Chapter 7 Problem Solutions

Nb problem 79 given the superposition of adjacent

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Unformatted text preview: erefore and we have X p 0\ = Hm p — wL-1ê4 ‰ - pa p2 2mw— Printed by Mathematica for Students p „p p2 = ‰2mw— 2 Ch7ProblemSetKey.nb Problem 7.9 Given the superposition of adjacent energy states (7.63) yHtL\ = ‰- Hn + 1ê2L w t Bcn n\ + cn + 1 ‰- w t n + 1\F Using ` x= ` `† — ê H2 m wL Ja + a N ` p x = - ` `† m w — ê 2 Ja - a N we calculate the expectation values ` `† — ê H2 m wL Ja + a N y HtL\ = Xx\ = Xy HtL X px \ = Xy HtL -  ` `† — ê H2 m wL BYyHtL a yHtL] + ZyHtL a yHtL^F ` `† m w — ê 2 Ja - a N yHtL\ = - ` `† m w — ê 2 BYyHtL a yHtL] - ZyHtL a yHtL^F where ` a n] = n n - 1] `† a n^ = n + 1 n + 1^ Thus ` ` YyHtL a yHtL] = Bc* Xn + c* + 1 ‰Â w t Xn + 1 F aBcn n\ + cn + 1 ‰- w t n + 1\F n n = Bc* Xn + c* + 1 ‰Â w t Xn + 1 FBcn n n n n - 1] + cn + 1 ‰- w t n+1 Ø 0 in inner prod ` YyHtL a yHtL] = n + 1 c* cn + 1 ‰- w t n and similarly `† ` * ZyHtL a yHtL^ = YyHtL a yHtL] = n + 1 cn c* + 1 ‰Â w t n This gives Xx\ = — Hn + 1L ê H2 m wL Ic* cn + 1 ‰- w t + cn c* + 1 ‰Â w t M n n X p x \ = - Xx\ = 2 X px \ = 2 m w — Hn + 1L ê 2 Ac* cn + 1 ‰- w t - cn c* + 1 ‰Â w t E n n — Hn + 1L ê H2 m wL ReAc* cn + 1 ...
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This note was uploaded on 02/01/2014 for the course PHYC 491/496 taught by Professor Akimasamiyake during the Fall '13 term at New Mexico.

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