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Derive the Schroedinger equation (for harmonic oscillator) in momentum space.

The attempt at a solution

We have

[tex]ih \frac{\partial}{\partial t} \langle p' | \alpha \rangle = \langle p' | \frac{p^2}{2m} | \alpha \rangle + \langle p' | V(x) | \alpha \rangle[/tex]

[tex]\iff ih \frac{\partial}{\partial t} \langle p' | \alpha \rangle = \frac{p'^2}{2m}\langle p' | \alpha \rangle + \left(i\hbar\frac{\partial}{\partial p'}\right)^2 \langle p' | \alpha \rangle[/tex]

[tex]

\[

\iff ih\frac{\partial \Phi(p)}{\partial t} = \frac{p^2}{2m}\Phi(p) - \hbar^2 \frac{\partial^2 \Phi(p)}{\partial p^2}

\]

[/tex]

Is the above correctly done? If so, is this the generalized Schroedinger equation (without electrodynamics, etc.. just p^2/2m kinetic energy)?:

[tex]

\[

\iff ih\frac{\partial \Phi(p)}{\partial t} = \frac{p^2}{2m}\Phi +\displaystyle\sum_{n = 0}^\infty \frac{V^{(n)}(0)}{n!}\left(i\hbar \frac{\partial}{\partial p}}\right)^n \Phi

\]

[/tex]

Thanks very much for your help :)

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# Homework Help: Schroedinger Equation in Momentum Space

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