harmonic oscillator ladder operators

Harmonic Oscillator Ladder Operators

The Hamiltonian for the 1-d harmonic oscillator can be stated as

{% \hat{H} = - \frac{\hbar ^2}{2m} \frac{\partial ^2}{\partial x^2} + \frac{1}{2} K x^2 %}

Which can restated using the standard position and momentum operators.

{% \hat{H} = \frac{\hat{p}^2}{2m} +\frac{1}{2}m \omega^2 \hat{x}^2 %}

{% \hat{a} = \sqrt{\frac{m \omega}{2\hbar}} (\hat{x} + \frac{i}{m \omega} \hat{p}) %}

{% \hat{a} ^\dagger = \sqrt{\frac{m \omega}{2\hbar}} (\hat{x} - \frac{i}{m \omega} \hat{p}) %}

{% \hat{x} = \sqrt{\frac{\hbar}{2 m \omega}} (\hat{a} + \hat{a}^{\dagger}) %}

{% \hat{p} = -i \sqrt{\frac{\hbar m \omega}{2 }} (\hat{a} - \hat{a}^{\dagger}) %}

{% \hat{H} = \hbar \omega (\hat{a}^{\dagger}\hat{a} + \frac{1}{2}) %}

The commutator can be evaluated as

{% [\hat{a}, \hat{a}^{\dagger}] = \frac{m \omega}{2 \hbar} (-\frac{i}{m \omega} [\hat{x}, \hat{p}] + \frac{i}{m \omega} [\hat{p}, \hat{x}]) = 1 %}

Number operator is defined to be

{% \hat{n} = \hat{a}^{\dagger}\hat{a} %}