Angular Momentum

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Fundamentals

In classical mechanics, the angular momentum $\mathit{L}$ of a particle of position $\mathit{r}$ and momentum $\mathit{p}$ is given by

$$\mathit{L}=\mathit{r}\times \mathit{p}$$

According to the correspondance principle, the angular-momentum observable is

$$\hat{\mathit{L}}=\hat{\mathit{r}}\times \hat{\mathit{p}}$$

and hence has following commutation relations

$$\hat{\mathit{L}}\times \hat{\mathit{L}}=i\hslash \hat{\mathit{L}}$$

We use these relations as the fundamental definition of an angular-momentum observable $\hat{\mathit{J}}$

$$\hat{\mathit{J}}\times \hat{\mathit{J}}=i\hslash \hat{\mathit{J}}$$

The observable denoted $\hat{J}{}^2$ is defined as $\hat{J}{}^2=\hat{J}{}_x^2+\hat{J}{}_y^2+\hat{J}{}_z^2$.

We observe that $[\hat{J}{}^2,\hat{\mathit{J}}]=0$ and that we can construct a CSCO made of $\{\hat{J}{}^2,\widehat{J_z}\}$ i.e. these two operators has a common eigenbasis. We can write their eigenvalues, without loss of generality, using two dimensionless numbers $j$ and $m$ such that

$$\begin{array}{rl}& \hat{J}{}^2|j,m⟩=j(j+1){\hslash }^2|j,m⟩\\ & \widehat{J_z}|j,m⟩=m\hslash |j,m⟩\end{array}$$

where the vectors $|j,m⟩$ form the set of eigenvectors.

It can be shown that the numbers $j$ and $m$ are quantized following the rules:

• $j$ is a positive (or zero) integer or half integer

• the only possible values of m are the $2j+1$ numbers $-j,-j+1,...,j-1,j$.

Orbital angular momentum

We consider here a particle moving in space, described by a wave function, and its orbital angular momentum $\hat{\mathit{L}}=\hat{\mathit{r}}\times \hat{\mathit{p}}$. In spherical coordinates the operator ${\hat{L}}_z$ has a simple form

$${\hat{L}}_z=\frac{\hslash }{i}\frac{\partial }{\partial \varphi }$$

The required periodicity in $\varphi$ with period $\pi$ leads to the conclusion that for an orbital angular momentum, $m$ must be an integer, and as a consequence $l$ is also an integer.

The eigenfunctions common to the observables $\hat{L}{}^2$ and ${\hat{L}}_z$ are called the spherical harmonics and are denoted $Y_{l,m}(\theta ,\varphi )$

$$\begin{array}{r}\begin{array}{c}\\ \begin{array}{rl}& \hat{L}{}^2{Y}_{l,m}(\theta ,\varphi )=l(l+1){\hslash }^2{Y}_{l,m}(\theta ,\varphi ),\\ & {\hat{L}}_z{Y}_{l,m}(\theta ,\varphi )=m\hslash Y_{l,m}(\theta ,\varphi )\end{array}\end{array}\end{array}$$

Motion in a central potential

In spherical coordinates, the Laplacian operator $\mathrm{\Delta }$ can be expressed in terms of the angular momentum $\hat{\mathit{L}}$

$$\mathrm{\Delta }=\frac{1}{r}\frac{{\partial }^2}{\partial r^2}r-\frac{1}{r^2{\hslash }^2}\hat{L}{}^2$$

Thus the operator associated to the kinetic energy of a particle, and hence its Hamiltonian $\hat{H}$, can be expressed using $\hat{\mathit{L}}$ too.

Furthermore one can show that for a particle in a central potential, they commute

$$[\hat{H},\hat{\mathit{L}}]=0$$

The wave function can be written as

$${\psi }_{l,m}(\mathit{r})=R_l(r)Y_{l,m}(\theta ,\varphi )$$

where $R(r)$ depend only on $l$.

The eigenvalues of the Hamiltonian can be labeled by the two quantum numbers $l$ and $n^{\prime }$, the radial quantum number. They do not depend on $m$, as a consequence of the rotation invariance of the system (central potential).

In a Coulomb potential, the energy levels depend only on the quantity $n^{\prime }+l+1$, and we can alternatively label the energies using this principal quantum number $n$.

In spectroscopic notation we associate the levels of $l$ to the letters s, p, d, f, g, h etc. and $n$ is denoted by a number preceding this letter, e.g. $n=1,l=0:\mathrm{state}1s$.

Magnetic Moment

We postulate that if the particle has a magnetic moment $\mathit{\mu }$, the corresponding observable $\hat{\mathit{\mu }}$ is proportional to $\hat{\mathit{L}}$, and we call the factor gyromagnetic ratio $\gamma$

$$\hat{\mathit{\mu }}=\gamma \hat{\mathit{L}}$$

Spin

A spin observable $\hat{\mathit{S}}$ acts in the 2-dimensional Hilbert space of spin 1/2 denoted as ${\mathcal{E}}_{\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}}$ and obeys the commutation relation of an angular momentum:

$$\hat{\mathit{S}}\times \hat{\mathit{S}}=i\hslash \hat{\mathit{S}}$$

with each of the observables $S_i$ having eigenvalues $\pm \hslash /2$. The observable $\hat{S}{}^2$ has one eigenvalue $3{\hslash }^2/4$.

This spin observable can be expressed in terms of the Pauli operators as

$$\hat{\mathit{S}}=\frac{\hslash }{2}\hat{\mathit{\sigma }}$$

References

• This story is basically a summary of Quantum Mechanics [B1], chapter 10 to 12. Some of the phrases are reproduced literally.