Angular Momentum

A.2 Angular Momentum

We omit hats on operators, and don’t always distinguish between operators and their position-space representation.

Orbital angular momentum

\begin{array}{rcl} L = r \times p \Rightarrow L_{z} = - i ℏ (x \frac{\partial}{\partial y} - y \frac{\partial}{\partial x}) etc \\ L^{2} = - ℏ^{2} (\frac{1}{sin θ} \frac{\partial}{\partial θ} (sin θ \frac{\partial}{\partial θ}) + \frac{1}{{sin}^{2} θ} \frac{\partial^{2}}{\partial ϕ^{2}}); L_{z} = - i ℏ \frac{\partial}{\partial ϕ} \\ \nabla^{2} = \nabla_{r}^{2} - \frac{1}{ℏ^{2} r^{2}} L^{2}; \nabla_{r}^{2} ψ \equiv \frac{1}{r^{2}} \frac{\partial}{\partial r} (r^{2} \frac{\partial ψ}{\partial r}) = \frac{1}{r} \frac{\partial^{2}}{\partial r^{2}} r ψ \end{array}

Eigenfunctions of $L^{2}$ and $L_{z}$ are spherical harmonics $Y_{l}^{m} (θ, ϕ)$ with eigenvalues $ℏ^{2} l (l + 1)$ and $ℏ m$ respectively; $l$ and $m$ are integers and must satisfy $l \geq 0$ and $m = - l, - l + 1, \dots l$ . In Dirac notation, the eigenstates are $| l m ⟩$ and $Y_{l}^{m} = ⟨ r | l m ⟩$ .

\begin{array}{l} Y_{0}^{0} (θ, ϕ) & = \sqrt{\frac{1}{4 π}} & Y_{1}^{\pm 1} (θ, ϕ) & = \mp \sqrt{\frac{3}{8 π}} sin θ e^{\pm i ϕ} \\ Y_{1}^{0} (θ, ϕ) & = \sqrt{\frac{3}{4 π}} cos θ & Y_{2}^{\pm 2} (θ, ϕ) & = \sqrt{\frac{15}{32 π}} {sin}^{2} θ e^{\pm 2 i ϕ} \\ Y_{2}^{\pm 1} (θ, ϕ) & = \mp \sqrt{\frac{15}{8 π}} sin θ cos θ e^{\pm i ϕ} & Y_{2}^{0} (θ, ϕ) & = \sqrt{\frac{5}{16 π}} (3 {cos}^{2} θ - 1) \end{array}

The Mathematica function to obtain them is SphericalHarmonicY[l,m, $θ$ , $ϕ$ ]. These are normalised and orthogonal:

\int {(Y_{l^{'}}^{m^{'}})}^{*} Y_{l}^{m} d Ω = δ_{l l^{'}} δ_{m m^{'}} where d Ω = sin θ d θ d ϕ

Rules obeyed by any angular momentum (eg $J$ can be replaced by $L$ or $S$ ):

\begin{array}{rcl} [J_{x}, J_{y}] = i ℏ J_{z} etc; [J^{2}, J_{i}] = 0; J_{\pm} \equiv J_{x} \pm i J_{y}; [J_{+}, J_{-}] = 2 ℏ J_{z}; [J_{z}, J_{\pm}] = \pm ℏ J_{\pm} \\ J^{2} = J_{x}^{2} + J_{y}^{2} + J_{z}^{2} = \frac{1}{2} (J_{+} J_{-} + J_{-} J_{+}) + J_{z}^{2} = J_{+} J_{-} + J_{z}^{2} - ℏ J_{z} \\ J^{2} | j, m ⟩ = ℏ^{2} j (j + 1) | j, m ⟩; J_{z} | j, m ⟩ = ℏ m | j, m ⟩; J_{\pm} | j, m ⟩ = ℏ \sqrt{(j \mp m) (j \pm m + 1)} | j, m \pm 1 ⟩ \end{array}

In the last line $j$ and $m$ must be integer or half-integer, and $m = - j, - j + 1, \dots j$ .

For the special case of a spin- $\frac{1}{2}$ particle (such as a proton, neutron or electron), the eigenstates of $S^{2}$ and $S_{z}$ are $| \frac{1}{2}, \frac{1}{2} ⟩$ and $| \frac{1}{2}, - \frac{1}{2} ⟩$ , often simply written $| \frac{1}{2} ⟩$ and $| - \frac{1}{2} ⟩$ or even $| ↑ ⟩$ and $| ↓ ⟩$ ; then $S^{2} | \pm \frac{1}{2} ⟩ = \frac{3}{4} ℏ^{2}$ and $S_{z} | \pm \frac{1}{2} ⟩ = \pm \frac{1}{2} ℏ | \pm \frac{1}{2} ⟩$ . In this basis, with $| \frac{1}{2} ⟩ \equiv (\binom{1}{0})$ and $| - \frac{1}{2} ⟩ \equiv (\binom{0}{1})$ , the components of the spin operator are given by ${\hat{S}}_{i} = \frac{1}{2} ℏ σ_{i}$ , where $σ_{i}$ are the Pauli matrices

σ_{x} = (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}) σ_{y} = (\begin{matrix} 0 & - i \\ i & 0 \end{matrix}) σ_{z} = (\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix})

If a sytem has two contributions to its angular momentum, with operators $J_{1}$ and $J_{2}$ and eigenstates $| j_{1} m_{1} ⟩$ and $| j_{2} m_{2} ⟩$ , the total angular momentum operator is $J = J_{1} + J_{2}$ . The quantum number $j$ of the combined system satisﬁes $j_{1} + j_{2} \geq j \geq |j_{1} - j_{2}|$ , and $m = m_{1} + m_{2}$ . Since enumerating states by ${m_{1}, m_{2}}$ gives $(2 j_{1} + 1) (2 j_{2} + 1)$ possible states, the number must be unchanged in the ${j, m}$ basis, which is veriﬁed as follows: labelling such that $j_{2} > j_{1}$ we have

\sum_{j = j_{2} - j_{1}}^{j_{2} + j_{1}} 2 j + 1 = ((j_{2} + j_{1}) (j_{2} + j_{1} + 1) - (j_{2} - j_{1}) (j_{2} - j_{1} - 1)) + (j_{2} + j_{1}) - (j_{2} - j_{1}) + 1 = (2 j_{1} + 1) (2 j_{2} + 1)

Depending on basis, we write the states either as $| j_{1} m_{1} ⟩ \otimes | j_{2} m_{2} ⟩$ or $| j_{1}, j_{2}; j m ⟩$ , and they must be linear combinations of each other as both span the space:

\begin{array}{rcl} | j_{1}, j_{2}; j m ⟩ = \sum_{m_{1} m_{2}} ⟨ j_{1} m_{1}; j_{2} m_{2} | j m ⟩ (| j_{1} m_{1} ⟩ \otimes | j_{2} m_{2} ⟩) and \\ | j_{1} m_{1} ⟩ \otimes | j_{2} m_{2} ⟩ = \sum_{j m} ⟨ j_{1} m_{1}; j_{2} m_{2} | j m ⟩ | j_{1}, j_{2}; j m ⟩ \end{array}

where the numbers denoted by $⟨ j_{1} m_{1}; j_{2} m_{2} | j m ⟩$ are called Clebsch-Gordan coeﬃcients; they vanish unless $j_{1} + j_{2} \geq j \geq |j_{1} - j_{2}|$ , and $m = m_{1} + m_{2}$ . These are tabulated in various places including the Particle Data Group site (see here for examples of how to use them); the Mathematica function to obtain them is ClebschGordan[ ${j_{1}, m_{1}}, {j_{2}, m_{2}}, {j, m}$ ]. There is also an on-line calculator at Wolfram Alpha which is simple to use if you only have a few to calculate. We use the “Condon-Shortley” phase convention, which is the most common; in this convention they are real which is why we have not written ${⟨ j_{1} m_{1}; j_{2} m_{2} | j m ⟩}^{*}$ in the second line above.

As an example we list the states arising from coupling angular momenta 1 and $\frac{1}{2}$ (as in p-wave states of the hydrogen atom):

\begin{array}{rcl} | 1, \frac{1}{2}; \frac{1}{2} \frac{1}{2} ⟩ & = & \sqrt{\frac{2}{3}} | 1 1 ⟩ \otimes | \frac{1}{2} - \frac{1}{2} ⟩ - \sqrt{\frac{1}{3}} | 1 0 ⟩ \otimes | \frac{1}{2} \frac{1}{2} ⟩ \\ | 1, \frac{1}{2}; \frac{1}{2} - \frac{1}{2} ⟩ & = & \sqrt{\frac{1}{3}} | 1 0 ⟩ \otimes | \frac{1}{2} - \frac{1}{2} ⟩ - \sqrt{\frac{2}{3}} | 1 - 1 ⟩ \otimes | \frac{1}{2} \frac{1}{2} ⟩ \\ | 1, \frac{1}{2}; \frac{3}{2} \frac{3}{2} ⟩ & = & | 1 1 ⟩ \otimes | \frac{1}{2} \frac{1}{2} ⟩ \\ | 1, \frac{1}{2}; \frac{3}{2} \frac{1}{2} ⟩ & = & \sqrt{\frac{1}{3}} | 1 1 ⟩ \otimes | \frac{1}{2} - \frac{1}{2} ⟩ + \sqrt{\frac{2}{3}} | 1, 0 ⟩ \otimes | \frac{1}{2} \frac{1}{2} ⟩ \\ | 1, \frac{1}{2}; \frac{3}{2} - \frac{1}{2} ⟩ & = & \sqrt{\frac{2}{3}} | 1 0 ⟩ \otimes | \frac{1}{2} - \frac{1}{2} ⟩ + \sqrt{\frac{1}{3}} | 1 - 1 ⟩ \otimes | \frac{1}{2} \frac{1}{2} ⟩ \\ | 1, \frac{1}{2}; \frac{3}{2} - \frac{3}{2} ⟩ & = & | 1 - 1 ⟩ \otimes | \frac{1}{2} - \frac{1}{2} ⟩ \end{array}

Somewhat more generally, the coupling of $l$ and $\frac{1}{2}$ to give $j = l \pm \frac{1}{2}$ is

| l, \frac{1}{2}; l \pm \frac{1}{2} m ⟩ = \sqrt{\frac{l \mp m + \frac{1}{2}}{2 l + 1}} | l m + \frac{1}{2} ⟩ \otimes | \frac{1}{2} - \frac{1}{2} ⟩ \pm \sqrt{\frac{l \pm m + \frac{1}{2}}{2 l + 1}} | l m - \frac{1}{2} ⟩ \otimes | \frac{1}{2} \frac{1}{2} ⟩ .

(The following material is not revision, but is asserted without proof. See Shankar 15.3 for a partial proof.)
Similarly we can write expressions for products of spherical harmonics:

Y_{k}^{q} Y_{l}^{m} = \sum_{l^{'} m^{'}} f (k, l, l^{'}) ⟨ k q; l m | l^{'} m^{'} ⟩ Y_{l^{'}}^{m^{'}} \Rightarrow \int {(Y_{l^{'}}^{m^{'}})}^{*} Y_{k}^{q} Y_{l}^{m} d Ω = f (k, l, l^{'}) ⟨ k q; l m | l^{'}, m^{'} ⟩

where $f (k, l, l^{'}) = \sqrt{1 ∕ 4 π} \sqrt{(2 l + 1) (2 k + 1) ∕ (2 l^{'} + 1)} ⟨ k 0; l 0 | l^{'} 0 ⟩$ . The factor $⟨ k 0; l 0 | l^{'} 0 ⟩$ is the one which vanishes unless parity is conserved, ie unless $k + l$ and $l^{'}$ are both odd or both even.

If an operator $Ω$ is a scalar, $[J_{i}, Ω] = 0$ . If $V$ is a triplet of operators which form a vector, then

\begin{array}{rcl} [J_{x}, V_{y}] = i ℏ V_{z}; \Rightarrow [J_{z}, V_{m}] = m ℏ V_{m}, [J_{\pm}, V_{m}] = ℏ \sqrt{(1 \mp m) (2 \pm m)} V_{m \pm 1} \\ where V_{\pm 1} = \mp \sqrt{\frac{1}{2}} (V_{x} \pm V_{y}), V_{0} \equiv V_{z} \end{array}

The operator $r$ is a vector operator which obeys these rules. Its components in the spherical basis are $\sqrt{4 π} r Y_{1}^{m}$ .

The following rules are obeyed by the matrix elements of a vector operator, by the Wigner-Eckart theorem

⟨ j^{'} m^{'} | V_{q} | j m ⟩ = ⟨ j^{'} ∥V∥ j ⟩ ⟨ j^{'} m^{'}; 1 q | j m ⟩

where $⟨ j^{'} ∥V∥ j ⟩$ is called the reduced matrix element and is independent of the $m$ , $m^{'}$ and $q$ . The Clebsch-Gordan coeﬃcient will vanish unless $j + 1 \geq j^{'} \geq |j - 1|$ , in other words operating with a vector operator is like adding one unit of angular momentum to the system. This is the origin of electric dipole selection rules for angular momentum. (In the rule for combining spherical harmonics the factor $f (k, l, l^{'})$ is a reduced matrix element.)

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