385lec14 - PHYS 385 Lecture 14 - Vibrating systems Lecture...

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14 - 1 ©2003 by David Boal, Simon Fraser University. All rights reserved; further copying or resale is strictly prohibited. Lecture 14 - Vibrating systems What's important : oscillator wavefunctions and classical limit application: quark model Text : Gasiorowicz, Chap. 5 and other sources In this lecture we look at oscillator wavefunctions in some detail, first examining the classical of the quantum oscillator, then applying the quantum oscillator to the quark model. Classical limit of the quantum oscillator The probability distribution of the quantum oscillator can be found from the complex square of u ( x ), where u n ( x ) = N n exp(- 2 /2) H n ( ) (1) where the normalization constant N n is N n = π 1/2 1 2 n n ! 1/2 (2) and H o = 1 H 1 = 2 (3) H 2 = 4 2 - 2 ... Recall the definitions: = x , = m / h . (4) Let's start with n = 0, the quantum ground state. Here, H o = 1, and N o = ( / π ) 1/4 , so that | u o | 2 = ( / π ) 1/2 exp(- 2 ). (5) This distribution is a Gaussian centered at x = 0: Now, the energy of this state is E o = h /2. What is the classical turning point for this energy? Solving kx 2 /2 = h /2 -> x 2 = h / k . But
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385lec14 - PHYS 385 Lecture 14 - Vibrating systems Lecture...

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