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knowlewj01
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Homework Statement
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.
Homework Equations
position / momentum
[itex]x \rightarrow i\hbar \frac{d}{dp}[/itex]
[itex]-i\hbar \frac{d}{dx} \rightarrow p[/itex]
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 don't think that helps.
Any ideas on where i could get a second equation to find B and E?
Thanks