Introduction to Quantum Mechanics

David J. Griffiths, Darrell F. Schroeter

Chapter 6 Symmetries & Conservation Laws - all with Video Answers

Educators

Chapter Questions

08:02

Problem 1

Consider the parity operator in three dimensions.
(a) Show that $\hat{\Pi} \psi(\mathbf{r})=\psi^{\prime}(\mathbf{r})=\psi(-\mathbf{r})$ is equivalent to a mirror reflection followed by a rotation.
(b) Show that, for $\psi$ expressed in polar coordinates, the action of the parity operator is $\hat{\Pi} \psi(r, \theta, \phi)=\psi(r, \pi-\theta, \phi+\pi).$
(c) Show that for the hydrogenic orbitals,
$\hat{\Pi} \psi_{n \ell m}(r, \theta, \phi)=(-1)^{\ell} \psi_{n \ell m}(r, \theta, \phi).$
That is, $\psi_{n \ell m}$ is an eigenstate of the parity operator, with eigenvalue $(-1)^{\ell} .$ Note: This result actually applies to the stationary states of any central potential $V(\mathbf{r})=V(r) .$ For a central potential, the eigenstates may be written in the separable form $R_{n \ell}(r) Y_{\ell}^{m}(\theta, \phi)$ where only the radial function $R_{n \ell}-$ which plays no role in determining the parity of the state-depends on the specific functional form of $V(r).$

Hafiz Shahzaib

Numerade Educator

02:15

Problem 2

Show that, for a Hermitian operator $\hat{Q},$ the operator $\hat{U}=\exp [i \hat{Q}]$ is unitary. Hint: First you need to prove that the adjoint is given by $\hat{U}^{\dagger}=\exp [-i \hat{Q}] ;$ then prove that $\hat{U}^{\dagger} \hat{U}=1 .$ Problem $\underline{3.5}$ may help.

Nick Johnson

Numerade Educator

01:31

Problem 3

Show that the operator $\hat{p}^{\prime}$ obtained by applying a translation to the operator $\hat{p}$ is $\hat{p}^{\prime}=\hat{T}^{\dagger} \hat{p} \hat{T}=\hat{p}.$

Allison Stroman

Numerade Educator

02:51

Problem 4

Prove Equation $6.8 .$ You may assume that $Q(\hat{x}, \hat{p})$ can be written in a power series $\hat{Q}(\hat{x}, \hat{p})=\sum_{m=0}^{\infty} \sum_{n=0}^{\infty} a_{m n} \hat{x}^{m} \hat{p}^{n}$ for some constants $a_{m n}.$

Linh Vu

Numerade Educator

01:04

Problem 5

Show that Equation 6.12 follows from Equation $6.11 .$ Hint: First write $\psi(x)=e^{i q x} u(x),$ which is certainly true for some $u(x),$ and then show that $u(x)$ is necessarily a periodic function of $x.$

Chai Santi

Numerade Educator

23:54

Problem 6

Consider a particle of mass $m$ moving in a potential $V(x)$ with period $a .$ We know from Bloch's theorem that the wave function can be written in the form of Equation $6.12 .$ Note: It is conventional to label the states with quantum numbers $n$ and $q$ as $\psi_{n q}(x)=e^{-i q x} u_{n q}(x)$ where $E_{n q}$ is the $n$ th energy for a given value of $q.$
(a) Show that $u$ satisfies the equation
$$
-\frac{\hbar^{2}}{2 m} \frac{d^{2} u_{n q}}{d x^{2}}-\frac{i \hbar^{2} q}{m} \frac{d u_{n q}}{d x}+V(x) u_{n q}=\left(E_{n q}-\frac{\hbar^{2} q^{2}}{2 m}\right) u_{n q}
$$
(b) Use the technique from Problem 2.61 to solve the differential equation for $u_{n q}$. You need to use a two-sided difference for the first derivative so that you have a Hermitian matrix to diagonalize:
$\frac{d \psi}{d x} \approx \frac{\psi_{j+1}-\psi_{j-1}}{2 \Delta x} .$ For the potential in the interval 0 to $a$ let $V(x)=\left\{\begin{array}{ll}-V_{0} & \text { a/ } 4<x<3 a / 4 \\ 0 & \text { else }\end{array}\right.$
with $V_{0}=20 \hbar^{2} / 2 m a^{2} .$ (You will need to modify the technique slightly to account for the fact that the function $u_{n q}$ is periodic.) Find the lowest two energies for the following values of the crystal momentum: $q a=-\pi$
$-\pi / 2,0, \pi / 2, \pi .$ Note that $q$ and $q+2 \pi / a$ describe the same wave function (Equation 6.12 ), so there is no reason to consider values of $q a$ outside of the interval from $-\pi$ to $\pi$. In solid state physics, the values of
$q$ inside this range constitute the first Brillouin zone.
(c) Make a plot of the energies $E_{1 q}$ and $E_{2 q}$ for values of $q$ between $-\pi / a$ and $\pi / a .$ If you've automated the code that you used in part (b), you should be able to show a large number of $q$ values in this range. If not, simply plot the values that you computed in (b).

Mirza Aslam Beig

Numerade Educator

11:54

Problem 7

Consider two particles of mass $m_{1}$ and $m_{2}$ (in one dimension) that interact via a potential that depends only on the distance between the particles $V\left(\left|x_{1}-x_{2}\right|\right),$ so that the Hamiltonian is $\hat{H}=-\frac{\hbar^{2}}{2 m_{1}} \frac{\partial^{2}}{\partial x_{1}^{2}}-\frac{\hbar^{2}}{2 m_{2}} \frac{\partial^{2}}{\partial x_{2}^{2}}+V\left(\left|x_{1}-x_{2}\right|\right)$
Acting on a two-particle wave function the translation operator would be
$\hat{T}(a) \psi\left(x_{1}, x_{2}\right)=\psi\left(x_{1}-a, x_{2}-a\right)$
(a) Show that the translation operator can be written $\hat{T}(a)=e^{-\frac{i a}{\hbar} \hat{P}}$ where $\hat{P}=\hat{p}_{1}+\hat{p}_{2}$ is the total momentum.
(b) Show that the total momentum is conserved for this system.

Ren Jie Tuieng

Numerade Educator

03:12

Problem 8

Problem 6.8
(a) Show that the parity operator $\hat{\Pi}$ is Hermitian.
(b) Show that the eigenvalues of the parity operator are ±1.

Victor Salazar

Numerade Educator

10:59

Problem 9

(a) Under parity, a "true" scalar operator does not change:
$\hat{\Pi}^{\dagger} \hat{f} \hat{\Pi}=\hat{f}$ whereas a pseudoscalar changes sign. Show therefore that $[\hat{\Pi}, \hat{f}]=0$ for a "true" scalar, whereas $\{\hat{\Pi}, \hat{f}\}=0$ for a pseudoscalar. Note: the anti-commutator of two operators $\hat{A}$ and $\hat{B}$ is defined as
$$
\{\hat{A}, \hat{B}\} \equiv \hat{A} \hat{B}+\hat{B} \hat{A}
$$
(b) Similarly, a "true" vector changes sign
$\hat{\Pi}^{\dagger} \hat{\mathbf{V}} \hat{\Pi}=-\hat{\mathbf{V}}$
whereas a pseudovector is unchanged. Show therefore that $\{\hat{\Pi}, \hat{\mathbf{V}}\}=\mathbf{0}$ for a "true" vector and $[\hat{\Pi}, \hat{\mathbf{V}}]=\mathbf{0}$ for a pseudovector.

Dr. Rajveer Singh

Numerade Educator

01:39

Problem 10

Show that the position and momentum operators are odd under parity. That is, prove Equations $6.18,6.19,$ and, by extension, 6.21 and 6.22.

Dominador Tan

Numerade Educator

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Problem 11

Consider the matrix elements of $\hat{\mathbf{L}}$ between two definite-parity states: $\left\langle n^{\prime} \ell^{\prime} m^{\prime}|\hat{\mathbf{L}}| n \ell m\right\rangle .$ Under what conditions is this matrix element guaranteed to vanish? Note that the same selection rule would apply to any pseudovector
operator, or any "true" scalar operator.

Nick Johnson

Numerade Educator

07:17

Problem 12

Spin angular momentum, $\hat{\mathbf{S}},$ is even under parity, just like orbital angular momentum $\hat{\mathbf{L}}$
$\hat{\Pi}^{\dagger} \hat{\mathbf{S}} \hat{\Pi}=\hat{\mathbf{S}} \quad$ or $\quad[\hat{\Pi}, \hat{\mathbf{S}}]=0$
Acting on a spinor written in the standard basis (Equation 4.139 ), the parity operator becomes a $2 \times 2$ matrix. Show that, due to Equation $6.27,$ this matrix must be a constant times the identity matrix. As such, the parity of a spinor isn't very interesting since both spin states are parity eigenstates with the same eigenvalue. We can arbitrarily choose that parity to be +1 , so the parity operator has no effect on the spin portion of the wave function.

Mahnoor Amin

Numerade Educator

05:19

Problem 13

Consider an electron in a hydrogen atom.
(a) Show that if the electron is in the ground state, then necessarily $\left\langle\mathbf{p}_{e}\right\rangle=0$ No calculation allowed.
(b) Show that if the electron is in an $n=2$ state, then $\left\langle\mathbf{p}_{e}\right\rangle$ need not vanish.
Give an example of a wave function for the energy level $n=2$ that has a non vanishing $\left\langle\mathbf{p}_{e}\right\rangle$ and compute $\left\langle\mathbf{p}_{e}\right\rangle$ for this state.

Robert Zaballa

Numerade Educator

05:05

Problem 14

In this problem you will establish the correspondence between Equations 6.30 and 6.31.
(a) Diagonalize the matrix $^{16}$
$M=\left(\begin{array}{cc}1 & -\varphi / N \\ \varphi / N & 1\end{array}\right)$
to obtain the matrix $\mathrm{M}^{\prime}=\mathrm{SMS}^{-1}$
where $S^{-1}$ is the unitary matrix whose columns are the (normalized) eigenvectors of M. (b) Use the binomial expansion to show that $\lim _{N \rightarrow \infty}\left(\mathrm{M}^{\prime}\right)^{N}$ is a diagonal matrix with entries $e^{-i \varphi \text { and } e^{i \varphi} \text { on the diagonal. }}$
(c) Transform back to the original basis to show that $\lim _{N \rightarrow \infty} \mathrm{M}^{N}=\mathrm{S}^{-1}\left[\lim _{N \rightarrow \infty}\left(\mathrm{M}^{\prime}\right)^{N}\right] \mathrm{S}$
agrees with the matrix in Equation 6.31.

Jack Chen

Numerade Educator

02:12

Problem 15

Show how Equation 6.34 guarantees that a scalar is unchanged by a rotation: $\hat{f}^{\prime}=\hat{R}^{\dagger} \hat{f} \hat{R}=\hat{f}.$

Foster Wisusik

Numerade Educator

03:14

Problem 16

Working from Equation $6.33,$ find how the vector operator $\hat{\mathbf{V}}$ transforms for an infinitesimal rotation by an angle $\delta$ about the $y$ axis. That is, find the matrix D in $\hat{\mathbf{V}}^{\prime}=\mathrm{D} \hat{\mathbf{V}}.$

Andrija Isakov

Numerade Educator

03:14

Problem 17

Consider the action of an infinitesimal rotation about the n axis of
an angular momentum eigenstate $\psi_{n \ell m}$. Show that
$$
\hat{R}_{\mathbf{n}}(\delta) \psi_{n \ell m}=\sum_{m^{\prime}} D_{m^{\prime} m} \psi_{n \ell m^{\prime}}
$$
and find the complex numbers $D_{m^{\prime} m}$ (they will depend on $\delta$, n, and $\ell$ as well as $m$ and $m^{\prime}$ ). This result makes sense: a rotation doesn't change the magnitude of the angular momentum (specified by $\ell$ ) but does change its projection along the $z$ axis (specified by $m$ ).

Manish Jain

Numerade Educator

17:54

Problem 18

Consider the free particle in one dimension: $\hat{H}=\hat{p}^{2} / 2 m .$ This Hamiltonian has both translational symmetry and inversion symmetry.
(a) Show that translations and inversion don't commute.
(b) Because of the translational symmetry we know that the eigenstates of $\hat{H}$ can be chosen to be simultaneous eigenstates of momentum, namely $f_{p}(x)$ (Equation 3.32 ). Show that the parity operator turns $f_{p}(x)$ into $f_{-p}(x) ;$ these two states must therefore have the same energy.
(c) Alternatively, because of the inversion symmetry we know that the eigenstates of $\hat{H}$ can be chosen to be simultaneous eigenstates of parity, namely $\frac{1}{\sqrt{\pi \hbar}} \cos \left(\frac{p x}{\hbar}\right)$ and $\frac{1}{\sqrt{\pi \hbar}} \sin \left(\frac{p x}{\hbar}\right).$
Show that the translation operator mixes these two states together; they therefore must be degenerate.
Note: Both parity and translational invariance are required to explain the degeneracy in the free-particle spectrum. Without parity, there is no reason for $f_{p}(x)$ and $f_{-p}(x)$ to have the same energy (I mean no reason based on symmetries discussed thus far ...obviously you can plug them in to the timeindependent Schrödinger equation and show it's true).

Mahnoor Amin

Numerade Educator

09:05

Problem 19

For any vector operator $\hat{\mathbf{V}}$ one can define raising and lowering
operators as $\hat{V}_{\pm}=\hat{V}_{x} \pm i \hat{V}_{y}.$
(a) Using Equation 6.33 , show that $\left[\hat{L}_{z}, \hat{V}_{\pm}\right]=\pm \hbar \hat{V}_{+}$ $\left[\hat{L}^{2}, \hat{V}_{\pm}\right]=2 \hbar^{2} \hat{V}_{\pm} \pm 2 \hbar \hat{V}_{\pm} \hat{L}_{z} \mp 2 \hbar \hat{V}_{z} \hat{L}_{\pm}$
(b) Show that, if $\psi$ is an eigenstate of $\hat{L}^{2}$ and $\hat{L}_{z}$ with eigenvalues $\ell(\ell+1) \hbar^{2}$ and $\ell \hbar$ respectively, then either $\hat{V}_{+} \psi$ is zero or $\hat{V}_{+} \psi$ is also an eigenstate of $\hat{L}^{2}$ and $\hat{L}_{z}$ with eigenvalues $(\ell+1)(\ell+2) \hbar^{2}$ and $(\ell+1) \hbar$ respectively. This means that, acting on a state with maximal $m_{\ell}=\ell,$ the operator $\hat{V}_{+}$ either "raises" both the $\ell$ and $m$ values by 1 or destroys the state.

Abhijit Das

Numerade Educator

08:43

Problem 20

Show that the commutator $\left[\hat{L}_{-}, \hat{f}\right]=0$ leads to the same rule, Equation $6.46,$ as does the commutator $\left[\hat{L}_{+}, \hat{f}\right]=0.$

Mahnoor Amin

Numerade Educator

03:26

Problem 21

For an electron in the hydrogen state $\psi=\frac{1}{\sqrt{2}}\left(\psi_{211}+\psi_{21-1}\right)$ find $\langle r\rangle$ after first expressing it in terms of a single reduced matrix element.

Sheh Lit Chang

University of Washington

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Problem 22

(a) Show that the commutation relations, Equations $6.50-6.54$, follow from the definition of a vector operator, Equation $6.33 .$ If you did Problem 6.19 you already derived one of these.
(b) Derive Equation 6.57

Victor Salazar

Numerade Educator

10:39

Problem 23

The Clebsch-Gordan coefficients are defined by Equation 4.183 Adding together two states with angular momentum $j_{1}$ and $j_{2}$ produces a state with total angular momentum $J$ according to
$|J M\rangle=\sum_{m_{1}, m_{2}} C_{m_{1}, m_{2} M}^{j_{1} j_{2} J}\left|j_{1} j_{2} m_{1} m_{2}\right\rangle$
(a) From Equation 6.64 , show that the Clebsch-Gordan coefficients satisfy
$C_{m_{1} m_{2} M}^{j_{1} j_{2} J}=\left\langle j_{1} j_{2} m_{1} m_{2} | J M\right\rangle$
$(6.6=$
(b) $\quad$ Apply $\hat{J}_{\pm}=\hat{J}_{\pm}^{(1)}+\hat{J}_{\pm}^{(2)}$ to Equation $\underline{6.64}$ to derive the recursion relations for Clebsch-Gordan coefficients:
$A_{J}^{M} C_{m_{1} m_{2} M+1}^{j_{1} j_{2} J}=B_{j_{1}}^{m_{1}} C_{m_{1}-1 m_{2} M}^{j_{1} j_{2} J}+B_{j_{2}}^{m_{2}} C_{m_{1} m_{2}-1 M}^{j_{1} j_{2} J}$
$B_{J}^{M} C_{m_{1} m_{2} M-1}^{j_{1} j_{2} J}=A_{j_{1}}^{m_{1}} C_{m+1}^{j_{1} j_{2} J}+A_{j_{2}}^{m_{2}} C_{m_{1} m_{2} M}^{j_{1} j_{2} J}$

Yaqub Khan

Numerade Educator

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Problem 24

(a) Sandwich each of the six commutation relations in Equations $6.52-6.54$ between $\left\langle n^{\prime} \ell^{\prime} m^{\prime}\right|$ and $|n \ell m\rangle$ to obtain relations between matrix elements of $\hat{\mathbf{V}} .$ As an example, Equation 6.52 with the upper signs gives
$$
B_{\ell^{\prime}}^{m^{\prime}}\left\langle n^{\prime} \ell^{\prime}\left(m^{\prime}-1\right)\left|V_{+}\right| n \ell m\right\rangle=A_{\ell}^{m}\left\langle n^{\prime} \ell^{\prime} m^{\prime}\left|V_{+}\right| n \ell(m+1)\right\rangle
$$
(b) Using the results in Problem $6.23,$ show that the six expressions you wrote down in part (a) are satisfied by Equations $6.59-6.61.$

Victor Salazar

Numerade Educator

02:06

Problem 25

Express the expectation value of the dipole moment $\mathbf{p}_{e}$ for an electron in the hydrogen state $\psi=\frac{1}{\sqrt{2}}\left(\psi_{211}+\psi_{210}\right)$ in terms of a single reduced matrix element, and evaluate the expectation value. Note: this is the expectation value of a vector so you need to compute all three components. Don't forget Laporte's rule!

Narayan Hari

Numerade Educator

01:14

Problem 26

Work out $\hat{p}_{H}(t)$ for the system in Example 6.7 and comment on the correspondence with the classical equation of motion.

James Kiss

Numerade Educator

01:16

Problem 27

Consider a free particle of mass $m .$ Show that the position and momentum operators in the Heisenberg picture are given by
$\hat{x}_{H}(t)=\hat{x}_{H}(0)+\frac{1}{m} \hat{p}_{H}(0) t$
$\hat{p}_{H}(t)=\hat{p}_{H}(0).$
Comment on the relationship between these equations and the classical equations of motion. Hint: you will first need to evaluate the commutator $\left[\hat{x}, \hat{H}^{n}\right] ;$ this will allow you to evaluate the commutator $[\hat{x}, \hat{U}].$

Lottie Adams

Numerade Educator

05:40

Problem 28

Show that Equations 6.75 and 6.76 are the solution to the Schrödinger equation for an infinitesimal time $\delta .$ Hint: expand $\Psi(x, t)$ in a Taylor series.

Nathan Silvano

Numerade Educator

06:45

Problem 29

Differentiate Equation 6.72 to obtain the Heisenberg equations of motion $(\text { for } \hat{Q} \text { and } \hat{H} \text { independent of time }) .^{39}$ Plug in $\hat{Q}=\hat{x}$ to obtain the differential equations for $\hat{x}_{H}$ and $\hat{p}_{H}$ in the Heisenberg picture for a single particle of mass $m$ moving in a potential $V(x).$

Lucas Finney

Numerade Educator

23:54

Problem 30

Consider a time-independent Hamiltonian for a particle moving in one dimension that has stationary states $\psi_{n}(x)$ with energies $E_{n}$
(a) Show that the solution to the time-dependent Schrödinger equation can be written $\Psi(x, t)=\hat{U}(t) \Psi(x, 0)=\int K\left(x, x^{\prime}, t\right) \Psi\left(x^{\prime}, 0\right) d x^{\prime}$ where $K\left(x, x^{\prime}, t\right),$ known as the propagator, is $K\left(x, x^{\prime}, t\right)=\sum_{n} \psi_{n}^{*}\left(x^{\prime}\right) e^{-i E_{n} t / \hbar} \psi_{n}(x)$
Here $\left|K\left(x, x^{\prime}, t\right)\right|^{2}$ is the probability for a quantum mechanical particle to travel from position $x^{\prime}$ to position $x$ in time $t.$
(b) Find $K$ for a particle of mass $m$ in a simple harmonic oscillator potential of frequency $\omega .$ You will need the identity
$$\frac{1}{\sqrt{1-z^{2}}} \exp \left[-\frac{\xi^{2}+\eta^{2}-2 \xi \eta z}{1-z^{2}}\right]=e^{-\xi^{2}} e^{-\eta^{2}} \sum_{n=0}^{\infty} \frac{z^{n}}{2^{n} n !} H_{n}(\xi) H_{n}(\eta)$$
(c) Find $\Psi(x, t)$ if the particle from part (a) is initially in the state $^{40}$
$\Psi(x, 0)=\left(\frac{2 a}{\pi}\right)^{1 / 4} e^{-a\left(x-x_{0}\right)^{2}}$
Compare your answer with Problem $2.49 .$ Note: Problem 2.49 is a special case with $a=m \omega / 2 \hbar.$
(d) Find $K$ for a free particle of mass $m$. In this case the stationary states are
continuous, not discrete, and one must make the replacement
$\sum_{n} \rightarrow \int_{-\infty}^{\infty} d p$ in Equation 6.79.
(e) Find $\Psi(x, t)$ for a free particle that starts out in the state
$\Psi(x, 0)=\left(\frac{2 a}{\pi}\right)^{1 / 4} e^{-a x^{2}}$
Compare your answer with Problem 2.21.

Mirza Aslam Beig

Numerade Educator

03:27

Problem 31

Problem 6.31 In deriving Equation 6.3 we assumed that our function had a Taylor series. The result holds more generally if we define the exponential of an
operator by its spectral decomposition, $$\hat{T}(a)=\int e^{-i a p / \hbar}|p\rangle\langle p| d p$$ rather than its power series. Here I've given the operator in Dirac notation; acting on a position-space function (see the discussion on page 123 ) this means $$\hat{T}(a) \psi(x)=\int_{-\infty}^{\infty} e^{-i a p / \hbar} f_{p}(x) \Phi(p) d p$$ where $\Phi(p)$ is the momentum space wave function corresponding to $\psi(x)$ and $f_{p}(x)$ is defined in Equation $3.32 .$ Show that the operator $\hat{T}(a),$ as given by Equation $6.81,$ applied to the function $\psi(x)=\sqrt{\lambda} e^{-\lambda|x|}$ (whose first derivative is undefined at $x=0$ ) gives the correct result.

James Kiss

Numerade Educator

07:17

Problem 32

Problem 6.32 Rotations on spin states are given by an expression identical to Equation $6.32,$ with the spin angular momentum replacing the orbital angular
momentum:
$\mathrm{R}_{\mathrm{n}}(\varphi)=\exp \left[-i \frac{\varphi}{\hbar} \mathbf{n} \cdot \mathrm{S}\right]$
In this problem we will consider rotations of a spin-1/2 state.
(a) Show that
$(\mathbf{a} \cdot \boldsymbol{\sigma})(\mathbf{b} \cdot \boldsymbol{\sigma})=\mathbf{a} \cdot \mathbf{b}+i(\mathbf{a} \times \mathbf{b}) \cdot \boldsymbol{\sigma}$
where the $\sigma_{i}$ are the Pauli spin matrices and a and $\mathbf{b}$ are ordinary vectors.
Use the result of Problem 4.29
(b) Use your result from part (a) to show that
$\exp \left[-i \frac{\varphi}{\hbar} \mathbf{n} \cdot \mathbf{S}\right]=\cos \left(\frac{\varphi}{2}\right)-i \sin \left(\frac{\varphi}{2}\right) \mathbf{n} \cdot \boldsymbol{\sigma}$
Recall that $\mathrm{S}=(\hbar / 2) \boldsymbol{\sigma}$
(c) Show that your result from part (b) becomes, in the standard basis of spin up and spin down along the $z$ axis, the matrix
$\mathrm{R}_{\mathrm{n}}=\cos \left(\frac{\varphi}{2}\right)\left(\begin{array}{cc}1 & 0 \\ 0 & 1\end{array}\right)-i \sin \left(\frac{\varphi}{2}\right)\left(\begin{array}{cc}\cos \theta & \sin \theta e^{-i \phi} \\ \sin \theta e^{i \phi} & -\cos \theta\end{array}\right)$
where $\theta$ and $\phi$ are the polar coordinates of the unit vector $\mathbf{n}$ that describes
the axis of rotation.
(d) Verify that the matrix $R_{n}$ in part (c) is unitary.
(e) Compute explicitly the matrix $S_{x}^{\prime}=R^{\dagger} S_{x} R$ where $R$ is a rotation by an angle $\varphi$ about the $z$ axis and verify that it returns the expected result. Hint: rewrite your result for $\mathrm{S}_{x}^{\prime}$ in terms of $\mathrm{S}_{x}$ and $\mathrm{S}_{y}$
(f) Construct the matrix for a $\pi$ rotation about the $x$ axis and verify that it
turns an up spin into a down spin.
(g) Find the matrix describing a $2 \pi$ rotation about the $z$ axis. Why is this answer surprising?

Mahnoor Amin

Numerade Educator

03:26

Problem 33

Consider a particle of mass $m$ in a two-dimensional infinite square well with sides of length $L$. With the origin placed at the center of the well,
the stationary states can be written as
$$
\psi_{n_{x} n_{y}}(x, y)=\frac{2}{L} \sin \left[\frac{n_{x} \pi}{L}\left(x-\frac{L}{2}\right)\right] \sin \left[\frac{n_{y} \pi}{L}\left(y-\frac{L}{2}\right)\right]
$$
with energies
$E_{n_{x} n_{y}}=\frac{\pi^{2} \hbar^{2}}{2 m L^{2}}\left(n_{x}^{2}+n_{y}^{2}\right)$
for positive integers $n_{x}$ and $n_{y}$
(a) The two states $\psi_{a b}$ and $\psi_{b a}$ for $a \neq b$ are clearly degenerate. Show that a rotation by 90 counterclockwise about the center of the square carries one
into the other,
$\hat{R} \psi_{a b} \propto \psi_{b a}$
and determine the constant of proportionality. Hint: write $\psi_{a b}$ in polar
coordinates.
(b) Suppose that instead of $\psi_{a b}$ and $\psi_{b a}$ we choose the basis $\psi_{+}$ and $\psi_{-}$ for our two degenerate states:
$\psi_{\pm}=\frac{\psi_{a b} \pm \psi_{b a}}{\sqrt{2}}$
Show that if $a$ and $b$ are both even or both odd, then $\psi_{+}$ and $\psi_{-}$ are eigenstates of the rotation operator.
(c) Make a contour plot of the state $\psi_{-}$ for $a=5$ and $b=7$ and verify (visually) that it is an eigenstate of every symmetry operation of the square (rotation by an integer multiple of $\pi / 2$, reflection across a diagonal, or reflection along a line bisecting two sides). The fact that $\psi_{+}$ and $\psi_{-}$ are not connected to each other by any symmetry of the square means that there must be additional symmetry explaining the degeneracy of these two states. 42.

Stanley Enemuo

Numerade Educator

16:18

Problem 34

The Coulomb potential has more symmetry than simply rotational invariance. This additional symmetry is manifest in an additional conserved quantity, the Laplace-Runge-Lenz vector $\hat{\mathbf{M}}=\frac{\hat{\mathbf{p}} \times \hat{\mathbf{L}}-\hat{\mathbf{L}} \times \hat{\mathbf{p}}}{2 m}+V(r) \mathbf{r}$
where $V(\mathbf{r})$ is the potential energy, $V(r)=-e^{2} / 4 \pi \epsilon_{0} r .^{43}$ The complete set of commutators for the conserved quantities in the hydrogen atom is
(i) $\left.\quad \hat{H}, \hat{M}_{i}\right]=0$
(ii) $\quad\left[\hat{H}, \hat{L}_{i}\right]=0$
(iii) $\quad\left[\hat{L}_{i}, \hat{L}_{j}\right]=i \hbar \epsilon_{i j k} \hat{L}_{k}$
(iv) $\quad\left[\hat{L}_{i}, \hat{M}_{j}\right]=i \hbar \epsilon_{i j k} \hat{M}_{k}$
(v) $\quad\left[\hat{M}_{i}, \hat{M}_{j}\right]=\frac{\hbar}{i} \epsilon_{i j k} L_{k} \frac{2}{m} \hat{H}$
The physical content of these equations is that (i) $\mathbf{M}$ is a conserved quantity,
(ii) $\mathbf{L}$ is a conserved quantity, (iii) $\mathbf{L}$ is a vector, and (iv) $\mathbf{M}$ is a vector ( (v) has no obvious interpretation). There are two additional relations between the quantities $\hat{\mathbf{L}}, \hat{\mathbf{M}},$ and $\hat{H} .$ They are
$(\mathrm{vi}) \quad \hat{M}^{2}=\left(\frac{e^{2}}{4 \pi \epsilon_{0}}\right)^{2}+\frac{2}{m} \hat{H}\left(\hat{L}^{2}+\hbar^{2}\right)$
(vii) $\quad \hat{\mathbf{M}} \cdot \hat{\mathbf{L}}=0$
(a) From the result of Problem 6.19 , and the fact that $\hat{\mathbf{M}}$ is a conserved quantity, we know that $\hat{M}_{+} \psi_{n \ell \ell}=c_{n \ell} \psi_{n(\ell+1)(\ell+1)}$ for some constants $c_{n} \ell .$ Apply (vii) to the state $\psi_{n \ell \ell}$ to show that $\hat{M}_{z} \psi_{n \ell \ell}=-\frac{1}{\sqrt{2}} \frac{1}{\sqrt{\ell+1}} c_{n} \ell \psi_{n(\ell+1) \ell}$
(b) Use (vi) to show that
$\hat{M}_{-} \hat{M}_{+} \psi_{n \ell \ell}=\left(\frac{e^{2}}{4 \pi \epsilon_{0}}\right)^{2}\left[1-\left(\frac{\ell+1}{n}\right)^{2}\right] \psi_{n \ell \ell}-\hat{M}_{z}^{2} \psi_{n \ell \ell}$
(c) From your results to parts (a) and (b), obtain the constants $c_{n \ell}$. You should find that $c_{n \ell}$ is nonzero unless $\ell=n-1$. Hint: Consider
$\int\left|M_{+} \psi_{n \ell m}\right|^{2} d^{3} \mathbf{r}$ and use the fact that $M_{\pm}$ are Hermitian conjugates. Figure 6.9 shows how the degenerate states of hydrogen are related by the generators $\hat{\mathbf{L}}$ and $\hat{\mathbf{M}}.$

Amit Srivastava

Numerade Educator

03:47

Problem 35

Problem 6.35 A Galilean transformation performs a boost from a reference frame $\mathcal{S}$ to a reference frame $\mathcal{S}^{\prime}$ moving with velocity $-v$ with respect to $\mathcal{S}$ (the origins of the two frames coincide at $t=0$ ). The unitary operator that carries out a Galilean transformation at time $t$ is $\hat{\Gamma}(v, t)=\exp \left[-\frac{i}{\hbar} v(t \hat{p}-m \hat{x})\right]$
(a) Find $\hat{x}^{\prime}=\hat{\Gamma}^{\dagger} \hat{x} \hat{\Gamma}$ and $\hat{p}^{\prime}=\hat{\Gamma}^{\dagger} \hat{p} \hat{\Gamma}$ for an infinitesimal transformation with velocity $\delta$. What is the physical meaning of your result?
(b) Show that $$\begin{aligned}
\hat{\Gamma}(v, t) &=\exp \left[\frac{i}{\hbar}\left(m x v-\frac{1}{2} m v^{2} t\right)\right] \hat{T}(v t) \\
&=\hat{T}(v t) \exp \left[\frac{i}{\hbar}\left(m x v+\frac{1}{2} m v^{2} t\right)\right]
\end{aligned}$$
where $\hat{T}$ is the spatial translation operator (Equation 6.3 ). You will need to use the Baker-Campbell-Hausdorff formula (Problem 3.29 ).
(c) Show that if $\Psi$ is a solution to the time-dependent Schrödinger equation with Hamiltonian
$\hat{H}=\frac{\hat{p}^{2}}{2 m}+V(x)$
then the boosted wave function $\Psi^{\prime}=\hat{\Gamma}(v, t) \Psi$ is a solution to the time-dependent Schrödinger equation with the potential $V(x)$ in motion:
$\hat{H}=\frac{\hat{p}^{2}}{2 m}+V(x-v t)$
$$
\text {Note:}(d / d t) e^{\hat{A}}=e^{\hat{A}}(d \hat{A} / d t) \text { only if }[\hat{A},(d \hat{A} / d t)]=0
$$
(d) Show that the result of Problem $2.50(\mathrm{a})$ is an example of this result.

Khoobchandra Agrawal

Numerade Educator

04:22

Problem 36

A ball thrown through the air leaves your hand at position $\mathbf{r}_{0}$ with a velocity of $\mathbf{v}_{0}$ and arrives a time $t$ later at position $\mathbf{r}_{1}$ traveling with a velocity $\mathbf{v}_{1}(\text { Figure } 6.10) .$ Suppose we could instantaneously reverse the ball's velocity when it reaches $\mathbf{r}_{1}$. Neglecting air resistance, it would retrace the path that took it from $\mathbf{r}_{0}$ to $\mathbf{r}_{1}$ and arrive back at $\mathbf{r}_{0}$ after another time $t$ had passed, traveling with a velocity $-\mathbf{v}_{0} .$ This is an example of time-reversal invariance reverse the motion of a particle at any point along its trajectory and it will retrace its path with an equal and opposite velocity at all positions.

Zulfiqar Ali

Numerade Educator

05:19

Problem 37

As an angular momentum, a particle's spin must flip under time reversal (Problem $\underline{6.36}$ ). The action of time-reversal on a spinor (Section $4.4 .1)$ is in fact $$\hat{\Theta}\left(\begin{array}{c}
a \\
b
\end{array}\right)=\left(\begin{array}{c}
-b^{*} \\
a^{*}
\end{array}\right)$$
so that, in addition to the complex conjugation, the up and down components are interchanged. 47
(a) Show that $\hat{\Theta}^{2}=-1$ for a spin-1/2 particle.
(b) Consider an eigenstate $\left|\psi_{n}\right\rangle$ of a time-reversal invariant Hamiltonian (Equation 6.83 ) with energy $E_{n}$ We know that $\left|\psi_{n}^{\prime}\right\rangle=\hat{\Theta}\left|\psi_{n}\right\rangle$ is also an eigenstate of $\hat{H}$ with the same energy $E_{n}$. There two possibilities: either $\left|\psi_{n}^{\prime}\right\rangle$ and $\left|\psi_{n}\right\rangle$ are the same state (meaning $\left|\psi_{n}^{\prime}\right\rangle=c\left|\psi_{n}\right\rangle$ for some
complex constant $c$ ) or they are distinct states. Show that the first case leads to a contradiction in the case of a spin-1/2 particle, meaning the energy level must be (at least) two-fold degenerate in that case.
Comment: What you have proved is a special case of Kramer's degeneracy: for an odd number of spin-1/2 particles (or any half-integer spin for that matter), every energy level (of a time-reversal-invariant Hamiltonian) is at least twofold degenerate. This is because- as you just showed-for half-integer spin a state and its time-reversed state are necessarily distinct. 48 .

Eduard Sanchez

Numerade Educator