Angular momentum (quantum)

Orbital angular momentum

The classical angular momentum of a point mass is,

$\mathbf {L} =\mathbf {r} \times \mathbf {p} ,$

where r is the position and p the (linear) momentum of the point mass. The simplest and oldest example of an angular momentum operator is obtained by applying the quantization rule:

$\mathbf {p} \rightarrow -i\hbar \mathbf {\nabla } ,$

where $\scriptstyle \hbar$ is Planck's constant (divided by 2π) and ∇ is the gradient operator. This rule applied to the classical angular momentum vector gives a vector operator with the following three components,

${\begin{aligned}L_{x}&=-i\hbar {\Big (}y{\frac {\partial }{\partial z}}-z{\frac {\partial }{\partial y}}{\Big )}\\L_{y}&=-i\hbar {\Big (}z{\frac {\partial }{\partial x}}-x{\frac {\partial }{\partial z}}{\Big )}\\L_{z}&=-i\hbar {\Big (}x{\frac {\partial }{\partial y}}-y{\frac {\partial }{\partial x}}{\Big )}.\\\end{aligned}}$

The following commutation relations can be proved,

$[L_{x},\,L_{y}]=i\hbar L_{z},\quad [L_{z},\,L_{x}]=i\hbar L_{y},\quad [L_{y},\,L_{z}]=i\hbar L_{x}.$

The square brackets indicate the commutator of two operators, defined for two arbitrary operators A and B as

$[A,\,B]\equiv AB-BA.$

For instance,

${\begin{aligned}{\big [}L_{x},\,L_{y}{\big ]}=&-\hbar ^{2}\left[{\Big (}y{\frac {\partial }{\partial z}}-z{\frac {\partial }{\partial y}}{\Big )}{\Big (}z{\frac {\partial }{\partial x}}-x{\frac {\partial }{\partial z}}{\Big )}-{\Big (}z{\frac {\partial }{\partial x}}-x{\frac {\partial }{\partial z}}{\Big )}{\Big (}y{\frac {\partial }{\partial z}}-z{\frac {\partial }{\partial y}}{\Big )}\right]\\=&-\hbar ^{2}\left[y{\frac {\partial }{\partial x}}-x{\frac {\partial }{\partial y}}\right]=i\hbar \left[-i\hbar {\Big (}x{\frac {\partial }{\partial y}}-y{\frac {\partial }{\partial x}}{\Big )}\right]=i\hbar L_{z},\\\end{aligned}}$

where we used that all the terms of the kind

$yx{\frac {\partial ^{2}}{\partial z\partial x}},\quad {\hbox{etc.}}$

mutually cancel. Since the three components of L do not commute, they do not have a common set of eigenfunctions.

The total angular momentum squared is defined by

$\mathbf {L} ^{2}\equiv L_{x}^{2}+L_{y}^{2}+L_{z}^{2}.$

This operator commutes with any of the three components,

$[\mathbf {L} ^{2},\,L_{k}]=0\quad {\hbox{for}}\quad k=x,y,z,$

so that common eigenfunctions of L² and one of its components can be found.

Note, parenthetically, that eigenfunctions of L² have been known since the nineteenth century, long before quantum mechanics was born. They are spherical harmonic functions. Indeed, in terms of spherical polar coordinates the operator is,

$L^{2}=-\hbar ^{2}\left[{\frac {1}{\sin \theta }}{\frac {\partial }{\partial \theta }}\sin \theta {\frac {\partial }{\partial \theta }}+{\frac {1}{\sin ^{2}\theta }}{\frac {\partial ^{2}}{\partial \varphi ^{2}}}\right],$

and this expression appears in the associated Legendre equation.