Overview
The angular momentum in classical mechanics is defined to be
{% \vec{L} = \vec{r} \times \vec{p} %}
Which reduces to the following components
{% L_x = yp_z - zp_y %}
{% L_y = zp_x - xp_z %}
{% L_z = xp_y - yp_x %}
Definition
In quantum mechanics, we follow the prescription of replacing position and momentum as operators to get the following quantum mechanical definition
{% \hat{L} = \hat{X} \times \hat {P} %}
This reduces to the following for {% \hat{L}_x %}
{% \hat{L}_x = \hat{y}\hat{p}_z - \hat{z}\hat{p}_y %}
Commutation Relations
{% [\hat{L}_x, \hat{L}_y] = i \hbar \hat{L}_z %}
{% [\hat{L}_y, \hat{L}_z] = i \hbar \hat{L}_x %}
{% [\hat{L}_z, \hat{L}_x] = i \hbar \hat{L}_y %}