Modern Quantum Mechanics

J. J. Sakurai, Jim Napolitano

Chapter 3 Theory of Angular Momentum - all with Video Answers

Educators

Chapter Questions

07:36

Problem 1

Use the specific form of $S_{x}$ given by (3.25) to evaluate (3.23) and show that $S_{x}$ rotates as expected through an angle $\phi$ about the $z$-axis.

Anthony Ramos

Numerade Educator

01:24

Find the eigenvalues and eigenvectors of $\sigma_{y}=\left(\begin{array}{cc}0 & -i \\ i & 0\end{array}\right)$. Suppose an electron is in the spin state $\left(\begin{array}{c}\alpha \\ \beta\end{array}\right)$. If $s_{y}$ is measured, what is the probability of the result $\hbar / 2 ?$

Anand Jangid

Numerade Educator

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Problem 3

Find, by explicit construction using Pauli matrices, the eigenvalues for the Hamiltonian
$$
H=-\frac{2 \mu}{\hbar} \mathbf{S} \cdot \mathbf{B}
$$
for a spin $\frac{1}{2}$ particle in the presence of a magnetic field $\mathbf{B}=B_{x} \hat{\mathbf{x}}+B_{y} \hat{\mathbf{y}}+B_{z} \hat{\mathbf{z}}$

Victor Salazar

Numerade Educator

02:58

Problem 4

Consider the $2 \times 2$ matrix defined by
$$
U=\frac{a_{0}+i \sigma \cdot \mathbf{a}}{a_{0}-i \sigma \cdot \mathbf{a}}
$$
where $a_{0}$ is a real number and a is a three-dimensional vector with real components.
a. Prove that $U$ is unitary and unimodular.
b. In general, a $2 \times 2$ unitary unimodular matrix represents a rotation in three dimensions. Find the axis and angle of rotation appropriate for $U$ in terms of $a_{0}, a_{1}, a_{2}$, and $a_{3} .$

Victor Salazar

Numerade Educator

08:06

Problem 5

The spin-dependent Hamiltonian of an electron-positron system in the presence of a uniform magnetic field in the $z$-direction can be written as
$$
H=A \mathbf{S}^{\left(e^{-}\right)} \cdot \mathbf{S}^{\left(e^{+}\right)}+\left(\frac{e B}{m c}\right)\left(S_{z}^{\left(e^{-}\right)}-S_{z}^{\left(e^{+}\right)}\right)
$$
Suppose the spin function of the system is given by $\chi_{+}^{\left(e^{-}\right)} \chi_{-}^{\left(e^{+}\right)} .$
a. Is this an eigenfunction of $H$ in the limit $A \rightarrow 0, e B / m c \neq 0$ ? If it is, what is the energy eigenvalue? If it is not, what is the expectation value of $H$ ?
b. Same problem when $e B / m c \rightarrow 0, A \neq 0$.

Khoobchandra Agrawal

Numerade Educator

05:50

Problem 6

Consider a spin 1 particle. Evaluate the matrix elements of
$$
S_{z}\left(S_{z}+\hbar\right)\left(S_{z}-\hbar\right) \quad \text { and } \quad S_{x}\left(S_{x}+\hbar\right)\left(S_{x}-\hbar\right) .
$$

Mahnoor Amin

Numerade Educator

02:08

Problem 7

Let the Hamiltonian of a rigid body be
$$
H=\frac{1}{2}\left(\frac{K_{1}^{2}}{I_{1}}+\frac{K_{2}^{2}}{I_{2}}+\frac{K_{3}^{2}}{I_{3}}\right),
$$
where $\mathbf{K}$ is the angular momentum in the body frame. From this expression obtain the Heisenberg equation of motion for $\mathbf{K}$ and then find Euler's equation of motion in the correspondence limit.

Bhavesh Kumar

Numerade Educator

17:10

Problem 8

Let $U=e^{i G_{3} \alpha} e^{i G_{2} \beta} e^{i G_{3} \gamma}$, where $(\alpha, \beta, \gamma)$ are the Eulerian angles. In order that $U$ represent a rotation $(\alpha, \beta, \gamma)$, what are the commutation rules satisfied by the $G_{k}$ ? Relate $\mathbf{G}$ to the angular-momentum operators.

Khoobchandra Agrawal

Numerade Educator

01:04

Problem 9

What is the meaning of the following equation:
$$
U^{-1} A_{k} U=\sum R_{k l} A_{l},
$$
where the three components of $\mathbf{A}$ are matrices? From this equation show that matrix elements $\left\langle m\left|A_{k}\right| n\right\rangle$ transform like vectors.

Fuzail Shakir

Numerade Educator

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Problem 10

Consider a sequence of Euler rotations represented by
$$
\begin{aligned}
\mathscr{D}^{(1 / 2)}(\alpha, \beta, \gamma) &=\exp \left(\frac{-i \sigma_{3} \alpha}{2}\right) \exp \left(\frac{-i \sigma_{2} \beta}{2}\right) \exp \left(\frac{-i \sigma_{3} \gamma}{2}\right) \\
&=\left(\begin{array}{cc}
e^{-i(a+\gamma) / 2} \cos \frac{\beta}{2} & -e^{-i(a-\gamma) / 2} \sin \frac{\beta}{2} \\
e^{i(\alpha-\gamma) / 2} \sin \frac{\beta}{2} & e^{i(\alpha+\gamma) / 2} \cos \frac{\beta}{2}
\end{array}\right)
\end{aligned}
$$
Because of the group properties of rotations, we expect that this sequence of operations is equivalent to a single rotation about some axis by an angle $\theta$. Find $\theta$.

Victor Salazar

Numerade Educator

06:22

Problem 11

Use the triangle inequality (1.147) and the definition (3.100) of the density operator $\rho$ to prove that $0 \leq \operatorname{Tr}\left(\rho^{2}\right) \leq 1$.

Rashmi Sinha

Numerade Educator

05:50

Problem 12

A large collection of spin $\frac{1}{2}$ particles is in a mixture of the two states $\left|S_{z} ;+\right\rangle$ and $\left|S_{y} ;-\right\rangle$. The fraction of particles in the state $\left|S_{z} ;+\right\rangle$ is $a$. Find the ensemble averages $\left[S_{x}\right],\left[S_{y}\right]$, and $\left[S_{z}\right]$ in terms of $a$. Confirm that your expression gives the answers you expect for $a=0$ and $a=1$.

Mahnoor Amin

Numerade Educator

03:54

Problem 13

a. Consider a pure ensemble of identically prepared spin $\frac{1}{2}$ systems. Suppose the expectation values $\left\langle S_{x}\right\rangle$ and $\left\langle S_{z}\right\rangle$ and the sign of $\left\langle S_{y}\right\rangle$ are known. Show how we may determine the state vector. Why is it unnecessary to know the magnitude of $\left\langle S_{y}\right\rangle$ ?
b. Consider a mixed ensemble of spin $\frac{1}{2}$ systems. Suppose the ensemble averages $\left[S_{x}\right],\left[S_{y}\right]$, and $\left[S_{z}\right]$ are all known. Show how we may construct the $2 \times 2$ density matrix that characterizes the ensemble.

Lazarus Arnau

Numerade Educator

10:42

Problem 14

Consider a one-dimensional simple harmonic oscillator with frequency $\omega$ and eigenstates $|0\rangle,|1\rangle,|2\rangle, \ldots$. A mixed ensemble is formed with equal parts of each of the three states
$|\alpha\rangle \equiv \frac{1}{\sqrt{2}}[|0\rangle+|1\rangle], \quad|\beta\rangle \equiv \frac{1}{\sqrt{2}}[|1\rangle+|2\rangle], \quad$ and $\quad|2\rangle .$
Find the density operator $\rho$ and calculate the ensemble average of the energy.

Mirza Aslam Beig

Numerade Educator

03:43

Problem 15

a. Prove that the time evolution of the density operator $\rho$ (in the Schrödinger picture) is given by
$$
\rho(t)=\mathscr{U}\left(t, t_{0}\right) \rho\left(t_{0}\right) \mathscr{U}^{\dagger}\left(t, t_{0}\right) .
$$
b. Suppose we have a pure ensemble at $t=0$. Prove that it cannot evolve into a mixed ensemble as long as the time evolution is governed by the Schrödinger equation.

Mehdi Hatefipour

Numerade Educator

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Problem 16

Consider an ensemble of spin 1 systems. The density matrix is now a $3 \times 3$ matrix. How many independent (real) parameters are needed to characterize the density matrix? What must we know in addition to $\left[S_{x}\right],\left[S_{y}\right]$, and $\left[S_{z}\right]$ to characterize the ensemble completely?

Nick Johnson

Numerade Educator

07:45

Problem 17

An angular-momentum eigenstate $\left|j, m=m_{\max }=j\right\rangle$ is rotated by an infinitesimal angle $\varepsilon$ about the $y$-axis. Without using the explicit form of the $d_{m^{\prime} m}^{(j)}$ function, obtain an expression for the probability for the new rotated state to be found in the original state up to terms of order $\varepsilon^{2}$.

Amit Srivastava

Numerade Educator

11:46

Problem 18

Show that the $3 \times 3$ matrices $G_{i}(i=1,2,3)$ whose elements are given by
$$
\left(G_{i}\right)_{j k}=-i \hbar \varepsilon_{i j k},
$$
where $j$ and $k$ are the row and column indices, satisfy the angular-momentum commutation relations. What is the physical (or geometric) significance of the transformation matrix that connects $G_{i}$ to the more usual $3 \times 3$ representations of the angular-momentum operator $J_{i}$ with $J_{3}$ taken to be diagonal? Relate your result to
$$
\mathbf{V} \rightarrow \mathbf{V}+\hat{\mathbf{n}} \delta \phi \times \mathbf{V}
$$
under infinitesimal rotations. (Note: This problem may be helpful in understanding the photon spin.)

Mahnoor Amin

Numerade Educator

11:46

Problem 19

a. Using the fact that $J_{x}, J_{y}, J_{z}$, and $J_{\pm} \equiv J_{x} \pm i J_{y}$ satisfy the usual angular-momentum commutation relations, prove that
$$
\mathbf{J}^{2}=J_{z}^{2}+J_{+} J_{-}-\hbar J_{z} .
$$
b. Using this result, or otherwise, derive the coefficient $c_{-}$that appears in
$$
J_{-}|j m\rangle=c_{-}\lfloor j, m-1\rangle .
$$

Mahnoor Amin

Numerade Educator

05:50

Problem 20

Construct the matrix representations of the operators $J_{x}$ and $J_{y}$ for a spin 1 system, in the $J_{z}$ basis, spanned by the kets $|+\rangle \equiv|1,1\rangle,|0\rangle \equiv|1,0\rangle$, and $|-\rangle \equiv|1,-1\rangle$. Use these matrices to find the three analogous eigenstates for each of the two operators $J_{x}$ and $J_{y}$ in terms of $|+\rangle,|0\rangle$, and $|-\rangle$.

Mahnoor Amin

Numerade Educator

04:20

Problem 21

Show that the orbital angular-momentum operator $\mathbf{L}$ commutes both with the operators $\mathbf{p}^{2}$ and $\mathbf{x}^{2}$, that is prove (3.264).

Hafiz Shahzaib

Numerade Educator

19:29

Problem 22

For the orbital angular-momentum operator $\mathbf{L}=\mathbf{x} \times \mathbf{p}$, derive (3.218), that is
$$
\left\langle\mathbf{x}^{\prime}\left|L_{z}\right| \alpha\right\rangle=-i \hbar \frac{\partial}{\partial \phi}\left\langle\mathbf{x}^{\prime} \mid \alpha\right\rangle
$$
by using the standard spherical coordinate transformation
$$
x^{\prime}=r \cos \phi \sin \theta, \quad y^{\prime}=r \sin \phi \sin \theta, \quad z^{\prime}=r \cos \theta .
$$

Hafiz Shahzaib

Numerade Educator

37:13

Problem 23

The wave function of a particle subjected to a spherically symmetrical potential $V(r)$ is given by
$$
\psi(\mathbf{x})=(x+y+3 z) f(r)
$$
a. Is $\psi$ an eigenfunction of $\mathbf{L}^{2}$ ? If so, what is the $l$-value? If not, what are the possible values of $l$ we may obtain when $\mathbf{L}^{2}$ is measured?
b. What are the probabilities for the particle to be found in various $m_{l}$ states?
c. Suppose it is known somehow that $\psi(\mathbf{x})$ is an energy eigenfunction with eigenvalue $E$. Indicate how we may find $V(r)$.

Susan Hallstrom

Numerade Educator

02:29

Problem 24

A particle in a spherically symmetrical potential is known to be in an eigenstate of $L^{2}$ and $L_{z}$ with eigenvalues $\hbar^{2} l(l+1)$ and $m \hbar$, respectively. Prove that the expectation values between $|l m\rangle$ states satisfy
$$
\left\langle L_{x}\right\rangle=\left\langle L_{y}\right\rangle=0, \quad\left\langle L_{x}^{2}\right\rangle=\left\langle L_{y}^{2}\right\rangle=\frac{\left[l(l+1) \hbar^{2}-m^{2} \hbar^{2}\right]}{2} .
$$
Interpret this result semiclassically.

Keshav Singh

Numerade Educator

03:04

Problem 25

Suppose a half-integer $l$-value, say $\frac{1}{2}$, were allowed for orbital angular momentum. From
$$
L_{+} Y_{1 / 2}^{1 / 2}(\theta, \phi)=0,
$$
we may deduce, as usual,
$$
Y_{1 / 2}^{1 / 2}(\theta, \phi) \propto e^{j \phi / 2} \sqrt{\sin \theta} .
$$
Now try to construct $Y_{1 / 2}^{-1 / 2}(\theta, \phi)$ (a) by applying $L_{-}$to $Y_{1 / 2}^{1 / 2}(\theta, \phi)$ and (b) using $L_{-} Y_{1 / 2}^{-1 / 2}(\theta, \phi)=0$. Show that the two procedures lead to contradictory results. (This gives an argument against half-integer $l$-values for orbital angular momentum.)

Keshav Singh

Numerade Educator

01:04

Problem 26

Consider an orbital angular-momentum eigenstate $|l=2, m=0\rangle .$ Suppose this state is rotated by an angle $\beta$ about the $y$-axis. Find the probability for the new state to be found in $m=0, \pm 1$, and $\pm 2$. (The spherical harmonics for $l=0,1$, and 2 given in Section B. 5 in Appendix B may be useful.)

Chai Santi

Numerade Educator

02:09

Problem 27

Show that by keeping both terms in $(3.275)$, the origin becomes a source of probability, provided that there is a relative phase between the constants $A$ and $B$.

Christopher Stanley

Numerade Educator

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Problem 28

Consider the energy eigenvalues for a spherically symmetric "box" of radius $a$.
a. For the box with infinite walls, check the eigenvalues for the $l=0, l=1$, and $l=2$ states, given in $(3.287),(3.288)$, and $(3.289)$.
b. Find the lowest energy eigenvalues with $l=0$ for a finite spherical box with potential wall height $V_{0}=\hbar^{2} \beta^{2} / 2 m a^{2}$ where $\beta=4,10,25$, and 100 , and show that your numerical results approach the appropriate value given in (a).

Victor Salazar

Numerade Educator

01:31

Problem 29

The goal of this problem is to determine degenerate eigenstates of the threedimensional isotropic harmonic oscillator written as eigenstates of $\mathbf{L}^{2}$ and $L_{z}$, in terms of the Cartesian eigenstates $\left|n_{x} n_{y} n_{z}\right\rangle$.
a. Show that the angular-momentum operators are given by
$$
\begin{aligned}
L_{i} &=i \hbar \varepsilon_{i j k} a_{j} a_{k}^{\dagger} \\
\mathbf{L}^{2} &=\hbar^{2}\left[N(N+1)-a_{k}^{\dagger} a_{k}^{\dagger} a_{j} a_{j}\right]
\end{aligned}
$$
where summation is implied over repeated indices, $\varepsilon_{i j k}$ is the totally antisymmetric symbol, and $N \equiv a_{j}^{\dagger} a_{j}$ counts the total number of quanta.
b. Use these relations to express the states $|q l m\rangle=|01 m\rangle, m=0, \pm 1$, in terms of the three eigenstates $\left|n_{x} n_{y} n_{z}\right\rangle$ that are degenerate in energy. Write down the representation of your answer in coordinate space, and check that the angular and radial dependences are correct.
c. Repeat for $|q l m\rangle=|100\rangle$.
d. Repeat for $|q l m\rangle=|02 m\rangle$, with $m=0,1$, and 2 .

Suzanne W.

Numerade Educator

01:35

Problem 30

By considering the associated Laguerre polynomials $L_{p}^{q}(x)$, which are also solutions to Kummer's equation $(3.308)$, we can derive the normalization constant in $(3.320)$.
a. First consider the Laguerre polynomials $L_{p}(x)$, which are defined according to a generating function as
$$
g(x, t)=\frac{e^{-x t(1-t)}}{1-t}=\sum_{p=0}^{\infty} L_{p}(x) \frac{t^{p}}{p !}
$$
where $0<t<1$. Prove that $L_{n}(0)=n !$ and $L_{0}(x)=1$. Differentiate $g(x, t)$ with respect to $x$ to show that
$$
L_{p}^{\prime}(x)-p L_{p-1}^{\prime}(x)=-p L_{p-1}(x)
$$
Then differentiate $g(x, t)$ with respect to $t$ to show that
$$
L_{p+1}(x)-(2 p+1-x) L_{p}(x)+p^{2} L_{p-1}(x)=0 .
$$
Combine these equations to derive the differential equation for the $L_{p}(x)$, namely
$$
x L_{p}^{\prime \prime}(x)+(1-x) L_{p}^{\prime}(x)+p L_{p}(x)=0 .
$$
b. The associated laguerre Polynomials $L_{p}^{q}(x)$ are defined from the $L_{p}(x)$ as
$$
L_{p}^{q}(x)=(-1)^{q} \frac{d^{q}}{d x^{q}}\left[L_{p+q}(x)\right] .
$$
Use this to show that the $L_{p}^{q}(x)$ satisfy the differential equation
$$
x L_{p}^{q^{\prime \prime}}(x)+(q+1-x) L_{p}^{q^{\prime}}(x)+p L_{p}^{q}(x)=0
$$
which is the same as (3.308) where $q=c-1$ and $p=-a$. Also show that
$$
h(x, t)=\frac{(-1)^{q}}{(1-t)^{q+1}} e^{-x t(1-t)}=\sum_{p=0}^{\infty} L_{p}^{q}(x) \frac{t^{p}}{(p+q) !}
$$
which gives $L_{p}^{q}(0)=(-1)^{q}[(p+q) !]^{2} / p ! q !$ through a generating function $h(x, t)$.
c. Now find the normalization constant in (3.320). Start with the relationship
$$
\int_{0}^{\infty} x^{q+1} e^{-x} h(x, t) h(x, s) d x=\sum_{p=0}^{\infty} \sum_{p^{\prime}=0}^{\infty} \frac{t^{p}}{(p+q) !} \frac{s^{p^{\prime}}}{\left(p^{\prime}+q\right) !} I_{p p^{\prime}}^{A}
$$
where $I_{p p^{\prime}}^{f} \equiv \int_{0}^{\infty} x^{q+1} e^{-x} L_{p}^{q}(x) L_{p^{\prime}}^{q}(x) d x$, and determine $I_{p p}^{q}$ by isolating the terms with the same powers of $s$ and $t$. Make use of the generalized binomial expansion
$$
\frac{1}{(1-x)^{n}}=\sum_{p=0}^{\infty}\left(\begin{array}{c}
n+p-1 \\
p
\end{array}\right) x^{p}=\sum_{p=0}^{\infty} \frac{(n+p-1) !}{p !(n-1) !} x^{p} .
$$

Manik Pulyani

Numerade Educator

04:20

Problem 31

Consider the Coulomb potential $V(\mathbf{x})=-Z e^{2} / r$ and define the (quantum-mechanical operator analogue of the) Runge-Lenz vector
$$
\mathbf{M}=\frac{1}{2 m}(\mathbf{p} \times \mathbf{L}-\mathbf{L} \times \mathbf{p})-\frac{Z e^{2}}{r} \mathbf{x} .
$$
Prove that $\mathbf{M}$ is Hermitian and that it commutes with the Hamiltonian. We will return to $\mathbf{M}$ when we go through Pauli's algebraic solution for this Hamiltonian in Section 4.1.4.

Hafiz Shahzaib

Numerade Educator

04:20

Problem 32

What is the physical significance of the operators
$$
K_{+} \equiv a_{+}^{\dagger} a_{-}^{\dagger} \quad \text { and } \quad K_{-} \equiv a_{+} a_{-}
$$
in Schwinger's scheme for angular momentum? Give the nonvanishing matrix elements of $K_{\pm}$.

Hafiz Shahzaib

Numerade Educator

02:08

Problem 33

Carry through the argument outlined on p. 208 for adding two spin $\frac{1}{2}$ particles by diagonalizing the $4 \times 4$ matrix corresponding to the operator $\mathbf{S}^{2}$ given in (3.339). That is, construct the matrix representation of $\mathbf{S}^{2}$ in the $|\pm \pm\rangle$ basis, and find the eigenvalues and eigenvectors. Your result should agree with (3.335).

Manik Pulyani

Numerade Educator

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Problem 34

Find all nine states $\lfloor j, m\rangle$ for $j=2,1$, and 0 formed by adding $j_{1}=1$ and $j_{2}=1$. Use a simplified notation, where $|j, m\rangle$ is explicit and $\pm, 0$ stand for $m_{1,2}=\pm 1,0$, respectively, for example
$$
|1,1\rangle=\frac{1}{\sqrt{2}}|+0\rangle-\frac{1}{\sqrt{2}}|0+\rangle
$$
You may also want to make use of the ladder operators $J_{\pm}$, or recursion relations, as well as orthonormality. Check your answers by finding a table of Clebsch-Gordan coefficients for comparison; see Appendix E.

Victor Salazar

Numerade Educator

02:19

Problem 35

A spin $\frac{1}{2}$ particle is in an orbital angular momentum $l=1$ state.
a. Starting with the total angular momentum state $|j, m\rangle=|j, j\rangle$ for $j=3 / 2$, use the operator $\mathbf{J}=\mathbf{L}+\mathbf{S}$ to construct all the states $\left\lfloor j, m_{j}\right\rangle$ in terms of the eigenstates $\left|l, m_{l}\right\rangle$ for $\mathbf{L}^{2}$ and $L_{z}$, and the eigenstates $|1 / 2, \pm 1 / 2\rangle$ for $\mathbf{S}^{2}$ and $S_{z}$. Check your answers against any available table of Clebsch-Gordan coefficients; see Appendix E.
b. If the particle is in a total angular-momentum eigenstate with $z$-component $+h / 2$, calculate the probability of finding the $z$-component of the spin of the particle to have the value $m_{s}=+1 / 2$.

Dominador Tan

Numerade Educator

07:17

Problem 36

The "spin-angular functions" (aka "spinor spherical harmonics") are defined as
$$
\mathscr{Y}_{l}^{j=l \pm 1 / 2, m}=\frac{1}{\sqrt{2 l+1}}\left[\begin{array}{c}
\pm \sqrt{l \pm m+\frac{1}{2}} Y_{l}^{m-1 / 2}(\theta, \phi) \\
\sqrt{l \mp m+\frac{1}{2}} Y_{l}^{m+1 / 2}(\theta, \phi)
\end{array}\right]
$$
See (3.384). These were constructed to be properly normalized eigenfunctions of $\mathbf{L}^{2}$, $\mathbf{S}^{2}, \mathbf{J}^{2}$, and $J_{z}$, where $\mathbf{J} \equiv \mathbf{L}+\mathbf{S}$. Use explicit calculations to prove the following:
a. The $\mathscr{Q}_{l}^{j=l \pm 1 / 2, m}$ are normalized, that is
$$
\int_{0}^{2 \pi} d \phi \int_{0}^{\pi} \sin \theta d \theta\left(\mathscr{Y}_{l}^{j=l \pm 1 / 2, m}\right)^{\dagger} \mathscr{Y}_{l}^{j=l \pm 1 / 2, m}=1
$$
b. The $\mathscr{Y}_{l}^{j=l \pm 1 / 2, m}$ have the correct eigenvalues for $\mathbf{J}^{2}=\mathbf{L}^{2}+\mathbf{S}^{2}+2 L_{z} S_{z}+L_{+} S_{-}+$ $L_{-} S_{+}$, that is see $(3.349)$, and $J_{z}$.

Mahnoor Amin

Numerade Educator

02:54

Problem 37

a. Evaluate
$$
\sum_{m=-j}^{j}\left|d_{m m^{\prime}}^{(j)}(\beta)\right|^{2} m
$$
for $a n y j$ (integer or half-integer); then check your answer for $j=\frac{1}{2}$.
b. Prove, for any $j$,
$$
\sum_{m=-j}^{j} m^{2}\left|d_{m^{\prime} m}^{(j)}(\beta)\right|^{2}=\frac{1}{2} j(j+1) \sin ^{2} \beta+m^{2} \frac{1}{2}\left(3 \cos ^{2} \beta-1\right)
$$
[Hint: This can be proved in many ways. You may, for instance, examine the rotational properties of $J_{z}^{2}$ using the spherical (irreducible) tensor language.]

Narayan Hari

Numerade Educator

08:50

Problem 38

a. Consider a system with $j=1$. Explicitly write
$$
\left\langle j=1, m^{\prime}\right| J_{y}\lfloor j=1, m\rangle
$$
in $3 \times 3$ matrix form.
b. Show that for $j=1$ only, it is legitimate to replace $e^{-i_{y} \beta / \hbar}$ by
$$
1-i\left(\frac{J_{y}}{\hbar}\right) \sin \beta-\left(\frac{J_{y}}{\hbar}\right)^{2}(1-\cos \beta)
$$
c. Using (b), prove

Anthony Ramos

Numerade Educator

Problem 39

Express the matrix element $\left\langle\alpha_{2} \beta_{2} \gamma_{2}\left|J_{3}^{2}\right| \alpha_{1} \beta_{1} \gamma_{1}\right\rangle$ in terms of a series in
$$
\mathscr{D}_{m n}^{j}(\alpha \beta \gamma)=\langle\alpha \beta \gamma \mid j m n\rangle .
$$

Check back soon!

03:54

Problem 40

Consider a system made up of two spin $\frac{1}{2}$ particles. Observer A specializes in measuring the spin components of one of the particles $\left(s_{1 z}, s_{1 x}\right.$ and so on), while observer B measures the spin components of the other particle. Suppose the system is known to be in a spin-singlet state, that is, $S_{\text {total }}=0$.
a. What is the probability for observer A to obtain $s_{1 z}=\hbar / 2$ when observer B makes no measurement? Same problem for $s_{1 x}=\hbar / 2$.
b. Observer B determines the spin of particle 2 to be in the $s_{2 z}=\hbar / 2$ state with certainty. What can we then conclude about the outcome of observer A's measurement if (i) A measures $s_{1 z}$ and (ii) A measures $s_{1, x}$ ? Justify your answer.

Lazarus Arnau

Numerade Educator

01:16

Problem 41

Using the defining property (3.451) for a vector operator, prove that the momentum operator $\mathbf{p}$ is a vector, based on its commutation relations with angular momentum $\mathbf{L}$.

Lottie Adams

Numerade Educator

01:01

Problem 42

Consider a spherical tensor of rank 1 (that is, a vector)
$$
V_{\pm 1}^{(1)}=\mp \frac{V_{x} \pm i V_{y}}{\sqrt{2}}, \quad V_{0}^{(1)}=V_{z} .
$$
Using the expression for $d^{(j=1)}$ given in Problem $3.38$, evaluate
$$
\sum_{q^{\prime}} d_{q q^{\prime}}^{(1)}(\beta) V_{q^{\prime}}^{(1)}
$$
and show that your results are just what you expect from the transformation properties of $V_{x y, z}$ under rotations about the $y$-axis.

Raj Bala

Numerade Educator

01:09

Problem 43

a. Construct a spherical tensor of rank 1 out of two different vectors $\mathbf{U}=\left(U_{x}, U_{y}, U_{z}\right)$ and $\mathbf{V}=\left(V_{x}, V_{y}, V_{z}\right)$. Explicitly write $T_{\pm 1,0}^{(1)}$ in terms of $U_{x, y, z}$ and $V_{x, y, z}$.
b. Construct a spherical tensor of rank 2 out of two different vectors $\mathbf{U}$ and $\mathbf{V}$. Write down explicitly $T_{\pm 2, \pm 1,0}^{(2)}$ in terms of $U_{x, y, z}$ and $V_{x, y, z}$.

Raj Bala

Numerade Educator

11:12

Problem 44

Consider a spinless particle bound to a fixed center by a central force potential.
a. Relate, as much as possible, the matrix elements
$$
\left\langle n^{\prime}, l^{\prime}, m^{\prime}\left|\mp \frac{1}{\sqrt{2}}(x \pm i y)\right| n, l, m\right\rangle \quad \text { and } \quad\left\langle n^{\prime}, l^{\prime}, m^{\prime}|z| n, l, m\right\rangle
$$
using only the Wigner-Eckart theorem. Make sure to state under what conditions the matrix elements are nonvanishing.
b. Do the same problem using wave functions $\psi(\mathbf{x})=R_{n l}(r) Y_{l}^{m n}(\theta, \phi)$.

Isaac Huidobro

Numerade Educator

04:19

Problem 45

a. Write $x y, x z$, and $\left(x^{2}-y^{2}\right)$ as components of the components of a spherical (irreducible) tensor of rank 2 .
b. The expectation value
$$
Q \equiv e\left\langle\alpha, j, m=j\left|\left(3 z^{2}-r^{2}\right)\right| \alpha, j, m=j\right\rangle
$$
is known as the quadrupole moment. Evaluate
$$
e\left\langle\alpha, j, m^{\prime}\left|\left(x^{2}-y^{2}\right)\right| \alpha, j, m=j\right\rangle
$$
(where $m^{\prime}=j, j-1, j-2, \ldots$ ) in terms of $Q$ and appropriate Clebsch-Gordan coefficients.

Jacob Fry

Numerade Educator

00:59

Problem 46

A spin $\frac{3}{2}$ nucleus situated at the origin is subjected to an external inhomogeneous electric field. The basic electric quadrupole interaction may by taken to be
$$
H_{\mathrm{int}}=\frac{e Q}{2 s(s-1) \hbar^{2}}\left[\left(\frac{\partial^{2} \phi}{\partial x^{2}}\right)_{0} s_{x}^{2}+\left(\frac{\partial^{2} \phi}{\partial y^{2}}\right)_{0} s_{y}^{2}+\left(\frac{\partial^{2} \phi}{\partial z^{2}}\right)_{0} s_{z}^{2}\right]
$$
where $\phi$ is the electrostatic potential satisfying Laplace's equation and the coordinate axes are so chosen that
$$
\left(\frac{\partial^{2} \phi}{\partial x \partial y}\right)_{0}=\left(\frac{\partial^{2} \phi}{\partial y \partial z}\right)_{0}=\left(\frac{\partial^{2} \phi}{\partial x \partial z}\right)_{0}=0
$$
Show that the interaction energy can be written as
$$
A\left(3 S_{z}^{2}-\mathbf{S}^{2}\right)+B\left(S_{+}^{2}+S_{-}^{2}\right)
$$
and express $A$ and $B$ in terms of $\left(\partial^{2} \phi / \partial x^{2}\right)_{0}$ and so on. Determine the energy eigenkets (in terms of $|m\rangle$, where $m=\pm \frac{3}{2}, \pm \frac{1}{2}$ ) and the corresponding energy eigenvalues. Is there any degeneracy?

Raj Bala

Numerade Educator