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9. In this problem you are asked to use Eqs. (3.25) and (3.27) to show that the expectation values of the position and momentum of a particle with mass m are related by (x)p" dr The method is similar to that used in the previous problem. (a) By noting that the time dependence of the wave function is governed by the Schrödinger equation, show that d(V*x¥) _ ih dr 2m and show that this can be rewritten as d(V*xV)_ ih a ih di 2m dx ax ax 2m (b) Assuming that the wave function tends to zero sufficiently rapidly at x = tox, show that ih V*xV dx dx. 2m ax (c) Now integrate by parts and show that m d V*(x, 1)( –ih - x,n(-ih)vcx, o) dx. V*x¥ dx =

9. In this problem you are asked to use Eqs. (3.25) and (3.27) to show that the expectation values of the position and momentum of a particle with mass m are related by (x)p" dr The method is similar to that used in the previous problem. (a) By noting that the time dependence of the wave function is governed by the Schrödinger equation, show that d(V*x¥) _ ih dr 2m and show that this can be rewritten as d(V*xV)_ ih a ih di 2m dx ax ax 2m (b) Assuming that the wave function tends to zero sufficiently rapidly at x = tox, show that ih V*xV dx dx. 2m ax (c) Now integrate by parts and show that m d V*(x, 1)( –ih - x,n(-ih)vcx, o) dx. V*x¥ dx =

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9. In this problem you are asked to use Eqs. (3.25) and (3.27) to show that the expectation values of the position and momentum of a particle with mass m are related by (x)p" dr The method is similar to that used in the previous problem. (a) By noting that the time dependence of the wave function is governed by the Schrödinger equation, show that d(V*x¥) _ ih dr 2m and show that this can be rewritten as d(V*xV)_ ih a ih di 2m dx ax ax 2m (b) Assuming that the wave function tends to zero sufficiently rapidly at x = tox, show that ih V*xV dx dx. 2m ax (c) Now integrate by parts and show that m d V*(x, 1)( –ih - x,n(-ih)vcx, o) dx. V*x¥ dx =
Transcribed Image Text:9. In this problem you are asked to use Eqs. (3.25) and (3.27) to show that the expectation values of the position and momentum of a particle with mass m are related by (x)p" dr The method is similar to that used in the previous problem. (a) By noting that the time dependence of the wave function is governed by the Schrödinger equation, show that d(V*x¥) _ ih dr 2m and show that this can be rewritten as d(V*xV)_ ih a ih di 2m dx ax ax 2m (b) Assuming that the wave function tends to zero sufficiently rapidly at x = tox, show that ih V*xV dx dx. 2m ax (c) Now integrate by parts and show that m d V*(x, 1)( –ih - x,n(-ih)vcx, o) dx. V*x¥ dx =
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