385lec15 - PHYS 385 Lecture 15 - Classical limit Lecture 15...

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PHYS 385 Lecture 15 - Classical limit 15 - 1 ©2003 by David Boal, Simon Fraser University. All rights reserved; further copying or resale is strictly prohibited. Lecture 15 - Classical limit What's important : uncertainty relations Ehrenfest's theorem Text : Gasiorowicz, App. B, Chap. 6 Having now solved a couple of the most common applications of quantum mechanics in one dimension, it's time to return to the foundations of the theory. We need to: determine the numerical factor in the uncertainty principle show how to regain the classical equations on motion for a potential. Uncertainty principle The numerical factors in the uncertainty principle can be extracted with little effort once we have established a few properties of Hermitian operators. The proof follows along Appendix B from Gasiorowicz. Consider two operators A and B , with expectations < A > and < B > respectively. Later, we will use position and momentum as specific examples, but for now the results are completely general. From these operators, construct two new operators U and V by subtracting a scalar from each, namely their mean values. That is, U = A - < A > V = B - < B > (1) The wavefunction on which the operators act is . Now, consider the linear combination of operators and wavefunctions : = U + i V . (2) Taking the complex square of and integrating over position x must result in a positive quantity, as | | 2 is not negative. In symbols, I ( ) = * dx 0 (3) Because A and B are Hermitian operators, so must be
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This note was uploaded on 09/07/2009 for the course PHYS 385 taught by Professor Davidboal during the Spring '09 term at Simon Fraser.

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385lec15 - PHYS 385 Lecture 15 - Classical limit Lecture 15...

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