Recap of the Harmonic Oscillator

The Hamiltonian of the harmonic oscillator

$$\hat{H}_{\rm osc}=\frac{\hat{p}^2}{2m}+\frac{1}{2}m\omega^2\hat{x}^2$$ (2.6)

can be re-written using the ladder operators

$$a \equiv \sqrt{\frac{m\omega}{2\hbar}}\hat{x}+\frac{i}{\sqrt{2m\hbar\omega... ...quiv \sqrt{\frac{m\omega}{2\hbar}}\hat{x}-\frac{i}{\sqrt{2m\hbar\omega}}\hat{p}$$ (2.7)

$$\hat{x} = \sqrt{\frac{\hbar}{2m\omega}}\left(a+a^{\dagger}\right),\quad \hat{p} =-i\sqrt{\frac{m\hbar\omega}{2}}\left(a-a^{\dagger}\right),$$ (2.8)

as

$$\hat{H}_{\rm osc}=\hbar\omega\left(a^{\dagger}a+\frac{1}{2}\right).$$ (2.9)

The commutation relation is

$$[\hat{x},\hat{p}]=i\hbar,\quad [a,a^{\dagger}]=1.$$ (2.10)

The eigenfunctions of the harmonic oscillator are $n$-phonon states,

$$\hat{H}_{\rm osc}\vert n\rangle = \varepsilon_n \vert n\rangle, \quad \varepsilon_n = \hbar\omega\left(n+\frac{1}{2}\right),\quad n=0,1,2,...$$

$$\vert n\rangle \leftrightarrow \psi_n(x) = \left(\frac{m\omega}{\pi\hbar}\right)^{1/4} \frac{1}{\sqrt{n! 2^n}}H_n\left(\sqrt{\frac{m\omega}{\hbar}}x\right)e^{-\frac{m\omega}{2\hbar}x^2},$$ (2.11)

where $H_n$ are the Hermite polynomials.

The ladder operators are also called creation ( $a^{\dagger}$) and annihiliation $(a)$ operators. They act on the states $\vert n\rangle$ as

$$a^{\dagger} \vert n\rangle = \sqrt{n+1} \vert n+1\rangle,\quad a\vert n\rangle = \sqrt{n} \vert n-1\rangle,\quad a\vert n\rangle =0.$$ (2.12)

The state $\vert\rangle$ is called ground state.