Angular Momentum

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Fundamentals

In classical mechanics, the angular momentum \(\bm L\) of a particle of position \(\bm r\) and momentum \(\bm p\) is given by

\[\bm L = \bm r \times \bm p\]

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

\[\hatb L = \hatb r \times \hatb p\]

and hence has following commutation relations

\[\hatb{L} \times \hatb{L} = i \hbar \hatb{L}\]

We use these relations as the fundamental definition of an angular-momentum observable \(\hatb J\)

\[\hatb{J} \times \hatb{J} = i \hbar \hatb{J}\]

The observable denoted \(\hatsup{J}{2}\) is defined as \(\hatsup{J}{2} = \hatsubsup{J}{x}{2} + \hatsubsup{J}{y}{2} + \hatsubsup{J}{z}{2}\).

We observe that \([ \hatsup{J}{2}, \hatb{J} ] = 0\) and that we can construct a CSCO made of \(\{ \hatsup{J}{2}, \hatsub{J}{z} \}\) 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

\[\begin{split}& \hatsup{J}{2} \ket{j, m} = j (j + 1) \, \hbar^2 \ket{j, m} \\ & \hatsub{J}{z} \ket{j, m} = m \hbar \ket{j, m}\end{split}\]

where the vectors \(\ket{j, m}\) 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 \(\hatb L = \hatb r \times \hatb p\). In spherical coordinates the operator \({\hat L}_z\) has a simple form

\[{\hat L}_z = \frac{\hbar}{i} \frac{\partial}{\partial \phi}\]

The required periodicity in \(\phi\) 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 \(\hatsup{L}{2}\) and \({\hat L}_z\) are called the spherical harmonics and are denoted \(Y_{l,m}(\theta, \phi)\)

\[ \begin{align}\begin{aligned}\newcommand{Ylm}{Y_{l,m}(\theta, \phi)}\\\begin{split}& \hatsup{L}{2} \, \Ylm = l (l + 1) \hbar^2 \, \Ylm , \\ & {\hat L}_z \, \Ylm = m \hbar \, \Ylm\end{split}\end{aligned}\end{align} \]

Motion in a central potential

In spherical coordinates, the Laplacian operator \(\Delta\) can be expressed in terms of the angular momentum \(\hatb{L}\)

\[\Delta = \frac{1}{r} \frac{\partial^2}{\partial r^2} r - \frac{1}{r^2 \hbar^2} \hatsup{L}{2}\]

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

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

\[[\hat H, \hatb L] = 0\]

The wave function can be written as

\[\psi_{l,m}(\bm r) = R_l(r) \, \Ylm\]

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

The eigenvalues of the Hamiltonian can be labeled by the two quantum numbers \(l\) and \(n'\), 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 \(n' + l + 1\), 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 \(\bm{\mu}\), the corresponding observable \(\hatb{\mu}\) is proportional to \(\hatb{L}\), and we call the factor gyromagnetic ratio \(\gamma\)

\[\hatb{\mu} = \gamma \hatb{L}\]

Spin

A spin observable \(\hatb{S}\) acts in the 2-dimensional Hilbert space of spin 1/2 denoted as \(\mathcal{E}_\rm{spin}\) and obeys the commutation relation of an angular momentum:

\[\hatb{S} \times \hatb{S} = i \hbar \hatb{S}\]

with each of the observables \(S_i\) having eigenvalues \(\pm \hbar / 2\). The observable \(\hatsup{S}{2}\) has one eigenvalue \(3 \hbar^2 / 4\).

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

\[\hatb{S} = \frac{\hbar}{2} \hatb \sigma\]

References

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