Angular Momentum

Abstract

In quantum mechanics angular momentum includes the usual angular momentum that we learn about in classical mechanics. This angular momentum is usually designated by $\boldsymbol{L}$ and defined as

$$\displaystyle{ \boldsymbol{L} =\boldsymbol{ r} \times \boldsymbol{ p} }$$

In quantum mechanics the term “angular momentum” has a much more general meaning. It is a “generalized angular momentum.” A vector operator $\boldsymbol{\hat{J }}$ is defined to be an angular momentum if its components obey the commutation rules

$$\displaystyle{ \left [\hat{J } _{i}, \hat{J } _{j}\right ] = i\hslash \hat{J } _{k}\epsilon _{ijk} }$$

where any of the i, j, and k represent Cartesian coordinates x, y, and z. The quantity ε _ijk  is known as the Levi-Cevita symbol . If the indexes i, j, and k are in cyclic order (e.g., jki), ε _ijk  = +1. If they are out of order (such as kji), then ε _ijk  = −1. If any two indexes are the same, ε _ijk  = 0.