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- Homework Statement
- please see below

- Relevant Equations
- please see below

We show by working backwards

$$\hbar w \Big(a^{\dagger}a+\frac{1}{2}\Big)=\hbar w \Big(\frac{mw}{2\hbar}(\hat{x}+\frac{i}{mw}\hat{p})(\hat{x}-\frac{i}{mw}\hat{p})+\frac{1}{2}\Big)$$

$$=\Big(\frac{mw^2}{2}(\hat{x}^2+\frac{i}{mw}[\hat{p},\hat{x}]+\frac{\hat{p}^2}{m^2w^2})+\frac{1}{2}\Big)=\frac{1}{2}mw^2\hat{x}^2-\frac{\hbar w}{2}+\frac{\hat{p}^2}{2m}+\frac{\hbar w}{2}$$

$$=\frac{1}{2}mw^2\hat{x}^2+\frac{\hat{p}^2}{2m}=\hat{H}$$

This shows the time-independent Schrödinger equation for the simple harmonic oscillator can be written as

$$\hbar w \Big(a^{\dagger}a+\frac{1}{2}\Big)|\psi>=E_n|\psi>$$

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