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Schiff QM chapter 3 prob 9

Let u1 (r) and u2 (r) be two eigenfunctions of the same hamiltonian that correspond to the same energy eigenvalue; they may be the same function, or they may be degenerate. Show that

Another eager mind and myself have been working on this problem for two weeks. We cannot seem to get the right things to line up. We have tried integration by parts, we have used some assumptions on the structure of the eigenfunctions, and we have even used the following identity in order to try and incorporate the hamiltonian into the integral

but this fails since the eigenfunctions are of the stationary equation and therefore

independent of time. If anyone could help please let us know.

Let u1 (r) and u2 (r) be two eigenfunctions of the same hamiltonian that correspond to the same energy eigenvalue; they may be the same function, or they may be degenerate. Show that

int[u∗1 (r) [xp + px] u2 (r)]dr = 0

where the momentum operator p = (−i/hbar)(∂ /∂ x) operates on everything to its right. Another eager mind and myself have been working on this problem for two weeks. We cannot seem to get the right things to line up. We have tried integration by parts, we have used some assumptions on the structure of the eigenfunctions, and we have even used the following identity in order to try and incorporate the hamiltonian into the integral

d/dt(<x^2>)=(1/m)(<xp>+<px>) where <> are the expectation value signs;

but this fails since the eigenfunctions are of the stationary equation and therefore

independent of time. If anyone could help please let us know.

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