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Modern Quantum Mechanics 2nd

J. J. Sakurai, Jim Napolitano

Chapter 2

Quantum Dynamics

Educators


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.

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Problem 2

Look again at the Hamiltonian of Chapter $1,$ Problem $1.11 .$ 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.)

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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 S $\cdot \hat{n}$ with eigenvalue $\hbar / 2,$ where $\hat{n}$ 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) $\beta \rightarrow \pi / 2$

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Problem 4

Derive the neutrino oscillation probability $(2.1 .65)$ 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 $\theta$

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Problem 5

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

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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 $E_{a^{\prime}}$

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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
\[
\frac{d}{d t}\langle\mathbf{x} \cdot \mathbf{p}\rangle=\left\langle\frac{\mathbf{p}^{2}}{m}\right\rangle-\langle\mathbf{x} \cdot \nabla V\rangle
\]
In order for us 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?

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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 minimum uncertainty wave packet you worked out in Chapter $1,$ Problem $1.18 .$ )

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Problem 9

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?

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Problem 10

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
\[
|\alpha\rangle=|R\rangle\langle R | \alpha\rangle+|L\rangle\langle L | \alpha\rangle
\]
where $\langle R | \alpha\rangle$ and $\langle L | \alpha\rangle$ 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 that 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 $\left\langle L | \alpha, t_{0}=0 ; t\right\rangle .$ 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.

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Problem 11

Using the one-dimensional simple harmonic oscillator as an example, illustrate the difference between the Heisenberg picture and the Schrödinger picture. Discuss in particular how (a) the dynamic variables $x$ and $p$ and (b) the most general state vector evolve with time in each of the two pictures.

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Problem 12

Consider a particle subject to a one-dimensional simple harmonic oscillator potential. Suppose that 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 and $a$ is some number with dimension of length. Using the Heisenberg picture, evaluate the expectation value $\langle x\rangle$ for $t \geq 0$

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Problem 13

(a) Write down the wave function (in coordinate space) for the state specified in Problem 2.12 at $t=0 .$ You may use
\[
\left\langle x^{\prime} | 0\right\rangle=\pi^{-1 / 4} x_{0}^{-1 / 2} \exp \left[-\frac{1}{2}\left(\frac{x^{\prime}}{x_{0}}\right)^{2}\right], \quad\left(x_{0} \equiv\left(\frac{\hbar}{m \omega}\right)^{1 / 2}\right)
\]
(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 $t>0 ?$

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Problem 14

Consider a one-dimensional simple harmonic oscillator.
(a) Using
\[
\left.\left.\begin{array}{l}
a \\
a^{\dagger}
\end{array}\right\}=\sqrt{\frac{m \omega}{2 \hbar}}\left(x \pm \frac{i p}{m \omega}\right), \quad \begin{array}{c}
a|n\rangle \\
a^{\dagger}|n\rangle
\end{array}\right\}=\left\{\begin{array}{l}
\sqrt{n}|n-1\rangle \\
\sqrt{n+1}|n+1\rangle
\end{array}\right.
\]
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) Check that the virial theorem holds for the expectation values of the kinetic energy and the potential energy taken with respect to an energy eigenstate.

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Problem 15

(a) Using
\[
\left\langle x^{\prime} | p^{\prime}\right\rangle=(2 \pi \hbar)^{-1 / 2} e^{i p^{\prime} x^{\prime} / \hbar} \quad \text { (one dimension) }
\]
prove
\[
\left\langle p^{\prime}|x| \alpha\right\rangle=i \hbar \frac{\partial}{\partial p^{\prime}}\left\langle p^{\prime} | \alpha\right\rangle
\]
(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?

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Problem 16

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.

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Problem 17

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.

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Problem 18

Show that 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.

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Problem 19

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
\[
|\lambda\rangle=e^{-|\lambda|^{2} / 2} e^{\lambda a^{\dagger}}|0\rangle
\]
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 of the Poisson form. Find the most probable value of $n,$ and hence of $E$
(d) Show that a coherent state can also be obtained by applying the translation (finite-displacement) operator $e^{-i p l / \hbar}$ (where $p$ is the momentum operator and $l$ is the displacement distance to the ground state. (See also Gottfried 1966 $262-64$

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Problem 20

Let
\[
J_{\pm}=\hbar a_{\pm}^{\dagger} a_{\mp}, \quad J_{z}=\frac{\hbar}{2}\left(a_{+}^{\dagger} a_{+}-a_{-}^{\dagger} a_{-}\right), \quad N=a_{+}^{\dagger} a_{+}+a_{-}^{\dagger} a_{-}
\]
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. Prove
\[
\left[J_{z}, J_{\pm}\right]=\pm \hbar J_{\pm}, \quad\left[\mathbf{J}^{2}, J_{z}\right]=0, \quad \mathbf{J}^{2}=\left(\frac{\hbar^{2}}{2}\right) N\left[\left(\frac{N}{2}\right)+1\right]
\]

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Problem 21

Derive the normalization constant $c_{n}$ in $(2.5 .28)$ by deriving the orthogonality relationship $(2.5 .29)$ 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.

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Problem 22

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

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Problem 23

A particle in one dimension is trapped between two rigid walls:
\[
V(x)=\left\{\begin{array}{ll}
0, & \text { for } 0<x<L \\
\infty, & \text { for } x<0, x>L
\end{array}\right.
\]
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.)

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Problem 24

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

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Problem 25

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.)

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Problem 26

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
\[
V=\lambda|x|
\]

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Problem 27

Derive an expression for the density of free-particle states in $t w o$ dimensions, normalized with periodic boundary conditions inside a box of side length $L .$ Your answer should be written as a function of $k(\text { or } E)$ times $d E d \phi,$ where $\phi$ is the polar angle that characterizes the momentum direction in two dimensions.

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Problem 28

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
\[
E_{l m n}=\left(\frac{\hbar^{2}}{2 m_{e}}\right)\left[k_{m n}^{2}+\left(\frac{l \pi}{L}\right)^{2}\right] \quad(l=1,2,3, \dots, m=0,1,2, \dots)
\]
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"
\[
\pi \rho_{a}^{2} B=\frac{2 \pi N \hbar c}{e}, \quad(N=0,\pm 1,\pm 2, \ldots)
\]

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Problem 29

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?

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Problem 30

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.

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Problem 31

Derive $(2.6 .16)$ and obtain the three-dimensional generalization of $(2.6 .16).$

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Problem 32

Define the partition function as
\[
Z=\left.\int d^{3} x^{\prime} K\left(\mathbf{x}^{\prime}, t ; \mathbf{x}^{\prime}, 0\right)\right|_{\beta=i t / \hbar}
\]
as in $(2.6 .20)-(2.6 .22) .$ 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.

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Problem 33

The propagator in momentum space analogous to $(2.6 .26)$ is given by $\left\langle\mathbf{p}^{\prime \prime}, t | \mathbf{p}^{\prime}, t_{0}\right\rangle$ Derive an explicit expression for $\left\langle\mathbf{p}^{\prime \prime}, t | \mathbf{p}^{\prime}, t_{0}\right\rangle$ for the free-particle case.

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Problem 34

(a) Write down an expression for the classical action for a simple harmonic oscillator for a finite time interval.
(b) Construct $\left\langle x_{n}, t_{n} | x_{n-1}, t_{n-1}\right\rangle$ 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 $(2.6 .26)$

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Problem 35

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

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Problem 36

Show that the wave-mechanical approach to the gravity-induced problem discussed in Section 2.7 also leads to phase-difference expression $(2.7 .17)$

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Problem 37

(a) Verify (2.7.25) and (2.7.27).
(b) Verify continuity equation $(2.7 .30)$ with $\mathbf{j}$ given by $(2.7 .31)$

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Problem 38

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
\[
\mathbf{p}_{\text {operator }} \rightarrow \mathbf{p}_{\text {operator }}-\frac{e \mathbf{A}}{c}
\]
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 $(e / 2 m c) \mathbf{L}$ with the magnetic field $\mathbf{B}$. 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.

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Problem 39

An electron moves in the presence of a uniform magnetic field in the $z$ -direction
\[
(\mathbf{B}=B \mathbf{2})
\]
(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.

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Problem 40

Consider the neutron interferometer.
Prove that the difference in the magnetic fields that produce two successive maxima in the counting rates is given by
\[
\Delta B=\frac{4 \pi \hbar c}{|e| g_{n} \bar{\lambda} l}
\]
where $g_{n}(=-1.91)$ is the neutron magnetic moment in units of $-e \hbar / 2 m_{n} c .$ (If you had solved this problem in $1967,$ you could have published your solution in Physical Review Letters!

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