How are creation and annihilation operators derived?

hfwang132@gmail.com (hfwang132) — Wed, 01 May 2024 03:04:30 +0800

The motivation for defining creation and annihilation operators is simple and can be entirely derived from classical mechanics.

Think about how we solve the classical harmonic oscillator. Since position and momentum are coupled:

$\begin{cases} \frac{\mathrm{d}x}{\mathrm{d}t} = \omega p \\ \frac{\mathrm{d}p}{\mathrm{d}t} = -\omega x \end{cases}$

That is,

$\frac{\mathrm{d}}{\mathrm{d}t} \begin{bmatrix} x \\ p \end{bmatrix} = \begin{bmatrix} 0 & \omega \\ -\omega & 0 \end{bmatrix} \begin{bmatrix} x \\ p \end{bmatrix}$

So, we just need to decouple them. By diagonalizing, we obtain the eigenvectors $a^{\pm} = x \pm \mathrm{i} p$, and the derivatives of $a^{\pm}$ only depend on themselves:

Mathematical Methods in Physics on Haifei's Home

How are creation and annihilation operators derived?