# 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
$$
\mathbf{S}_1 \cdot \mathbf{S}_2=S_{1 z} S_{2 z}+\frac{1}{2}\left(S_{1+} S_{2-}+S_{1-} S_{2+}\right)\tag{15.1.12}
$$



**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,
$$
H_{h f}=A \mathbf{S}_1 \cdot \mathbf{S}_2 \quad(A>0)\tag{15.1.22}
$$
(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:
$$
\begin{aligned}
E_{+} & =-\mathrm{Ry}+\frac{\hbar^2 A}{4} \\
E_{-} & =-\mathrm{Ry}-\frac{3 \hbar^2 A}{4}
\end{aligned}\tag{15.1.23}
$$
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.)
$$
\mathscr{H}_{h f} \cong \frac{\mu_e \cdot \mu_p}{a_0^3}
$$
Show that this implies that the constant in Eq. (15.1.22) is
$$
A \sim \frac{2 e}{2 m c} \frac{(5.6) e}{2 M c} \frac{1}{a_0^3}
$$
(where 5.6 is the $g$ factor for the proton), and that
$$
\Delta E=E_{+}-E_{-}=A \hbar^2
$$
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
$$
\langle\alpha j|\left|J_1\right|\left|\alpha^{\prime} j^{\prime}\right\rangle=\delta_{\alpha \alpha^{\prime}} \delta_{j j} \hbar[j(j+1)]^{1 / 2}
$$

(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
$$
\left\langle\alpha^{\prime} j m^{\prime}\right| \mathbf{J} \cdot \mathbf{A}|\alpha j m\rangle=c\left\langle\alpha^{\prime} j\mid |A| \mid \alpha j\right\rangle
$$
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
$$
\left\langle\alpha^{\prime} j m^{\prime}\right| A^q|\alpha j m\rangle=\frac{\left\langle\alpha^{\prime} j m\right| \mathbf{J} \cdot \mathbf{A}|\alpha j m\rangle}{\hbar^2 j(j+1)}\left\langle j m^{\prime}\right| J^q|j m\rangle
$$

**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
$$
\boldsymbol{\mu}=\gamma_1 \mathbf{J}_1+\gamma_2 \mathbf{J}_2
$$
In a state $\left|j m, j_1 j_2\right\rangle$ show, using Eq. (15.3.19), that
$$
\begin{aligned}
& \left\langle\mu_x\right\rangle=\left\langle\mu_y\right\rangle=0 \\
& \left\langle\mu_z\right\rangle=m \hbar\left[\frac{\gamma_1+\gamma_2}{2}+\frac{\left(\gamma_1-\gamma_2\right)}{2} \frac{j_1\left(j_1+1\right)-j_2\left(j_2+1\right)}{j(j+1)}\right]
\end{aligned}
$$

(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

Chapter 15 Addition of Angular Momenta

# Chapter 14 Spin
## 14.1 Introduction
## 14.2 What is the Nature of Spin?
## 14.3 Kinematics of Spin
**Exercise 14.3.1** Let us verify the above corollary explicitly. Take some spinor with components $\alpha=\rho_1 \mathrm{e}^{\mathrm{i} \phi_1}$ and $\beta=\rho_2 \mathrm{e}^{\mathrm{i} \phi_2}$. From $\langle\chi \mid \chi\rangle=1$, deduce that we can write $\rho_1=\cos (\theta / 2)$ and $\rho_2=\sin (\theta / 2)$ for some $\theta$. Next pull out a common phase factor so that the spinor takes the form in Eq. (14.3.28a). This verifies the corollary and also fixes $\hat{n}$.



**Exercise 14.3.2** (1) Show that the eigenvectors of $\boldsymbol{\sigma}\cdot\hat{n}$ are given by Eq. (14.3.28).

(2) Verify Eq. (14.3.29).

**Exercise 14.3.3** Using Eqs. (14.3.32) and (14.3.33) show that the Pauli matrices are traceless.

**Exercise 14.3.4** Derive Eq. (14.3.39) in two different ways.

(1) Write $\sigma_{1}\sigma_{2}$ in terms of $[\sigma_{i},\sigma_{j}]_{+}$ and $[\sigma_{i},\sigma_{j}]$.

(2) Use Eqs. (14.3.42) and (14.3.43).

**Exercise 14.3.5** Express the following matrix $M$ in terms of the Pauli matrices:
$$
M=\begin{pmatrix}
\alpha & \beta \\
\gamma & \delta
\end{pmatrix}
$$

**Exercise 14.3.6** (1) Argue that $|\hat{n},+\rangle=U[R(\phi \mathbf{k})] U[R(\theta \mathbf{j})]\left|s_z=\hbar / 2\right\rangle$.

(2) Verify by explicit calculation.

**Exercise 14.3.7** Express the following as linear combinations of the Pauli matrices and $I$:

(1) $\left(I+\mathrm{i} \sigma_x\right)^{1 / 2}$. (Relate it to half a certain rotation.)

(2) $\left(2 I+\sigma_x\right)^{-1}$.

(3) $\sigma_x^{-1}$.

**Exercise 14.3.8** (1) Show that any matrix that commutes with $\boldsymbol{\sigma}$ is a multiple of the unit matrix.

(2) Show that we cannot find a matrix that anticommutes with all three Pauli matrices. (If such a matrix exists, it must equal zero.)
## 14.4 Spin Dynamics
**Exercise 14.4.1** Show that if $H=-\gamma \mathbf{L} \cdot \mathbf{B}$, and $\mathbf{B}$ is position independent,
$$
\frac{\mathrm{d}\langle\mathbf{L}\rangle}{\mathrm{d} t}=\langle\boldsymbol{\mu} \times \mathbf{B}\rangle=\langle\boldsymbol{\mu}\rangle \times \mathbf{B}
$$

**Exercise 14.4.2** Derive (14.4.31) by studying Fig. 14.3.

**Exercise 14.4.3** We would like to study here the evolution of a state that starts out as $\binom{1}{0}$ and is subject to the $\mathbf{B}$ field given in Eq. (14.4.27). This state obeys
$$
\mathrm{i} \hbar \frac{\mathrm{d}}{\mathrm{d} t}|\psi(t)\rangle=H|\psi\rangle\tag{14.4.34}
$$
where $H=-\gamma \mathbf{S} \cdot \mathbf{B}$, and $\mathbf{B}$ is time dependent. Since classical reasoning suggests that in a frame rotating at frequency ($-\omega \mathbf{k}$) the Hamiltonian should be time independent and governed by $\mathbf{B}_r$ [Eq. (14.4.29)], consider the ket in the rotating frame, $\left|\psi_r(t)\right\rangle$, related to $|\psi(t)\rangle$ by a rotation angle $\omega t$:
$$
\left|\psi_r(t)\right\rangle=\mathrm{e}^{-\mathrm{i} \omega t S_z / \hbar}|\psi(t)\rangle\tag{14.4.35}
$$
Combine Eqs. (14.4.34) and (14.4.35) to derive Schrödinger's equation for $\left|\psi_r(t)\right\rangle$ in the $S_z$ basis and verify that the classical expectation is borne out. Solve for $\left|\psi_r(t)\right\rangle=U_r(t)\left|\psi_r(0)\right\rangle$ by computing $U_r(t)$, the propagator in the rotating frame. Rotate back to the lab and show that
$$
|\psi(t)\rangle \xrightarrow[S_{z}~\text{basis}]{ }\left[\begin{array}{c}
{\left[\cos \left(\frac{\omega_r t}{2}\right)+\mathrm{i} \frac{\omega_0-\omega}{\omega_r} \sin \left(\frac{\omega_r t}{2}\right)\right] \mathrm{e}^{+\mathrm{i} \omega t / 2}} \\
\frac{\mathrm{i} \gamma B}{\omega_r} \sin \left(\frac{\omega_r t}{2}\right) \mathrm{e}^{-\mathrm{i} \omega t / 2}
\end{array}\right]\tag{14.4.36}
$$
Compare this to the state $|\hat{\mathbf{n}},+\rangle$ and see what is happening to the spin for the case $\omega_0=\omega$.

Calculate $\left\langle\mu_z(t)\right\rangle$ and verify that it agrees with Eq. (14.4.31).

**Exercise 14.4.4** At $t=0$, an electron is in the state with $s_z=\hbar/2$. A steady field $\mathbf{B}=B \mathbf{i}$, $B=100 \mathrm{G}$, is turned on. How many seconds will it take for the spin to flip?

**Exercise 14.4.5** We would like to establish the validity of Eq. (14.4.26) when $\boldsymbol{\omega}$ and $\mathbf{B}_0$ are not parallel.

(1) Consider a vector $\mathbf{V}$ in the inertial (nonrotating) frame which changes by $\Delta \mathbf{V}$ in a time $\Delta t$. Argue, using the results from Exercise 12.4.3, that the change as seen in a frame rotating at an angular velocity $\boldsymbol{\omega}$, is $\Delta \mathbf{V}-\boldsymbol{\omega} \times \mathbf{V} \Delta t$. Obtain a relation between the time derivatives of $\mathbf{V}$ in the two frames.

(2) Apply this result to the case of $\mathbf{l}$ [Eq. (14.4.8)], and deduce the formula for the effective field in the rotating frame.

**Exercise 14.4.6** (A Density Matrix Problem)
(1) Show that the density matrix for an ensemble of spin-1/2 particles may be written as
$$
\rho=\frac{1}{2}(I+\mathbf{a} \cdot \boldsymbol{\sigma})
$$
where $\mathbf{a}$ is a $c$-number vector.

(2) Show that $\mathbf{a}$ is the mean polarization, $\langle\overline{\boldsymbol{\sigma}}\rangle$.

(3) An ensemble of electrons in a magnetic field $\mathbf{B}=B \mathbf{k}$, is in thermal equilibrium at temperature $T$. Construct the density matrix for this ensemble. Calculate $\langle\overline{\boldsymbol{\mu}}\rangle$.
## 14.5 Return of Orbital Degrees of Freedom
**Exercise 14.5.1** (1) Why is the coupling of the proton's intrinsic moment to B an order $m / M$ correction to Eq. (14.5.4)?

(2) Why is the coupling of its orbital motion an order $(m / M)^2$ correction? (You may reason classically in both parts.)

**Exercise 14.5.2** (1) Estimate the relative size of the level splitting in the $n=1$ state to the unperturbed energy of the $n=1$ state, when a field $\mathbf{B}=1000 \mathrm{kG}$ is applied.

(2) Recall that we have been neglecting the order $B^2$ term in $H$. Estimate its contribution in the $n=1$ state relative to the linear ($-\boldsymbol{\mu} \cdot \mathbf{B}$) term we have kept, by assuming the electron moves on a classical orbit of radius $a_0$. Above what $|\mathbf{B}|$ does it begin to be a poor approximation?

**Exercise 14.5.3** A beam of spin-1/2 particles moving along the $y$ axis goes through two collinear SG apparatuses, both with lower beams blocked. The first has its B field along the $z$ axis and the second has its $\mathbf{B}$ field along the $x$ axis (i.e., is obtained by rotating the first by an angle $\pi / 2$ about the $y$ axis). What fraction of particles leaving the first will exit the second? If a third filter that transmits only spin up along the $z$ axis is introduced, what fraction of particles leaving the first will exit the third? If the middle filter transmits both spins up and down (no blocking) the $x$ axis, but the last one transmits only spin down the $z$ axis, what fraction of particles leaving the first will leave the last?

**Exercise 14.5.4** A beam of spin-1 particles, moving along the $y$ axis, is incident on two collinear SG apparatuses, the first with $\mathbf{B}$ along the $z$ axis and the second with $\mathbf{B}$ along the $z^{\prime}$ axis, which lies in the $x-z$ plane at an angle $\theta$ relative to the $z$ axis. Both apparatuses transmit only the uppermost beams. What fraction leaving the first will pass the second?

Chapter 14 Spin

# Chapter 13 The Hydrogen Atom
## 13.1 The Eigenvalue Problem
**Exercise 13.1.1** Derive Eqs. (13.1.11) and (13.1.14) starting from Eqs. (13.1.8)-(13.1.10).



**Exercise 13.1.2** Derive the degeneracy formula, Eq. (13.1.18).

**Exercise 13.1.3** Starting from the recursion relation, obtain $\psi_{210}$ (normalized).

**Exercise 13.1.4** Recall from the last chapter [Eq. (12.6.19)] that as $r \rightarrow \infty$, $U_E \sim(r)^{m e^2 / \kappa \hbar^2} \mathrm{e}^{-\kappa r}$ in a Coulomb potential $V=-\mathrm{e}^2 / r$ $\left[\kappa=\left(2 m W / \hbar^2\right)^{1 / 2}\right]$. Show that this agrees with Eq. (13.1.26).

**Exercise 13.1.5** (Virial Theorem) Since $|n, l, m\rangle$ is a stationary state, $\langle\dot{\mathbf{\Omega}}\rangle=0$ for any $\mathbf{\Omega}$.\\ Consider $\mathbf{\Omega}=\mathbf{R} \cdot \mathbf{P}$ and use Ehrenfest's theorem to show that $\langle T\rangle=(-1 / 2)\langle V\rangle$ in the state $|n, l, m\rangle$.

## 13.2 The Degeneracy of the Hydrogen Spectrum
**Exercise 13.2.1** Let us see why the conservation of the Runge-Lenz vector $\mathbf{n}$ implies closed orbits.

(1) Express $\mathbf{n}$ in terms of $\mathbf{r}$ and $\mathbf{p}$ alone (get rid of $\mathbf{1}$).

(2) Since the particle is bound, it cannot escape to infinity. So, as we follow it from some arbitrary time onward, it must reach a point $\mathbf{r}_{\max}$ where its distance from the origin stops growing. Show that
$$
\mathbf{n}=\mathbf{r}_{\max}\left(2 E+\frac{\mathrm{e}^2}{r_{\max}}\right)
$$
## 13.3 Numerical Estimates and Comparison with Experiment
**Exercise 13.3.1** The pion has a range of $1~\mathrm{Fermi}=10^{-5} \text{\AA}$ as a mediator of nuclear force. Estimate its rest energy.

**Exercise 13.3.2** Estimate the de Broglie wavelength of an electron of kinetic energy $200~\mathrm{eV}$. (Recall $\lambda=2 \pi \hbar / p$.)
## 13.4 Multielectron Atoms and the Periodic Table
**Exercise 13.4.1** Show that if we ignore interelectron interactions, the energy levels of a multielectron atom go as $Z^2$. Since the Coulomb potential is $Z e / r$, why is the energy $\propto Z^2$?

**Exercise 13.4.2** Compare (roughly) the sizes of the uranium atom and the hydrogen atom. Assume levels fill in the order of increasing $n$, and that the nonrelativistic description holds. Ignore interelectron effects.

**Exercise 13.4.3** Visible light has a wavelength of approximately $5000 \text{\AA}$. Which of the series-Lyman, Balmer, Paschen-do you think was discovered first?

Chapter 13 The Hydrogen Atom

Chapter 12 Rotational Invariance and Angular Momentum

# Chapter 11 Symmetries and Their Consequences
## 11.1 Overview
## 11.2 Translational Invariance in Quantum Theory
**Exercise 11.2.1** Verify Eq. (11.2.11b).



**Exercise 11.2.2** Using $T^{\dagger}(\varepsilon)T(\varepsilon)=I$ to order $\varepsilon$, deduce that $G^{\dagger}=G$.

**Exercise 11.2.3** Recall that we found the finite rotation transformation from the infinitesimal one, by solving differential equations (Section 2.8). Verify that if, instead, you relate the transformed coordinates $\overline{x}$ and $\overline{y}$ to $x$ and $y$ by the infinite string of Poisson brackets, you get the same result, $\overline{x}=x \cos \theta-y \sin \theta$, etc. (Recall the series for $\sin \theta$, etc.)
## 11.3 Time Translational Invariance
## 11.4 Parity Invariance
**Exercise 11.4.1** Prove that if $[\Pi, H]=0$, a system that starts out in a state of even/odd parity maintains its parity. (Note that since parity is a discrete operation, it has no associated conservation law in classical mechanics.)

**Exercise 11.4.2** A particle is in a potential
$$
V(x)=V_0 \sin (2 \pi x / a)
$$
which is invariant under the translations $x \rightarrow x+m a$, where $m$ is an integer. Is momentum conserved? Why not?

**Exercise 11.4.3** You are told that in a certain reaction, the electron comes out with its spin always parallel to its momentum. Argue that parity is violated.

**Exercise 11.4.4** We have treated parity as a mirror reflection. This is certainly true in one dimension, where $x \rightarrow-x$ may be viewed as the effect of reflecting through a (point) mirror at the origin. In higher dimensions when we use a plane mirror (say lying on the $x-y$ plane), only one ($z$) coordinate gets reversed, whereas the parity transformation reverses all three coordinates.

Verify that reflection on a mirror in the $x-y$ plane is the same as parity followed by $180^{\circ}$ rotation about the $z$ axis. Since rotational invariance holds for weak interactions, noninvariance under mirror reflection implies noninvariance under parity.
## 11.5 Time-Reversal Symmetry

Chapter 15 Addition of Angular Momenta

15.1 A Simple Example

Chapter 14 Spin

14.1 Introduction

14.2 What is the Nature of Spin?

14.3 Kinematics of Spin

Chapter 13 The Hydrogen Atom

13.1 The Eigenvalue Problem

Chapter 12 Rotational Invariance and Angular Momentum

12.1 Translations in Two Dimensions

Chapter 11 Symmetries and Their Consequences

11.1 Overview

11.2 Translational Invariance in Quantum Theory