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.

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)