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Quantum Harmonic Oscillator

It follows from momentum space definition of position operator as $x=i\hbar {\frac {\partial }{\partial p}}$, that by using Schrodinger's equation and taking $m_{0}={\frac {1}{m\omega ^{2}}}$, we get:

${\begin{aligned}{\frac {1}{2m}}p^{2}\psi (p)-{\frac {1}{2}}m\omega ^{2}\hbar ^{2}{\frac {\partial ^{2}}{\partial p^{2}}}\psi (p)=E\psi (p)\\{\frac {1}{2}}{\frac {1}{m_{0}}}(-i\hbar \nabla _{p})^{2}\psi (p)+{\frac {1}{2}}m_{0}\omega ^{2}p^{2}\psi (p)=E\psi (p)\end{aligned}}$

Since momentum basis Schrodinger equation is same as that of position basis Schrodinger equation, the real-valued momentum wavefunction solution is same as momentum wavefunction corresponding to spatial wavefunction up-to an arbitrary phase which can be found to be $(-i)^{n}$ for $E_{n}$ energy eigenfunction. Hence the change $x\rightarrow p$, $m\rightarrow {\frac {1}{m\omega ^{2}}}$, $\omega \rightarrow \omega$ and $|n\rangle \rightarrow (-i)^{n}|n\rangle$ in any equation expressed in terms of those variables gives another valid equation.