Quantum Harmonic Oscillator

There is a symmetry between the momentum space and position space wavefunctions for the harmonic oscillator, as is evident by their matrix representations, that isn’t present in other bound state systems. To explore this, we need to find the momentum space representation of the energy eigenstates $|n⟩\dot{=}{\varphi }_n(p)=⟨p⟩n$. There are three ways to do this

• Take the approach of using the annihilation operator on the ground state and the termination condition combined.

$$\mathrm{a}|0⟩=0$$

• By using the combination of position and momentum operators that represent the annihilation operator, we can rewrite this as

$$\sqrt{\frac{m\omega }{2\hslash }}(\hat{x}-i\frac{\hat{p}}{m\omega }){\varphi }_0(p)=0$$