The Physics of Quantum Mechanics

James Binney, David Skinner

Chapter 7 Angular momentum - all with Video Answers

Educators

Chapter Questions

Problem 1

Show that $\left\langle j, j\left|J_{x}\right| j, j\right\rangle=\left\langle j, j\left|J_{y}\right| j, j\right\rangle=0$ and that $\langle j, j|\left(J_{x}^{2}+\right.$ $\left.J_{y}^{2}\right)|j, j\rangle=j$. Discuss the implications of these results for the uncertainty in the orientation of the classical angular momentum vector $\mathbf{J}$ for both small and large values of $j$.

Suzanne W.

Numerade Educator

06:45

Problem 2

In the rotation spectrum of ${ }^{12} \mathrm{C}^{16} \mathrm{O}$ the line arising from the transition $l=4 \rightarrow 3$ is at $461.04077 \mathrm{GHz}$, while that arising from $l=36 \rightarrow$ 35 is at $4115.6055 \mathrm{GHz}$. Show from these data that in a non-rotating CO molecule the intra-nuclear distance is $s \simeq 0.113 \mathrm{~nm}$, and that the electrons provide a spring between the nuclei that has force constant $\sim 1904 \mathrm{Nm}^{-1}$. Hence show that the vibrational frequency of CO should lie near $6.47 \times 10^{13} \mathrm{~Hz}$ (measured value is $6.43 \times 10^{13} \mathrm{~Hz}$ ). Hint: show from classical mechanics that the distance of $\mathrm{O}$ from the centre of mass is $\frac{3}{7} s$ and that the molecule's moment of inertia is $\frac{48}{7} m_{\mathrm{p}} s^{2}$. Recall also the classical relation $L=I \omega$.

Susan Hallstrom

Numerade Educator

01:49

Problem 3

Show that $L_{i}$ commutes with $\mathbf{x} \cdot \mathbf{p}$ and thus also with scalar functions of $\mathbf{x}$ and $\mathbf{p}$.

William Mead

Numerade Educator

03:42

Problem 4

Write down the expression for the commutator $\left[\sigma_{i}, \sigma_{j}\right]$ of two Pauli matrices. Show that the anticommutator of two Pauli matrices is
$$
\left\{\sigma_{i}, \sigma_{j}\right\}=2 \delta_{i j}
$$

Lucas Finney

Numerade Educator

01:29

Problem 5

Let $\mathbf{n}$ be any unit vector and $\boldsymbol{\sigma}=\left(\sigma_{x}, \sigma_{y}, \sigma_{z}\right)$ be the vector whose components are the Pauli matrices. Why is it physically necessary that $\mathbf{n} \cdot \boldsymbol{\sigma}$ satisfy $(\mathbf{n} \cdot \boldsymbol{\sigma})^{2}=I$, where $I$ is the $2 \times 2$ identity matrix? Let $\mathbf{m}$ be a unit vector such that $\mathbf{m} \cdot \mathbf{n}=0$. Why do we require that the commutator $[\mathbf{m} \cdot \boldsymbol{\sigma}, \mathbf{n} \cdot \boldsymbol{\sigma}]=2 \mathrm{i}(\mathbf{m} \times \mathbf{n}) \cdot \boldsymbol{\sigma} ?$ Prove that these relations follow from the algebraic properties of the Pauli matrices. You should be able to show that $[\mathbf{m} \cdot \boldsymbol{\sigma}, \mathbf{n} \cdot \boldsymbol{\sigma}]=2 \mathrm{i}(\mathbf{m} \times \mathbf{n}) \cdot \boldsymbol{\sigma}$ for any two vectors $\mathbf{n}$ and $\mathbf{m}$.

James Kiss

Numerade Educator

07:17

Problem 6

Let $\mathbf{n}$ be the unit vector in the direction with polar coordinates $(\theta, \phi) .$ Write down the matrix $\mathbf{n} \cdot \boldsymbol{\sigma}$ and find its eigenvectors. Hence show that the state of a spin-half particle in which a measurement of the component of spin along $\mathbf{n}$ is certain to yield $\frac{1}{2} \hbar$ is
$$
|+, \mathbf{n}\rangle=\sin (\theta / 2) \mathrm{e}^{\mathrm{i} \phi / 2}|-\rangle+\cos (\theta / 2) \mathrm{e}^{-\mathrm{i} \phi / 2}|+\rangle
$$
where $|\pm\rangle$ are the states in which $\pm \frac{1}{2}$ is obtained when $s_{z}$ is measured. Obtain the corresponding expression for $|-, \mathbf{n}\rangle$. Explain physically why the amplitudes in (7.171) have modulus $2^{-1 / 2}$ when $\theta=\pi / 2$ and why one of the amplitudes vanishes when $\theta=\pi$.

Mahnoor Amin

Numerade Educator

07:17

Problem 7

For a spin-half particle at rest, the rotation operator $\mathbf{J}$ is equal to the spin operator $\mathbf{S}$. Use the result of Problem $7.4$ to show that in this case the rotation operator $U(\boldsymbol{\alpha}) \equiv \exp (-\mathrm{i} \boldsymbol{\alpha} \cdot \mathbf{J})$ is
$$
U(\boldsymbol{\alpha})=I \cos \left(\frac{\alpha}{2}\right)-\mathrm{i} \hat{\boldsymbol{\alpha}} \cdot \boldsymbol{\sigma} \sin \left(\frac{\alpha}{2}\right)
$$
where $\hat{\boldsymbol{\alpha}}$ is the unit vector parallel to $\boldsymbol{\alpha}$. Comment on the value this gives for $U(\boldsymbol{\alpha})$ when $\alpha=2 \pi$.

Mahnoor Amin

Numerade Educator

05:50

Problem 8

Write down the $3 \times 3$ matrix that represents $S_{x}$ for a spin-one system in the basis in which $S_{z}$ is diagonal (i.e., the basis states are $|0\rangle$ and $|\pm\rangle$ with $S_{z}|+\rangle=|+\rangle$, etc. $)$
A beam of spin-one particles emerges from an oven and enters a Stern-Gerlach filter that passes only particles with $J_{z}=\hbar$. On exiting this filter, the beam enters a second filter that passes only particles with $J_{x}=\hbar$, and then finally it encounters a filter that passes only particles with $J_{z}=-\hbar$. What fraction of the particles stagger right through?

Mahnoor Amin

Numerade Educator

01:29

Problem 9

Repeat the analysis of Problem $7.8$ for spin-one particles coming on filters aligned successively along $+z, 45^{\circ}$ from $z$ towards $x$ [i.e. along $(1,0,1)]$, and along $x$.
Use classical electromagnetic theory to determine the outcome in the case that the spin-one particles were photons and the filters were Polaroid. Why do you get a different answer?

Dominador Tan

Numerade Educator

05:41

Problem 10

A system that has spin momentum $\sqrt{6} \hbar$ is rotated through an angle $\phi$ around the $z$-axis. Write down the $5 \times 5$ matrix that updates the amplitudes $a_{m}$ that $S_{z}$ will take the value $m$.

Shoukat Ali

Other Schools

08:06

Problem 11

Justify physically the claim that the Hamiltonian of a particle that precesses in a magnetic field $\mathbf{B}$ can be written
$$
H=-2 \mu \mathbf{s} \cdot \mathbf{B} \text {. }
$$
In a coordinate system oriented such that the $z$-axis is parallel to $\mathbf{B}$, a proton is initially in the eigenstate $|+, x\rangle$ of $s_{x}$. Obtain expressions for the expectation values of $s_{x}$ and $s_{y}$ at later times. Explain the physical content of your expressions.
Bearing in mind that a rotating magnetic field must be a source of radiation, do you expect your expressions to remain valid to arbitrarily late times? What really happens in the long run?

Khoobchandra Agrawal

Numerade Educator

02:17

Problem 12

Show that a classical top with spin angular momentum $\mathbf{S}$ which is subject to a torque $\mathbf{G}=\mu \mathbf{S} \times \mathbf{B} /|\mathbf{S}|$ precesses at angular velocity $\boldsymbol{\omega}=$ $\mu \mathbf{B} /|\mathbf{S}|$. Explain the relevance of this calculation to magnetic resonance imaging in general and equation (7.70b) in particular.

Chai Santi

Numerade Educator

05:58

Problem 13

Write a computer program that determines the amplitudes $a_{m}$ in
$$
|\mathbf{n} ; s, s\rangle=\sum_{m=-s}^{s} a_{m}|s, m\rangle
$$
where $\mathbf{n}=(\sin \theta, 0, \cos \theta)$ with $\theta$ any angle and $|\mathbf{n} ; s, s\rangle$ is the ket that solves the equation $(\mathbf{n} \cdot \mathbf{S})|\mathbf{n} ; s, s\rangle=s|\mathbf{n} ; s, s\rangle . \quad$ Explain physically the nature of this state.
Use your $a_{m}$ to evaluate the expectation values $\left\langle S_{x}\right\rangle$ and $\left\langle S_{x}^{2}\right\rangle$ for this state and hence show that the rms fluctuation in measurements of $S_{x}$ will be $\sqrt{s / 2} \cos \theta$.

Khoobchandra Agrawal

Numerade Educator

13:39

Problem 14

We have that
$$
L_{+} \equiv L_{x}+\mathrm{i} L_{y}=\mathrm{e}^{\mathrm{i} \phi}\left(\frac{\partial}{\partial \theta}+\mathrm{i} \cot \theta \frac{\partial}{\partial \phi}\right)
$$
From the Hermitian nature of $L_{z}=-\mathrm{i} \partial / \partial \phi$ we infer that derivative operators are anti-Hermitian. So using the rule $(A B)^{\dagger}=B^{\dagger} A^{\dagger}$ on equation (7.174), we infer that
$$
L_{-} \equiv L_{+}^{\dagger}=\left(-\frac{\partial}{\partial \theta}+\mathrm{i} \frac{\partial}{\partial \phi} \cot \theta\right) \mathrm{e}^{-\mathrm{i} \phi}
$$
This argument and the result it leads to is wrong. Obtain the correct result by integrating by parts $\int \mathrm{d} \theta \sin \theta \int \mathrm{d} \phi\left(f^{*} L_{+} g\right)$, where $f$ and $g$ are arbitrary functions of $\theta$ and $\phi .$ What is the fallacy in the given argument?

Samuel Smith

Numerade Educator

01:02

Problem 15

By writing $\hbar^{2} L^{2}=(\mathbf{x} \times \mathbf{p}) \cdot(\mathbf{x} \times \mathbf{p})=\sum_{i j k l m} \epsilon_{i j k} x_{j} p_{k} \epsilon_{i l m} x_{l} p_{m}$ show that
$$
p^{2}=\frac{\hbar^{2} L^{2}}{r^{2}}+\frac{1}{r^{2}}\left\{(\mathbf{r} \cdot \mathbf{p})^{2}-\mathrm{i} \hbar \mathbf{r} \cdot \mathbf{p}\right\}
$$
By showing that $\mathbf{p} \cdot \hat{\mathbf{r}}-\hat{\mathbf{r}} \cdot \mathbf{p}=-2 \mathrm{i} \hbar / r$, obtain $\mathbf{r} \cdot \mathbf{p}=r p_{r}+\mathrm{i} \hbar .$ Hence obtain
$$
p^{2}=p_{r}^{2}+\frac{\hbar^{2} L^{2}}{r^{2}}
$$
Give a physical interpretation of one over $2 m$ times this equation.

Nick Johnson

Numerade Educator

07:45

Problem 16

The angular part of a system's wavefunction is
$$
\langle\theta, \phi \mid \psi\rangle \propto\left(\sqrt{2} \cos \theta+\sin \theta \mathrm{e}^{-\mathrm{i} \phi}-\sin \theta \mathrm{e}^{\mathrm{i} \phi}\right) .
$$
What are the possible results of measurement of (a) $L^{2}$, and (b) $L_{z}$, and their probabilities? What is the expectation value of $L_{z} ?$

Nathan Silvano

Numerade Educator

03:12

Problem 17

A system's wavefunction is proportional to $\sin ^{2} \theta \mathrm{e}^{2 \mathrm{i} \phi}$. What are the possible results of measurements of (a) $L_{z}$ and (b) $L^{2} ?$

Ajay Singhal

Numerade Educator

01:42

Problem 18

A system's wavefunction is proportional to $\sin ^{2} \theta .$ What are the possible results of measurements of (a) $L_{z}$ and (b) $L^{2} ?$ Give the probabilities of each possible outcome.

Lottie Adams

Numerade Educator

01:23

Problem 19

Consider a stationary state $|E, l\rangle$ of a free particle of mass $m$ that has angular-momentum quantum number $l$. Show that $H_{l}|E, l\rangle=E|E, l\rangle$, where
$$
H_{l} \equiv \frac{1}{2 m}\left(p_{r}^{2}+\frac{l(l+1) \hbar^{2}}{r^{2}}\right)
$$
Give a physical interpretation of the two terms in the big bracket. Show that $H_{l}=A_{l}^{\dagger} A_{l}$, where
$$
A_{l} \equiv \frac{1}{\sqrt{2 m}}\left(\mathrm{i} p_{r}-\frac{(l+1) \hbar}{r}\right)
$$
Show that $\left[A_{l}, A_{l}^{\dagger}\right]=H_{l+1}-H_{l} .$ What is the state $A_{l}|E, l\rangle ?$ Show that for $E>0$ there is no upper bound on the angular momentum. Interpret this result physically.

Chai Santi

Numerade Educator

01:23

Problem 20

Show that $\left[J_{i}, L_{j}\right]=\mathrm{i} \sum_{k} \epsilon_{i j k} L_{k}$ and $\left[J_{i}, L^{2}\right]=0$ by eliminating $L_{i}$ using its definition $\mathbf{L}=\hbar^{-1} \mathbf{x} \times \mathbf{p}$, and then using the commutators of $J_{i}$ with $\mathrm{x}$ and $\mathbf{p}$.

Carson Merrill

Numerade Educator

06:35

Problem 21

In this problem you show that many matrix elements of the position operator $\mathbf{x}$ vanish when states of well-defined $l, m$ are used as basis states. These results will lead to selection rules for electric dipole radiation. First show that $\left[L^{2}, x_{i}\right]=\mathrm{i} \sum_{j k} \epsilon_{j i k}\left(L_{j} x_{k}+x_{k} L_{j}\right)$. Then show that $\mathbf{L} \cdot \mathbf{x}=0$ and using this result derive
$$
\left[L^{2},\left[L^{2}, x_{i}\right]\right]=\mathrm{i} \sum \epsilon_{j i k}\left(L_{j}\left[L^{2}, x_{k}\right]+\left[L^{2}, x_{k}\right] L_{j}\right)=2\left(L^{2} x_{i}+x_{i} L^{2}\right)
$$
By squeezing this equation between angular-momentum eigenstates $\langle l, m|$ and $\left|l^{\prime}, m^{\prime}\right\rangle$ show that
$$
0=\left\{\left(\beta-\beta^{\prime}\right)^{2}-2\left(\beta+\beta^{\prime}\right)\right\}\left\langle l, m\left|x_{i}\right| l^{\prime}, m^{\prime}\right\rangle
$$
where $\beta \equiv l(l+1)$ and $\beta^{\prime}=l^{\prime}\left(l^{\prime}+1\right)$. By equating the factor in front of $\left\langle l, m\left|x_{i}\right| l^{\prime}, m^{\prime}\right\rangle$ to zero, and treating the resulting equation as a quadratic equation for $\beta$ given $\beta^{\prime}$, show that $\left\langle l, m\left|x_{i}\right| l^{\prime}, m^{\prime}\right\rangle$ must vanish unless $l+l^{\prime}=$ 0 or $l=l^{\prime} \pm 1 .$ Explain why the matrix element must also vanish when $l=l^{\prime}=0$

Eduard Sanchez

Numerade Educator

04:20

Problem 22

Show that $l$ excitations can be divided amongst the $x, y$ or $z$ oscillators of a three-dimensional harmonic oscillator in $\left(\frac{1}{2} l+1\right)(l+1)$ ways. Verify in the case $l=4$ that this agrees with the number of states of well-defined angular momentum and the given energy.

Hafiz Shahzaib

Numerade Educator

09:36

Problem 23

Let
$$
A_{l} \equiv \frac{1}{\sqrt{2 m \hbar \omega}}\left(\mathrm{i} p_{r}-\frac{(l+1) \hbar}{r}+m \omega r\right)
$$
be the ladder operator of the three-dimensional harmonic oscillator and $|E, l\rangle$ be the stationary state of the oscillator that has energy $E$ and angular-momentum quantum number $l$. Show that if we write $A_{l}|E, l\rangle=$ $\alpha_{-}|E-\hbar \omega, l+1\rangle$, then $\alpha_{-}=\sqrt{\mathcal{L}-l}$, where $\mathcal{L}$ is the angular-momentum quantum number of a circular orbit of energy $E .$ Show similarly that if $A_{l-1}^{\dagger}|E, l\rangle=\alpha_{+}|E+\hbar \omega, l-1\rangle$, then $\alpha_{+}=\sqrt{\mathcal{L}-l+2}$

Robert Zaballa

Numerade Educator

02:34

Problem 24

Show that the probability distribution in radius of a particle that orbits in the three-dimensional harmonic oscillator potential on a circular orbit with angular-momentum quantum number $l$ peaks at $r / \ell=$ $\sqrt{2(l+1)}$, where
$$
\ell \equiv \sqrt{\frac{\hbar}{2 m \omega}}
$$
Derive the corresponding classical result.

Suzanne W.

Numerade Educator

01:49

Problem 25

A particle moves in the three-dimensional harmonic oscillator potential with the second largest angular-momentum quantum number possible at its energy. Show that the radial wavefunction is
$u_{1} \propto x^{l}\left(x-\frac{2 l+1}{x}\right) \mathrm{e}^{-x^{2} / 4}$ where $x \equiv r / \ell \quad$ with $\quad \ell \equiv \sqrt{\frac{\hbar}{2 m \omega}}$
How many radial nodes does this wavefunction have?

Ronald Prasad

Numerade Educator

02:19

Problem 26

A box containing two spin-one gyros $\mathrm{A}$ and $\mathrm{B}$ is found to have angular-momentum quantum numbers $j=2, m=1$. Determine the probabilities that when $J_{z}$ is measured for gyro $\mathrm{A}$, the values $m=\pm 1$ and 0 will be obtained.
What is the value of the Clebsch-Gordan coefficient $C(2,1 ; 1,1,1,0)$ ?

Dominador Tan

Numerade Educator

01:23

Problem 27

The angular momentum of a hydrogen atom in its ground state is entirely due to the spins of the electron and proton. The atom is in the state $|1,0\rangle$ in which it has one unit of angular momentum but none of it is parallel to the $z$-axis. Express this state as a linear combination of products of the spin states $|\pm, \mathrm{e}\rangle$ and $|\pm, \mathrm{p}\rangle$ of the proton and electron. Show that the states $|x \pm, \mathrm{e}\rangle$ in which the electron has well-defined spin along the $x$-axis are
$$
|x \pm, \mathrm{e}\rangle=\frac{1}{\sqrt{2}}(|+, \mathrm{e}\rangle \pm|-, \mathrm{e}\rangle)
$$
By writing
$$
|1,0\rangle=|x+, \mathrm{e}\rangle\langle x+, \mathrm{e} \mid 1,0\rangle+|x-, \mathrm{e}\rangle\langle x-, \mathrm{e} \mid 1,0\rangle
$$
express $|1,0\rangle$ as a linear combination of the products $|x \pm, \mathrm{e}\rangle|x \pm, \mathrm{p}\rangle .$ Explain the physical significance of your result.

Manik Pulyani

Numerade Educator

01:33

Problem 28

The interaction between neighbouring spin-half atoms in a crystal is described by the Hamiltonian
$$
H=K\left(\frac{\mathbf{S}^{(1)} \cdot \mathbf{S}^{(2)}}{a}-3 \frac{\left(\mathbf{S}^{(1)} \cdot \mathbf{a}\right)\left(\mathbf{S}^{(2)} \cdot \mathbf{a}\right)}{a^{3}}\right)
$$
where $K$ is a constant, $\mathbf{a}$ is the separation of the atoms and $\mathbf{S}^{(1)}$ is the first atom's spin operator. Explain what physical idea underlies this form of $H .$ Show that $S_{x}^{(1)} S_{x}^{(2)}+S_{y}^{(1)} S_{y}^{(2)}=\frac{1}{2}\left(S_{+}^{(1)} S_{-}^{(2)}+S_{-}^{(1)} S_{+}^{(2)}\right) .$ Show that the mutual eigenkets of the total spin operators $S^{2}$ and $S_{z}$ are also eigenstates of $H$ and find the corresponding eigenvalues.
At time $t=0$ particle 1 has its spin parallel to a, while the other particle's spin is antiparallel to a. Find the time required for both spins to reverse their orientations.

Dominador Tan

Numerade Educator