Modern Quantum Mechanics

J. J. Sakurai, Jim Napolitano

Chapter 2 Quantum Dynamics - all with Video Answers

Educators

Chapter Questions

02:27

Problem 1

Consider the spin-precession problem discussed in the text. It can also be solved in the Heisenberg picture. Using the Hamiltonian
$$
H=-\left(\frac{e B}{m c}\right) S_{z}=\omega S_{z}
$$
write the Heisenberg equations of motion for the time-dependent operators $S_{x}(t)$, $S_{y}(t)$, and $S_{z}(t)$. Solve them to obtain $S_{x, y, z}$ as functions of time.

Adriano Chikande

Numerade Educator

09:59

Problem 2

Look again at the Hamiltonian of Chapter 1, Problem 1.13. Suppose the typist made an error and wrote $H$ as
$$
H=H_{11}|1\rangle\left\langle 1\left|+H_{22}\right| 2\right\rangle\left\langle 2\left|+H_{12}\right| 1\right\rangle\langle 2| .
$$
What principle is now violated? Illustrate your point explicitly by attempting to solve the most general time-dependent problem using an illegal Hamiltonian of this kind. (You may assume $H_{11}=H_{22}=0$ for simplicity.)

Isaac Huidobro

Numerade Educator

08:06

Problem 3

An electron is subject to a uniform, time-independent magnetic field of strength $B$ in the positive $z$-direction. At $t=0$ the electron is known to be in an eigenstate of $\mathbf{S} \cdot \hat{\mathbf{n}}$ with eigenvalue $\hbar / 2$, where $\hat{\mathbf{n}}$ is a unit vector, lying in the $x z$-plane, that makes an angle $\beta$ with the $z$-axis.
a. Obtain the probability for finding the electron in the $S_{x}=\hbar / 2$ state as a function of time.
b. Find the expectation value of $S_{x}$ as a function of time.
c. For your own peace of mind show that your answers make good sense in the extreme cases (i) $\beta \rightarrow 0$ and (ii) $\beta \rightarrow \pi / 2$.

Khoobchandra Agrawal

Numerade Educator

02:41

Problem 4

Derive the neutrino oscillation probability (2.65) and use it, along with the data in Figure $2.2$, to estimate the values of $\Delta m^{2} c^{4}$ (in units of $\mathrm{eV}^{2}$ ) and $\theta$.

Penny Riley

Numerade Educator

02:23

Problem 5

Let $x(t)$ be the coordinate operator for a free particle in one dimension in the Heisenberg picture. Evaluate
$$
[x(t), x(0)]
$$

Sheh Lit Chang

University of Washington

09:59

Problem 6

Consider a particle in one dimension whose Hamiltonian is given by
$$
H=\frac{p^{2}}{2 m}+V(x) .
$$
By calculating $[[H, x], x]$ prove
$$
\sum_{a^{\prime}}\left|\left\langle a^{\prime \prime}|x| a^{\prime}\right\rangle\right|^{2}\left(E_{a^{\prime}}-E_{a^{\prime \prime}}\right)=\frac{\hbar^{2}}{2 m},
$$
where $\left|a^{\prime}\right\rangle$ is an energy eigenket with eigenvalue $E_{a^{\prime}}$.

Isaac Huidobro

Numerade Educator

17:54

Problem 7

Consider a particle in three dimensions whose Hamiltonian is given by
$$
H=\frac{\mathbf{p}^{2}}{2 m}+V(\mathbf{x})
$$
By calculating $[\mathbf{x} \cdot \mathbf{p}, H]$ obtain
$$
\frac{d}{d t}\langle\mathbf{x} \cdot \mathbf{p}\rangle=\left\langle\frac{\mathbf{p}^{2}}{m}\right\rangle-\langle\mathbf{x} \cdot \nabla V\rangle
$$
To identify the preceding relation with the quantum-mechanical analogue of the virial theorem it is essential that the left-hand side vanish. Under what condition would this happen?

Mahnoor Amin

Numerade Educator

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Problem 8

Consider a free-particle wave packet in one dimension. At $t=0$ it satisfies the minimum uncertainty relation
$$
\left\langle(\Delta x)^{2}\right\rangle\left\langle(\Delta p)^{2}\right\rangle=\frac{\hbar^{2}}{4} \quad(t=0)
$$
In addition, we know
$$
\langle x\rangle=\langle p\rangle=0 \quad(t=0)
$$
Using the Heisenberg picture, obtain $\left\langle(\Delta x)^{2}\right\rangle_{t}$ as a function of $t(t \geq 0)$ when $\left\langle(\Delta x)^{2}\right\rangle_{t=0}$ is given. (Hint: Take advantage of the property of the minimumuncertainty wave packet you worked out in Chapter 1, Problem 1.20.)

Lainey Roebuck

Numerade Educator

18:14

Problem 9

For a wave function $\left\langle x^{\prime} \mid \alpha\right\rangle=A\left(x^{\prime}-a\right)^{2}\left(x^{\prime}+a\right)^{2} e^{i k x^{\prime}}$ for $-a \leq x^{\prime} \leq a$ and zero otherwise, carry out the following.
a. Find the constant $A$.
b. Find the expectation values $\langle x\rangle,\langle p\rangle,\left\langle x^{2}\right\rangle$, and $\left\langle p^{2}\right\rangle$.
c. Find the expectation values $\left\langle(\Delta x)^{2}\right\rangle$ and $\left\langle(\Delta p)^{2}\right\rangle$, and compare their product to that for a Gaussian wave packet.

Dr. Rajveer Singh

Numerade Educator

20:49

Problem 10

Let $\left|a^{\prime}\right\rangle$ and $\left|a^{\prime \prime}\right\rangle$ be eigenstates of a Hermitian operator $A$ with eigenvalues $a^{\prime}$ and $a^{\prime \prime}$, respectively $\left(a^{\prime} \neq a^{\prime \prime}\right)$. The Hamiltonian operator is given by
$$
H=\left|a^{\prime}\right\rangle \delta\left\langle a^{\prime \prime}|+| a^{\prime \prime}\right\rangle \delta\left\langle a^{\prime}\right|
$$
where $\delta$ is just a real number.
a. Clearly, $\left|a^{\prime}\right\rangle$ and $\left|a^{\prime \prime}\right\rangle$ are not eigenstates of the Hamiltonian. Write down the eigenstates of the Hamiltonian. What are their energy eigenvalues?
b. Suppose the system is known to be in state $\left|a^{\prime}\right\rangle$ at $t=0$. Write down the state vector in the Schrödinger picture for $t>0$.
c. What is the probability for finding the system in $\left|a^{\prime \prime}\right\rangle$ for $t>0$ if the system is known to be in state $\left|a^{\prime}\right\rangle$ at $t=0$ ?
d. Can you think of a physical situation corresponding to this problem?

Mahnoor Amin

Numerade Educator

03:26

Problem 11

A box containing a particle is divided into a right and a left compartment by a thin partition. If the particle is known to be on the right (left) side with certainty, the state is represented by the position eigenket $|R\rangle(|L\rangle)$, where we have neglected spatial variations within each half of the box. The most general state vector can then be written as
$$
|\alpha\rangle=|R\rangle\langle R \mid \alpha\rangle+|L\rangle\langle L \mid \alpha\rangle
$$
where $\langle R \mid \alpha\rangle$ and $\langle L \mid \alpha\rangle$ can be regarded as "wave functions." The particle can tunnel through the partition; this tunneling effect is characterized by the Hamiltonian
$$
H=\Delta(|L\rangle\langle R|+| R\rangle\langle L|)
$$
where $\Delta$ is a real number with the dimension of energy.
a. Find the normalized energy eigenkets. What are the corresponding energy eigenvalues?
b. In the Schrödinger picture the base kets $|R\rangle$ and $|L\rangle$ are fixed, and the state vector moves with time. Suppose the system is represented by $|\alpha\rangle$ as given above at $t=0 .$ Find the state vector $\left|\alpha, t_{0}=0 ; t\right\rangle$ for $t>0$ by applying the appropriate time-evolution operator to $|\alpha\rangle$.
c. Suppose at $t=0$ the particle is on the right side with certainty. What is the probability for observing the particle on the left side as a function of time?
d. Write down the coupled Schrödinger equations for the wave functions $\langle R| \alpha, t_{0}=$ $0 ; t\rangle$ and $\left\langle L \mid \alpha, t_{0}=0 ; t\right\rangle$. Show that the solutions to the coupled Schrödinger equations are just what you expect from (b).
e. Suppose the printer made an error and wrote $H$ as
$$
H=\Delta|L\rangle\langle R|
$$
By explicitly solving the most general time-evolution problem with this Hamiltonian, show that probability conservation is violated.

Stanley Enemuo

Numerade Educator

09:46

Problem 12

A one-dimensional simple harmonic oscillator with natural frequency $\omega$ is in initial state
$$
|\alpha\rangle=\frac{1}{\sqrt{2}}|0\rangle+\frac{e^{j \delta}}{\sqrt{2}}|1\rangle
$$
where $\delta$ is a real number.
a. Find the time-dependent wave function $\left\langle x^{\prime} \mid \alpha ; t\right\rangle$ and evaluate the (time-dependent) expectation values $\langle x\rangle$ and $\langle p\rangle$ in the state $|\alpha ; t\rangle$, i.e. in the Schrödinger picture.
b. Now calculate $\langle x\rangle$ and $\langle p\rangle$ in the Heisenberg picture and compare the results.

Coleman Green

Numerade Educator

19:46

Problem 13

A particle with mass $m$ moves in one dimension and is acted on by a constant force $F$. Find the operators $x(t)$ and $p(t)$ in the Heisenberg picture, and find their expectation values for an arbitrary state $|\alpha\rangle$. Use $\langle x(0)\rangle=x_{0}$ and $\langle p(0)\rangle=p_{0}$. The result should be obvious. Comment on how to do this problem in the Schrödinger picture, but do not try to work it through.

Nathan Silvano

Numerade Educator

03:13

Problem 14

Consider a particle subject to a one-dimensional simple harmonic oscillator potential.
Suppose at $t=0$ the state vector is given by
$$
\exp \left(\frac{-i p a}{\hbar}\right)|0\rangle
$$
where $p$ is the momentum operator, $a$ is some number with dimension of length, and the state $|0\rangle$ is the one for which $\langle x\rangle=0=\langle p\rangle$. Using the Heisenberg picture, evaluate the expectation value $\langle x\rangle$ for $t \geq 0$.

Sheh Lit Chang

University of Washington

08:47

Problem 15

a. Write down the wave function (in coordinate space) for the state specified in Problem $2.14$ at $t=0$. You may use
$$
\left\langle x^{\prime} \mid 0\right\rangle=\pi^{-1 / 4} x_{0}^{-1 / 2} \exp \left[-\frac{1}{2}\left(\frac{x^{\prime}}{x_{0}}\right)^{2}\right], \quad x_{0} \equiv\left(\frac{\hbar}{m \omega}\right)^{1 / 2}
$$
b. Obtain a simple expression for the probability that the state is found in the ground state at $t=0$. Does this probability change for $t>0$ ?

Eduard Sanchez

Numerade Educator

15:48

Problem 16

Consider a one-dimensional simple harmonic oscillator.
a. Using
$$
\left.\left.\begin{array}{l}
a \\
a^{\dagger}
\end{array}\right\}=\sqrt{\frac{m \omega}{2 \hbar}}\left(x \pm \frac{i p}{m \omega}\right), \quad \begin{array}{l}
a|n\rangle \\
a^{\dagger}|n\rangle
\end{array}\right\}=\left\{\begin{array}{l}
\sqrt{n}|n-1\rangle \\
\sqrt{n+1}|n+1\rangle
\end{array}\right.
$$
evaluate $\langle m|x| n\rangle,\langle m|p| n\rangle,\langle m|\{x, p\}| n\rangle,\left\langle m\left|x^{2}\right| n\right\rangle$, and $\left\langle m\left|p^{2}\right| n\right\rangle$.
b. Translated from classical physics, the virial theorem states that
$$
\left\langle\frac{\mathbf{p}^{2}}{m}\right\rangle=\langle\mathbf{x} \cdot \nabla V\rangle \text { (3D) or }\left\langle\frac{p^{2}}{m}\right\rangle=\left\langle x \frac{d V}{d x}\right\rangle
$$
Check that the virial theorem holds for the expectation values of the kinetic and the potential energy taken with respect to an energy eigenstate.

Mahnoor Amin

Numerade Educator

05:35

Problem 17

a. Using
$$
\left\langle x^{\prime} \mid p^{\prime}\right\rangle=(2 \pi \hbar)^{-1 / 2} e^{i p^{\prime} x^{\prime / h}} \quad \text { (one dimension) }
$$
prove
$$
\left\langle p^{\prime}|x| \alpha\right\rangle=i \hbar \frac{\partial}{\partial p^{\prime}}\left\langle p^{\prime} \mid \alpha\right\rangle
$$
b. Consider a one-dimensional simple harmonic oscillator. Starting with the Schrödinger equation for the state vector, derive the Schrödinger equation for the momentum-space wave function. (Make sure to distinguish the operator $p$ from the eigenvalue $p^{\prime}$.) Can you guess the energy eigenfunctions in momentum space?

Eduard Sanchez

Numerade Educator

06:57

Problem 18

Consider a function, known as the correlation function, defined by
$$
C(t)=\langle x(t) x(0)\rangle
$$
where $x(t)$ is the position operator in the Heisenberg picture. Evaluate the correlation function explicitly for the ground state of a one-dimensional simple harmonic oscillator.

Ozenc Gungor

Numerade Educator

01:51

Problem 19

Consider again a one-dimensional simple harmonic oscillator. Do the following algebraically, that is, without using wave functions.
a. Construct a linear combination of $|0\rangle$ and $|1\rangle$ such that $\langle x\rangle$ is as large as possible.
b. Suppose the oscillator is in the state constructed in (a) at $t=0$. What is the state vector for $t>0$ in the Schrödinger picture? Evaluate the expectation value $\langle x\rangle$ as a function of time for $t>0$ using (i) the Schrödinger picture and (ii) the Heisenberg picture.
c. Evaluate $\left\langle(\Delta x)^{2}\right\rangle$ as a function of time using either picture.

Penny Riley

Numerade Educator

01:28

Problem 20

Show for the one-dimensional simple harmonic oscillator
$$
\left\langle 0\left|e^{i k x}\right| 0\right\rangle=\exp \left[-k^{2}\left\langle 0\left|x^{2}\right| 0\right\rangle / 2\right]
$$
where $x$ is the position operator.

Adriano Chikande

Numerade Educator

01:10

Problem 21

The problem covers some fundamental concepts in quantum optics. See Glauber, Phys. Rev., 84 (1951) 395 and his Nobel lecture, Rev. Mod. Phys., 78 (2006) 1267; Gottfried (1966), Section 31; Merzbacher (1998), Section 10.7; and Gottfried and Yan (2003), Section 4.2.

A coherent state of a one-dimensional simple harmonic oscillator is defined to be an eigenstate of the (non-Hermitian) annihilation operator $a$ :
$$
a|\lambda\rangle=\lambda|\lambda\rangle
$$
where $\lambda$ is, in general, a complex number.
a. Prove that
$$
|\lambda\rangle=e^{-|\lambda|^{2} / 2} e^{\lambda a^{7}}|0\rangle
$$
is a normalized coherent state.
b. Prove the minimum uncertainty relation for such a state.
c. Write $|\lambda\rangle$ as
$$
|\lambda\rangle=\sum_{n=0}^{\infty} f(n)|n\rangle
$$
Show that the distribution of $|f(n)|^{2}$ with respect to $n$ is in the form of a Poisson distribution, that is $P_{n}(\mu)=e^{-\mu} \mu^{n} / n !$ where $\mu$ is the mean of the distribution. Find the most probable (integer) value of $n$, hence of $E$.
d. Show that a coherent state can also be obtained by applying the translation (finite displacement) operator $e^{-i p l l h}$ (where $p$ is the momentum operator, and $l$ is the displacement distance) to the ground state. (See also Gottfried (1966), pp. 262264; Problem $2.13$ in Gottfried and Yan (2003), and Eq. 39 in Glauber's 1951 paper.)

Dominador Tan

Numerade Educator

03:43

Problem 22

Write a computer program to animate the time dependence of an arbitrary linear combination of stationary states for the simple harmonic oscillator in one dimension. The animation should display the time dependence of both the wave function and the probability density distribution for finding the particle, both as a function of position. Check your animation by using pure eigenstates as input, and consider combinations that would approximate classical motion. Also animate the coherent state $|\lambda\rangle$ in Problem $2.21$ above, for different mean values of the Poisson distribution.

Kajal Gautam

Numerade Educator

07:07

Problem 23

Make the definitions
$$
J_{\pm} \equiv \hbar a_{\pm}^{\dagger} a_{\mp}, \quad J_{z} \equiv \frac{\hbar}{2}\left(a_{+}^{\dagger} a_{+}-a_{-}^{\dagger} a_{-}\right), \quad N \equiv a_{+}^{\dagger} a_{+}+a_{-}^{\dagger} a_{-}
$$
where $a_{\pm}$and $a_{\pm}^{\dagger}$ are the annihilation and creation operators of two independent simple harmonic oscillators satisfying the usual simple harmonic oscillator commutation relations. Also make the definition
$$
\mathbf{J}^{2} \equiv J_{z}^{2}+\frac{1}{2}\left(J_{+} J_{-}+J_{-} J_{+}\right)
$$
Prove
$$
\left[J_{z}, J_{\pm}\right]=\pm \hbar J_{\pm}, \quad\left[\mathbf{J}^{2}, J_{z}\right]=0, \quad \mathbf{J}^{2}=\left(\frac{\hbar^{2}}{2}\right) N\left[\left(\frac{N}{2}\right)+1\right]
$$

Hafiz Shahzaib

Numerade Educator

01:00

Problem 24

This exercise has to do with proving conservation laws in classical and quantum physics.
a. Show that if a quantity $Q$ with a density $\rho(\mathbf{x}, t)$ in some region $\mathscr{R}$ can only be changed by a flux density $\mathbf{j}(\mathbf{x}, t)$ through the surface bordering $\mathscr{R}$, then
$$
\frac{\partial \rho}{\partial t}+\nabla \cdot \mathbf{j}=0
$$
b. Prove that Maxwell's equations imply that electric charge is conserved
c. Prove that Schrödinger's equation implies that probability is conserved, i.e. (2.190)

Raj Bala

Numerade Educator

01:00

Problem 25

Derive the normalization constant $c_{n}$ in (2.232) by deriving the orthogonality relationship (2.233) using generating functions. Start by working out the integral
$$
I=\int_{-\infty}^{\infty} g(x, t) g(x, s) e^{-x^{2}} d x
$$
and then consider the integral again with the generating functions in terms of series with Hermite polynomials.

Dominador Tan

Numerade Educator

05:34

Problem 26

Derive an expression for the action of the position operator on an arbitrary state in the momentum representation. That is, find $\left\langle p^{\prime}|x| \alpha\right\rangle$ in terms of $\left\langle p^{\prime} \mid \alpha\right\rangle$. Use this to solve the linear potential (2.234) problem in momentum space, and show that the
Fourier transform of your solution is indeed the appropriate Airy function. Note that an alternative form of the Airy function (see http://dlmf.nist.gov/9.5) is
$$
A i(x)=\frac{1}{\pi} \int_{0}^{\infty} \cos \left(\frac{1}{3} t^{3}+x t\right) d t .
$$
You need not be concerned with normalizing the wave function.

Ameer Said

Numerade Educator

04:57

Problem 27

Consider a particle of mass $m$ subject to a one-dimensional potential of the following form:
$$
V=\left\{\begin{array}{lll}
\frac{1}{2} k x^{2} & \text { for } & x>0 \\
\infty & \text { for } & x<0
\end{array}\right.
$$
a. What is the ground-state energy?
b. What is the expectation value $\left\langle x^{2}\right\rangle$ for the ground state?

Laszlo Zalavari

Numerade Educator

04:45

Problem 28

A particle in one dimension is trapped between two rigid walls:
$$
V(x)=\left\{\begin{array}{lll}
0 & \text { for } & 0<x<L \\
\infty & \text { for } & x<0, x>L .
\end{array}\right.
$$
At $t=0$ it is known to be exactly at $x=L / 2$ with certainty. What are the relative probabilities for the particle to be found in various energy eigenstates? Write down the wave function for $t \geq 0$. (You need not worry about absolute normalization, convergence, and other mathematical subtleties.)

Robert Zaballa

Numerade Educator

01:34

Problem 29

Consider a particle in one dimension bound to a fixed center by a $\delta$-function potential of the form
$$
V(x)=-v_{0} \delta(x)
$$
where $v_{0}$ is real and positive. Find the wave function and the binding energy of the ground state. Are there excited bound states?

Bettina Hanlon

Numerade Educator

11:54

Problem 30

A particle of mass $m$ in one dimension is bound to a fixed center by an attractive $\delta$-function potential:
$$
V(x)=-\lambda \delta(x) \quad(\lambda>0)
$$
At $t=0$, the potential is suddenly switched off (that is, $V=0$ for $t>0$ ). Find the wave function for $t>0$. (Be quantitative! But you need not attempt to evaluate an integral that may appear.)

Ren Jie Tuieng

Numerade Educator

02:17

Problem 31

A particle in one dimension $(-\infty<x<\infty)$ is subjected to a constant force derivable from
$$
V=\lambda x \quad(\lambda>0) .
$$
a. Is the energy spectrum continuous or discrete? Write down an approximate expression for the energy eigenfunction specified by $E$. Also sketch it crudely.
b. Discuss briefly what changes are needed if $V$ is replaced by
$$
V=\lambda|x| \text {. }
$$

Katie Mcalpine

Numerade Educator

11:38

Problem 32

A particle of mass $m$ is confined to a one-dimensional square well with finite walls. That is, a potential $V(x)=0$ for $-a \leq x \leq+a$, and $V(x)=V_{0}=\eta\left(\hbar^{2} / 2 m a^{2}\right)$ otherwise. You are to find the bound-state energy eigenvalues as $E=\varepsilon V_{0}$ along with their wave functions.
a. Set the problem up with a wave function $A e^{\alpha x}$ for $x \leq-a, D e^{-a x}$ for $x \geq+a$, and $B e^{i k x}+C e^{-i k x}$ inside the well. Match the boundary conditions at $x=\pm a$ and show that $k$ and $\alpha$ must satisfy $z=\pm z^{*}$ where $z \equiv e^{\text {iak }}(k-i \alpha) .$ Proceed to find a purely real or purely imaginary expression for $z$ in terms of $k$ and $\alpha$.
b. Find the wave functions for the two choices of $z$ and show that the purely real (imaginary) choice leads to a wave function that is even (odd) under the exchange $x \rightarrow-x$.
c. Find a transcendental equation for each of the two wave functions relating $\eta$ and $\varepsilon$. Show that even a very shallow well $(\eta \rightarrow 0)$ has at least one solution for the even wave function, but you are not guaranteed any solution for an odd wave function.
d. For $\eta=10$, find all the energy eigenvalues and plot their normalized wave functions.

Ameer Said

Numerade Educator

07:53

Problem 33

Consider a particle of mass $m$ moving in one dimension $x$ under the influence of a potential energy function $V(x)$.
a. Show that if $V(-x)=V(x)$, then a solution $u(x)$ to the time-independent Schrödinger equation must have the property $u(-x)=\pm u(x)$. (This is a simple example of parity symmetry, which will be discussed in more detail in Section 4.2.)
b. Consider a potential that is an infinite square well but with a rectangular barrier in the middle. That is $V(x)=V_{0}>0$ for $-b \leq x \leq b$, infinity for $|x|>a$, and zero for $b \leq|x| \leq a$. For $b / a=1 / 3$ and $V_{0}=20\left(\hbar^{2} / 2 m a^{2}\right)$, find the two energy eigenvalues along with their normalized eigenfunctions. (Plots of the eigenfunctions are shown in Figure 4.3.)
c. For $V_{0}=10\left(\hbar^{2} / 2 m a^{2}\right)$ show that there is only one energy eigenstate, and find the eigenvalue.

You will need to write a computer program to calculate the eigenvalues and plot the wave functions. It is easiest to scale the eigenvalues by $\hbar^{2} / 2 m a^{2}$ and find transcendental equations to solve (numerically) for the eigenvalues in the positive and negative parity cases.

Mahipal Kumawat

Numerade Educator

19:46

Problem 34

A particle of mass $m$ moves in one dimension $x$ under a potential energy $V(x)$.
a. For $V(x)=-V_{0} b \delta(x), V_{0}>0, b>0$, find the bound-state energy eigenvalue $E$.
b. Generalize this to the "double delta function" potential
$$
V(x)=-V_{0} \frac{b}{2}\left[\delta\left(x+\frac{a}{2}\right)+\delta\left(x-\frac{a}{2}\right)\right]
$$
and find the bound-state energy eigenvalues and plot the corresponding eigenfunctions. Also show that you get the expected results as $a \rightarrow 0$.

Nathan Silvano

Numerade Educator

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Problem 35

Derive an expression for the density of free-particle states in two dimensions, normalized with periodic boundary conditions inside a box of side length $L$.

Rashmi Sinha

Numerade Educator

03:47

Problem 36

Use the WKB method to find the (approximate) energy eigenvalues for the onedimensional simple harmonic oscillator potential $V(x)=m \omega^{2} x^{2} / 2$.

Lottie Adams

Numerade Educator

05:53

Problem 37

Consider an electron confined to the interior of a hollow cylindrical shell whose axis coincides with the $z$-axis. The wave function is required to vanish on the inner and outer walls, $\rho=\rho_{a}$ and $\rho_{b}$, and also at the top and bottom, $z=0$ and $L$.
a. Find the energy eigenfunctions. (Do not bother with normalization.) Show that the energy eigenvalues are given by
$$
E_{l m n}=\left(\frac{\hbar^{2}}{2 m_{e}}\right)\left[k_{m n}^{2}+\left(\frac{l \pi}{L}\right)^{2}\right] \quad(l=1,2,3, \ldots, m=0,1,2, \ldots)
$$
where $k_{m n}$ is the $n$th root of the transcendental equation
$$
J_{m}\left(k_{m n} \rho_{b}\right) N_{m}\left(k_{m n} \rho_{a}\right)-N_{m}\left(k_{m n} \rho_{b}\right) J_{m}\left(k_{m n} \rho_{a}\right)=0 .
$$
b. Repeat the same problem when there is a uniform magnetic field $\mathbf{B}=B \hat{\mathbf{z}}$ for $0<$ $\rho<\rho_{a}$. Note that the energy eigenvalues are influenced by the magnetic field even though the electron never "touches" the magnetic field.
c. Compare, in particular, the ground state of the $B=0$ problem with that of the $B \neq 0$ problem. Show that if we require the ground-state energy to be unchanged in the presence of $B$, we obtain "flux quantization"
$$
\pi \rho_{a}^{2} B=\frac{2 \pi N \hbar c}{e} \quad(N=0, \pm 1, \pm 2, \ldots)
$$

Khoobchandra Agrawal

Numerade Educator

04:39

Problem 38

Consider a particle moving in one dimension under the influence of a potential $V(x)$. Suppose its wave function can be written as $\exp [i S(x, t) / \hbar]$. Prove that $S(x, t)$ satisfies the classical Hamilton-Jacobi equation to the extent that $\hbar$ can be regarded as small in some sense. Show how one may obtain the correct wave function for a plane wave by starting with the solution of the classical Hamilton-Jacobi equation with $V(x)$ set equal to zero. Why do we get the exact wave function in this particular case?

Nathan Silvano

Numerade Educator

06:23

Problem 39

Using spherical coordinates, obtain an expression for $\mathbf{j}$ for the ground and excited states of the hydrogen atom. Show, in particular, that for $m_{l} \neq 0$ states, there is a circulating flux in the sense that $\mathbf{j}$ is in the direction of increasing or decreasing $\phi$, depending on whether $m_{l}$ is positive or negative.

Mirza Aslam Beig

Numerade Educator

02:25

Problem 40

Derive $(2.280)$ and obtain the three-dimensional generalization of $(2.280)$.

James Kiss

Numerade Educator

25:40

Problem 41

A particle of mass $m$ moves along one of two "paths" through space and time connecting the points $(x, t)=(0,0)$ and $(x, t)=(D, T) .$ One path is quadratic in time, i.e. $x_{1}(t)=\frac{1}{2} a t^{2}$ where $a$ is a constant. The second path is linear in time, i.e. $x_{2}(t)=v t$ where $v$ is a constant. The correct classical path is the quadratic path, that is $x_{1}(t)$.
a. Find the acceleration $a$ for the correct classical path. Use freshman physics to find the force $F=m a=-d V / d x$ and then the potential energy function $V(x)$ in terms of $m, D$, and $T .$ Also find the velocity $v$ for the linear (i.e. incorrect classical) path.
b. Calculate the classical action $S[x(t)]=\int_{0}^{T}\left[\frac{1}{2} m \dot{x}^{2}-V(x)\right] d t$ for each of the two paths $x_{1}(t)$ and $x_{2}(t)$. Confirm that $S_{1} \equiv S\left[x_{1}(t)\right]<S_{2} \equiv S\left[x_{2}(t)\right]$, and find $\Delta S=$ $S_{2}-S_{1}$.
c. Calculate $\Delta S / \hbar$ for a particle which moves $1 \mathrm{~mm}$ in $1 \mathrm{~ms}$ for two cases. The particle is a nanoparticle made up of 100 carbon atoms in one case. The other case is an electron. For which of these would you consider the motion "quantum mechanical" and why?

Zachary Warner

Numerade Educator

02:36

Problem 42

Define the partition function as
$$
Z=\left.\int d^{3} x^{\prime} K\left(\mathbf{x}^{\prime}, t ; \mathbf{x}^{\prime}, 0\right)\right|_{\beta=i t h},
$$
as in (2.284)-(2.286). Show that the ground-state energy is obtained by taking
$$
-\frac{1}{Z} \frac{\partial Z}{\partial \beta} \quad(\beta \rightarrow \infty)
$$
Illustrate this for a particle in a one-dimensional box.

Chai Santi

Numerade Educator

01:01

Problem 43

The propagator in momentum space analogous to $(2.290)$ is given by $\left\langle\mathbf{p}^{\prime \prime}, t \mid \mathbf{p}^{\prime}, t_{0}\right\rangle .$ Derive an explicit expression for $\left\langle\mathbf{p}^{\prime \prime}, t \mid \mathbf{p}^{\prime}, t_{0}\right\rangle$ for the free-particle case.

Raj Bala

Numerade Educator

11:37

Problem 44

a. Write down an expression for the classical action for a simple harmonic oscillator for a finite time interval.
b. Construct $\left\langle x_{n}, t_{n} \mid x_{n-1}, t_{n-1}\right\rangle$ for a simple harmonic oscillator using Feynman's prescription for $t_{n}-t_{n-1}=\Delta t$ small. Keeping only terms up to order $(\Delta t)^{2}$, show that it is in complete agreement with the $t-t_{0} \rightarrow 0$ limit of the propagator given by $(2.290)$.

Susan Hallstrom

Numerade Educator

05:46

Problem 45

State the Schwinger action principle (see Finkelstein (1973), p. 155). Obtain the solution for $\left\langle x_{2} t_{2} \mid x_{1} t_{1}\right\rangle$ by integrating the Schwinger principle and compare it with the corresponding Feynman expression for $\left\langle x_{2} t_{2} \mid x_{1} t_{1}\right\rangle$. Describe the classical limits of these two expressions.

Cheryl Coles

Numerade Educator

03:39

Problem 46

Show that the wave-mechanical approach to the gravity-induced problem discussed in Section $2.7$ also leads to phase-difference expression $(2.337)$.

Khoobchandra Agrawal

Numerade Educator

01:54

Problem 47

a. Verify $(2.345)$ and $(2.347)$
b. Verify continuity equation $(2.350)$ with $\mathbf{j}$ given by $(2.351)$.

Penny Riley

Numerade Educator

01:42

Problem 48

Consider the Hamiltonian of a spinless particle of charge $e$. In the presence of a static magnetic field, the interaction terms can be generated by
$$
\mathbf{p}_{\text {operator }} \rightarrow \mathbf{p}_{\text {operator }}-\frac{e \mathbf{A}}{c} \text {, }
$$
where $\mathbf{A}$ is the appropriate vector potential. Suppose, for simplicity, that the magnetic field $\mathbf{B}$ is uniform in the positive $z$-direction. Prove that the above prescription indeed leads to the correct expression for the interaction of the orbital magnetic moment (e/2mc)L with the magnetic field B. Show that there is also an extra term proportional to $B^{2}\left(x^{2}+y^{2}\right)$, and comment briefly on its physical significance.

Ajay Singhal

Numerade Educator

08:06

Problem 49

An electron moves in the presence of a uniform magnetic field in the $z$-direction $(\mathbf{B}=B \hat{\mathbf{z}})$
a. Evaluate
$$
\left[\Pi_{x}, \Pi_{y}\right]
$$
where
$$
\Pi_{x} \equiv p_{x}-\frac{e A_{x}}{c}, \quad \Pi_{y} \equiv p_{y}-\frac{e A_{y}}{c} .
$$
b. By comparing the Hamiltonian and the commutation relation obtained in (a) with those of the one-dimensional oscillator problem, show how we can immediately write the energy eigenvalues as
$$
E_{k, n}=\frac{\hbar^{2} k^{2}}{2 m}+\left(\frac{|e B| \hbar}{m c}\right)\left(n+\frac{1}{2}\right),
$$
where $\hbar k$ is the continuous eigenvalue of the $p_{z}$ operator and $n$ is a nonnegative integer including zero.

Khoobchandra Agrawal

Numerade Educator

02:44

Problem 50

Consider the neutron interferometer.
Prove that the difference in the magnetic fields that produce two successive maxima in the counting rates is given by
$$
B=\frac{4 \pi \hbar c}{|e| g_{n} \pi l}
$$
where $g_{n}(=-1.91)$ is the neutron magnetic moment in units of $-e \hbar / 2 m_{n} c$, and $\lambda \equiv$ $\lambda / 2 \pi$. This problem was in fact analyzed in the paper by Bernstein, Phys. Rev. Lett., 18 (1967) 1102 .

Guilherme Barros

Numerade Educator