0 Superposition of solutions 0 A combo of stationary states can have 111n1752 time dependent Stationary States prescription Take a given timeindependent potential Va Solve 1 D time independent Schrodinger eqn E2 dZwaE to get in nite number of solns Ilna t wna e z Enth Assuming initial wave function at 11152 0 we can solve for on such that Ila 0 Z cnwn r The general soln to 1 D timedependent Schrodinger eqn I1437715 Z qulnx7 t Z e ZEnth n21 n1 The in nite square well 14 a First mineindependent potential Va to solve V0 lt a lt a 0 and Velsewhere 00 VX a this implies that ibna 0 for elsewhere 0 inside the well we have Va 0 and need to solve 71 2 d2wna 2m das2 The in nite square well 24 h2 CplMac 2m dx2 Which is re written as a simple harmonic oscillator eqn 2 v2 En d kiwna Where kn 77 1 assuming En 2 0 cf Griffiths problem 22 The SHO has the general solution wna An sinkna Bn coskna To solve for coef cients impose boundary conditions assuming that wnw is always continuous and dwnada is continuous except Where Va oo The in nite square well 34 rms An sinkna Bn coskna impose boundary conditions at the edges Wm WW 0 that s at the edges to which we match at a 0 1M0 and matching at a a WW So either An 0 boring or sinkna 0 An sinkna Bn coskna Bn 0 An sinkna 0 kna 0 7T 27T7 37T7 but kn 0 is also boring since wnw 0 The in nite square well 44 kna 7T 27T7 37T7 o For the V6 solutions sin kna sinkna which we can simply absorb into A o Distinct solutions are kn mrai with n 1 2 3 Ma An sinkna 0 Where kn mra 00 WK wx l a X l A X l o Normalise foa An2 sin2kna da An2 1 2 h2k2 2 2712 MM Sin Where En n TNT Cl CL 2m 2ma2