Chapter 15 Addition of Angular Momenta

15.1 A Simple Example

Exercise 15.1.1 Derive Eqs. (15.1.10) and (15.1.11). It might help to use

Exercise 15.1.2 In addition to the Coulomb interaction, there exists another, called the hyperfine interaction, between the electron and proton in the hydrogen atom. The Hamiltonian describing this interaction, which is due to the magnetic moments of the two particles is,

(This formula assumes the orbital state of the electron is $|1,0,0\rangle$.) The total Hamiltonian is thus the Coulomb Hamiltonian plus $H_{h f}$.

\noindent (1) Show that $H_{h f}$ splits the ground state into two levels:

and that corresponding states are triplets and singlet, respectively.

(2) Try to estimate the frequency of the emitted radiation as the atom jumps from the triplet to the singlet. To do so, you may assume that the electron and proton are two dipoles $\mu_e$ and $\mu_p$ separated by a distance $a_0$, with an interaction energy of the order (The description here is oversimplified; both $\mathscr{H}_{hf}$ and $H_{hf}$ are rather tricky to derive. Our aim is just to estimate $|A|$ and not to get into its precise origin.)

Show that this implies that the constant in Eq. (15.1.22) is

(where 5.6 is the $g$ factor for the proton), and that

is a correction of order $(m / M) \alpha^2$ relative to the ground-state energy. Estimate that the frequency of emitted radiation is a few tens of centimeters, using the mnemonics from Chapter 13. The measured value is $21.4~\mathrm{cm}$. This radiation, called the $21-\mathrm{cm}$ line, is a way to detect hydrogen in other parts of the universe.

(3) Estimate the probability ratio $P(\text{triplet})/P(\text{singlet})$ of hydrogen atoms in thermal equilibrium at room temperature.

15.2 The General Problem

Exercise 15.2.1 (1) Verify that $\left|j_1 j_1, j_2 j_2\right\rangle$ is indeed a state of $j=j_1+j_2$ by letting $J^2=J_1^2+J_2^2+2 J_{1 z} J_{2 z}+J_{1+} J_{2-}+J_{1-} J_{2+}$ act on it.

(2) Verify that the right-hand side of Eq. (15.2.8) indeed has angular momentum $j=j_1+j_2-1$.

Exercise 15.2.2 Find the CG coefficients of

(1) $\dfrac{1}{2}\otimes 1=\dfrac{3}{2}\oplus \dfrac{1}{2}$

(2) $1\otimes 1=2\oplus 1\oplus 0$

Exercise 15.2.3 Argue that $\dfrac{1}{2}\otimes\dfrac{1}{2}\otimes\dfrac{1}{2}=\dfrac{3}{2}\oplus\dfrac{1}{2}\oplus\dfrac{1}{2}$.

Exercise 15.2.4 Derive Eqs. (15.2.19) and (15.2.20).

Exercise 15.2.5 (1) Show that $\mathbb{P}_1=\dfrac{3}{4} I+\left(\mathbf{S}_1 \cdot \mathbf{S}_2\right) / \hbar^2$ and $\mathbb{P}_0=\dfrac{1}{4} I-\left(\mathbf{S}_1 \cdot \mathbf{S}_2\right) / \hbar^2$ are projection operators, i.e., obey $\mathbb{P}_i \mathbb{P}_j=\delta_{i j} \mathbb{P}_j$ [use Eq. (14.3.39)].

(2) Show that these project into the spin-1 and spin-0 spaces in $\dfrac{1}{2} \otimes \dfrac{1}{2}=1 \oplus 0$.

Exercise 15.2.6 Construct the project operators $\mathbb{P}_{\pm}$ for the $j=l \pm 1 / 2$ subspaces in the addition $\mathbf{L}+\mathbf{S}=\mathbf{J}$.

Exercise 15.2.7 Show that when we add $j_1$ to $j_1$, the states with $j=2 j_1$ are symmetric. Show that the states with $j=2 j_1=1$ are antisymmetric. (Argue for the symmetry of the top states and show that lowering does not change symmetry.) This pattern of alternating symmetry continues as $j$ decreases, but is harder to prove.

15.3 Irreducible Tensor Operators

Exercise 15.3.1 (1) Show that Eq. (15.3.11) follows from Eq. (15.3.10) when one considers infinitesimal rotations. (Hint: $D_{q^{\prime} q}^{(k)}=\left\langle kq^{\prime}\right| I-(\mathrm{i}\delta\boldsymbol{\theta} \cdot \mathbf{J}) / \hbar|k q\rangle$. Pick $\delta \boldsymbol{\theta}$ along, say, the $x$ direction and then generalize the result to the other directions.)

(2) Verify that the spherical tensor $V_1^q$ constructed out of $V$ as in Eq. (15.3.15) obeys Eq. (15.3.11).

Exercise 15.3.2 It is claimed that $\sum\limits_q (-1)^q S_k^q T_k^{(-q)}$ is a scalar operator.

(1) For $k=1$, verify that this is just $\mathbf{S} \cdot \mathbf{T}$.

(2) Prove it in general by considering its response to a rotation. [Hint: $D_{-m,-m^{\prime}}^{(j)}=(-1)^{m-m^{\prime}}\left(D_{m, m^{\prime}}^{(j)}\right)^*$.]

Exercise 15.3.3 (1) Using $\langle j j \mid j j, 10\rangle=[j /(j+1)]^{1 / 2}$ show that

(2) Using $\boldsymbol{J} \cdot \mathbf{A}=J_z A_z+\dfrac{1}{2}\left(J_{-} A_{+}+J_{+} A_{-}\right)$(where $A_{\pm}=A_x \pm i A_y$ ) argue that

where $c$ is a constant independent of $\alpha, \alpha^{\prime}$ and $\mathbf{A}$. Show that $c=\hbar[j(j+1)]^{1 / 2} \delta_{m, m^{\prime}}$.

(3) Using the above, show that

Exercise 15.3.4 (1) Consider a system whose angular momentum consists of two parts $\mathbf{J}_1$ and $\mathbf{J}_2$ and whose magnetic moment is

In a state $\left|j m, j_1 j_2\right\rangle$ show, using Eq. (15.3.19), that

(2) Apply this to the problem of a proton $(g=5.6)$ in a ${ }^2 P_{1 / 2}$ state and show that $\left\langle\mu_z\right\rangle=$ $\pm 0.26$ nuclear magnetons.

(3) For an electron in a ${ }^2 P_{1 / 2}$ state show that $\left\langle\mu_z\right\rangle=\pm \dfrac{1}{3}$ Bohr magnetons.

Exercise 15.3.5 Show that $\langle j m\mid T_k^q\mid j m\rangle=0$ if $k>2 j$.

15.4 Explanation of Some “Accidental” Degeneracies