Angular momentum (quantum)

In quantum mechanics, angular momentum is a vector operator of which the three components have well-defined commutation relations.

Angular momentum entered quantum mechanics through atomic spectroscopy, where angular momentum theory—together with its connection to group theory—was able to put order to a perplexing number of spectroscopic observations, see, for instance, Wigner's seminal work. When in 1926 electron spin was discovered and it was realized that spin was a form of angular momentum, its importance rose even further. Now the quantum theory of angular momentum is an indispensable discipline for the working physicist, irrespective of his field of specialization, be it solid state physics, molecular-, atomic,- nuclear,- or even hadronic-structure physics.

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,

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

Writing r _i (i = 1,2,3) for x, y, and z, respectively, and using

${\frac {\partial r_{i}}{\partial r_{j}}}=\delta _{ij}={\begin{cases}&1\quad \mathrm {if} \quad i=j\\&0\quad \mathrm {if} \quad i\neq j\\\end{cases}}$

we find easily

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

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.

The total angular momentum squared is defined by

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

In terms of spherical polar coordinates the operator is,

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

Eigenfunctions of the latter operator have been known since the nineteenth century, long before quantum mechanics was born. They are spherical harmonic functions.

Abstract angular momentum operators

Angular momentum operators are Hermitian operators j_x, j_y, and j_z,that satisfy the commutation relations

$[j_{k},j_{l}]=i\sum _{m=x,y,z}\varepsilon _{klm}j_{m},$

where $\varepsilon _{klm}$ is the Levi-Civita symbol. Together the three components define a vector operator $\mathbf {j}$. The square of the length of $\mathbf {j}$ is defined as

$\mathbf {j} ^{2}=j_{x}^{2}+j_{y}^{2}+j_{z}^{2}.$

We also define raising $(j_{+})$ and lowering $(j_{-})$ operators

$j_{\pm }=j_{x}\pm ij_{y}.\,$

Angular momentum states

It can be shown from the above definitions that j² commutes with j_x, j_y, and j_z

$[\mathbf {j} ^{2},j_{k}]=0\quad \mathrm {for} \;\;k=x,y,z.$

When two Hermitian operators commute a common set of eigenfunctions exists. Conventionally j² and j_z are chosen. From the commutation relations the possible eigenvalues can be found. The result is

$\mathbf {j} ^{2}|jm\rangle =j(j+1)|jm\rangle ,\qquad j=0,1/2,1,3/2,2,\ldots$

$j_{z}|jm\rangle =m|jm\rangle ,\qquad \quad m=-j,-j+1,\ldots ,j.$

The raising and lowering operators change the value of $m$

$j_{\pm }|jm\rangle =C_{\pm }(j,m)|jm\pm 1\rangle$

with

$C_{\pm }(j,m)={\sqrt {j(j+1)-m(m\pm 1)}}={\sqrt {(j\mp m)(j\pm m+1)}}.$

A (complex) phase factor could be included in the definition of $C_{\pm }(j,m)$ The choice made here is in agreement with the Condon and Shortley phase conventions. The angular momentum states must be orthogonal (because their eigenvalues with respect to a Hermitian operator are distinct) and they are assumed to be normalized

$\langle jm|j'm'\rangle =\delta _{j,j'}\delta _{m,m'}.$

References

(to be continued)