(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Write down the time independant Schrodinger eqn in the momentum representation for a particle with mass m when the potential is given by [itex]V(x) = \frac{1}{2} \gamma x^2[/itex]

Given that a possible solution is given by [itex]\Phi(p) = e^{\frac{-Bp^2}{2}}[/itex]

determine B and the corresponding energy eigenvalue.

2. Relevant equations

position / momentum

[itex]x \rightarrow i\hbar \frac{d}{dp}[/itex]

[itex]-i\hbar \frac{d}{dx} \rightarrow p[/itex]

3. The attempt at a solution

in position representation the full time independant schrodinger eqn is:

[itex]- \frac{\hbar^2}{2m}\frac{d^2}{dx^2} \Psi(x) + \frac{1}{2}\gamma x^2 \Psi(x) = E\Psi(x)[/itex]

becomes

[itex]\frac{p^2}{2m}\Phi(p) - \frac{\hbar^2 \gamma}{2}\frac{d^2}{dp^2}\Phi(p) = E\Phi(p)[/itex]

in the momentum representation, where Phi is the FT of Psi.

After plugging in the Trial Solution I get:

[itex]E = \frac{p^2}{2m} - \frac{\hbar^2 \gamma}{2}\left( B^2 p^2 - B\right)[/itex]

Not sure what to do after this bit, I tried to normalise the wavefunction [itex]A = \left(\frac{B}{\pi}\right)^{\frac{1}{4}}[/itex]

But I dont think that helps.

Any ideas on where i could get a second equation to find B and E?

Thanks

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# Homework Help: QM - Position/Momentum representation problem

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