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Modern Quantum Mechanics

J. J. Sakurai, Jim Napolitano

Chapter 5

Approximation Methods - all with Video Answers

Educators

Chapter Questions

09:46

Problem 1

A simple harmonic oscillator (in one dimension) is subjected to a perturbation
$$
H_{1}=b x
$$
where $b$ is a real constant.
a. Calculate the energy shift of the ground state to lowest nonvanishing order.
b. Solve this problem exactly and compare with your result obtained in (a).

CG
Coleman Green
Numerade Educator
00:35

Problem 2

A one-dimensional potential well has infinite walls at $x=0$ and $x=L$. The bottom of the well is not flat, but rather increases linearly from 0 at $x=0$ to $V$ at $x=L$. Find the first-order shift in the energy levels as a function of principal quantum number $n$.

Salamat Ali
Salamat Ali
Numerade Educator
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Problem 3

A particle of mass $m$ moves in a potential well $V(x)=m \omega^{2} x^{2} / 2$. Treating relativistic effects to order $\beta^{2}=(p / m c)^{2}$, find the ground-state energy shift.

Lainey Roebuck
Lainey Roebuck
Numerade Educator
03:42

Problem 4

A diatomic molecule can be modeled as a rigid rotor with moment of inertia $I$ and an electric dipole moment $d$ along the axis of the rotor. The rotor is constrained to rotate in a plane, and a weak uniform electric field $\&$ lies in the plane. Write the classical Hamiltonian for the rotor, and find the unperturbed energy levels by quantizing the angular-momentum operator. Then treat the electric field as a perturbation, and find the first nonvanishing corrections to the energy levels.

Vishal Gupta
Vishal Gupta
Numerade Educator
11:22

Problem 5

In nondegenerate time-independent perturbation theory, what is the probability of finding in a perturbed energy eigenstate $(|k\rangle)$ the corresponding unperturbed eigenstate $\left(\left|k^{(0)}\right\rangle\right)$ ? Solve this up to terms of order $\lambda^{2}$.

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
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Problem 6

Consider a particle in a two-dimensional potential
$$
V_{0}= \begin{cases}0 & \text { for } 0 \leq x \leq L, 0 \leq y \leq L \\ \infty & \text { otherwise }\end{cases}
$$
Write the energy eigenfunctions for the ground and first excited states. We now add a time-independent perturbation of the form
$$
V_{1}= \begin{cases}\lambda x y & \text { for } 0 \leq x \leq L, 0 \leq y \leq L \\ 0 & \text { otherwise }\end{cases}
$$
Obtain the zeroth-order energy eigenfunctions and the first-order energy shifts for the ground and first excited states.

AP
Andreas Papavassiliou
Numerade Educator
03:29

Problem 7

Consider an isotropic harmonic oscillator in two dimensions. The Hamiltonian is
$$
H_{0}=\frac{p_{x}^{2}}{2 m}+\frac{p_{y}^{2}}{2 m}+\frac{m \omega^{2}}{2}\left(x^{2}+y^{2}\right) .
$$
a. What are the energies of the three lowest-lying states? Is there any degeneracy?
b. We now apply a perturbation
$$
V=\delta m \omega^{2} x y
$$
where $\delta$ is a dimensionless real number much smaller than unity. Find the zerothorder energy eigenket and the corresponding energy to first order [that is, the unperturbed energy obtained in (a) plus the first-order energy shift] for each of the three lowest-lying states.
c. Solve the $H_{0}+V$ problem exactly. Compare with the perturbation results obtained in (b).

Keshav Singh
Keshav Singh
Numerade Educator
23:12

Problem 8

Establish (5.54) for the one-dimensional harmonic oscillator given by (5.50) with an additional perturbation $V=\frac{1}{2} \varepsilon m \omega^{2} x^{2}$. Show that all other matrix elements $V_{k 0}$ vanish.

Dr. Rajveer Singh
Dr. Rajveer Singh
Numerade Educator
02:11

Problem 9

A slightly anisotropic three-dimensional harmonic oscillator has $\omega_{x}=\omega_{y} \equiv \omega$ and $\omega_{z}=(1+\varepsilon) \omega$ where $\varepsilon \ll 1$. (See Section $3.7 .3$ for nomenclature and wave functions.) A charged particle moves in the field of this oscillator and is at the same time exposed to a uniform magnetic field in the $x$-direction. Assuming that the Zeeman splitting is comparable to the splitting produced by the anisotropy, but small compared to $\hbar \omega$, calculate to first order the energies of the components of the first excited state. Discuss various limiting cases. (This is taken from Problem $17.7$ in Merzbacher (1970). You might find it useful to consult Problem $2.16$ in Chapter 2 and Problem $3.29$ in Chapter $3 .$ )

Zhuxi Luo
Zhuxi Luo
Numerade Educator
08:06

Problem 10

A one-electron atom whose ground state is nondegenerate is placed in a uniform electric field in the $z$-direction. Obtain an approximate expression for the induced electric dipole moment of the ground state by considering the expectation value of $e z$ with respect to the perturbed state vector computed to first order. Show that the same expression can also be obtained from the energy shift $\Delta=-\alpha|\mathbf{E}|^{2} / 2$ of the ground state computed to second order. (Note: $\alpha$ stands for the polarizability.) Ignore spin.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
08:34

Problem 11

Evaluate the matrix elements (or expectation values) given below. If any vanishes, explain why it vanishes using simple symmetry (or other) arguments.
a. $\langle n=2, l=1, m=0|x| n=2, l=0, m=0\rangle$.
b. $\left\langle n=2, l=1, m=0\left|p_{z}\right| n=2, l=0, m=0\right\rangle$.
[In (a) and (b), $|n l m\rangle$ stands for the energy eigenket of a nonrelativistic hydrogen atom with spin ignored.]
c. $\left\langle L_{z}\right\rangle$ for an electron in a central field with $j=\frac{9}{2}, m=\frac{7}{2}, l=4$.
d. $\left\langle\operatorname{singlet}, m_{s}=0\left|S_{z}^{(e-)}-S_{z}^{(c+)}\right|\right.$ triplet, $\left.m_{s}=0\right\rangle$ for an $s$-state positronium.
e. $\left\langle\mathbf{S}^{(1)} \cdot \mathbf{S}^{(2)}\right\rangle$ for the ground state of a hydrogen molecule.

Mahnoor Amin
Mahnoor Amin
Numerade Educator
05:34

Problem 12

A $p$-orbital electron characterized by $|n, l=1, m=\pm 1,0\rangle$ (ignore spin) is subjected to a potential
$$
V=\lambda\left(x^{2}-y^{2}\right) \quad(\lambda=\text { constant })
$$
a. Obtain the "correct" zeroth-order energy eigenstates that diagonalize the perturbation. You need not evaluate the energy shifts in detail, but show that the original threefold degeneracy is now completely removed.
b. Because $V$ is invariant under time reversal and because there is no longer any degeneracy, we expect each of the energy eigenstates obtained in (a) to go into itself (up to a phase factor or sign) under time reversal. Check this point explicitly.

Nathan Silvano
Nathan Silvano
Numerade Educator
07:06

Problem 13

Consider a spinless particle in a two-dimensional infinite square well:
$$
V= \begin{cases}0 & \text { for } 0 \leq x \leq a, 0 \leq y \leq a \\ \infty & \text { otherwise }\end{cases}
$$
a. What are the energy eigenvalues for the three lowest states? Is there any degeneracy?
b. We now add a potential
$$
V_{1}=\lambda x y, 0 \leq x \leq a, 0 \leq y \leq a
$$
Taking this as a weak perturbation, answer the following.
(i) Is the energy shift due to the perturbation linear or quadratic in $\lambda$ for each of the three states?
(ii) Obtain expressions for the energy shifts of the three lowest states accurate to order $\lambda$. (You need not evaluate integrals that may appear.)
(iii) Draw an energy diagram with and without the perturbation for the three energy states. Make sure to specify which unperturbed state is connected to which perturbed state.

Andrew Eddins
Andrew Eddins
Emory University
07:06

Problem 14

The Hamiltonian matrix for a two-state system can be written as
$$
\mathscr{H}=\left(\begin{array}{ll}
E_{1}^{0} & \lambda \Delta \\
\lambda \Delta & E_{2}^{0}
\end{array}\right)
$$
Clearly the energy eigenfunctions for the unperturbed problems $(\lambda=0)$ are given by
$$
\phi_{1}^{(0)}=\left(\begin{array}{l}
1 \\
0
\end{array}\right), \quad \phi_{2}^{(0)}=\left(\begin{array}{l}
0 \\
1
\end{array}\right) \text {. }
$$
a. Solve this problem exactly to find the energy eigenfunctions $\psi_{1}$ and $\psi_{2}$ and the energy eigenvalues $E_{1}$ and $E_{2}$.
b. Assuming that $\lambda|\Delta| \ll\left|E_{1}^{0}-E_{2}^{0}\right|$, solve the same problem using time-independent perturbation theory up to first order in the energy eigenfunctions and up to second order in the energy eigenvalues. Compare with the exact results obtained in (a).
c. Suppose the two unperturbed energies are "almost degenerate," that is,
$$
\left|E_{1}^{0}-E_{2}^{0}\right| \ll \lambda|\Delta|
$$
Show that the exact results obtained in (a) closely resemble what you would expect by applying degenerate perturbation theory to this problem with $E_{1}^{0}$ set exactly equal to $E_{2}^{0}$.

Andrew Eddins
Andrew Eddins
Emory University
07:06

Problem 15

(This is a tricky problem because the degeneracy between the first and the second state is not removed in first order. See also Gottfried (1966), p. 397, Problem 1.) This problem is from Schiff (1968), p. 295, Problem 4. A system that has three unperturbed states can be represented by the perturbed Hamiltonian matrix
$$
\left(\begin{array}{lll}
E_{1} & 0 & a \\
0 & E_{1} & b \\
a^{*} & b^{*} & E_{2}
\end{array}\right)
$$
where $E_{2}>E_{1}$. The quantities $a$ and $b$ are to be regarded as perturbations that are of the same order and are small compared with $E_{2}-E_{1}$. Use the second-order nondegenerate perturbation theory to calculate the perturbed eigenvalues. (Is this procedure correct?) Then diagonalize the matrix to find the exact eigenvalues. Finally, use the second-order degenerate perturbation theory. Compare the three results obtained.

Andrew Eddins
Andrew Eddins
Emory University
06:09

Problem 16

Use perturbation theory to calculate the effect of the proton's finite size on the $n=1$ and $n=2$ energy levels of the hydrogen atom. Assume the proton is a uniformly charged sphere of radius $R$. Give a physical explanation for why the $\ell=1$ shifts are so much smaller than for $\ell=0$.

Hafiz Shahzaib
Hafiz Shahzaib
Numerade Educator
02:47

Problem 17

This chapter derived two of the three relativistic corrections to the one-electron atom, namely $\Delta_{K}^{(1)}$ from "relativistic kinetic energy"" and $\Delta_{L S}^{(1)}$ from the spin-orbit interaction. A third term comes from the spread of the electron wave function in the region of changing electric field. The perturbation for this "Darwin term" is
$$
V_{D}=-\frac{1}{8 m^{2} c^{2}} \sum_{i=1}^{3}\left[p_{i},\left[p_{i}, e \phi(r)\right]\right]
$$
where $\phi(r)$ is the Coulomb potential. Find $\Delta_{D}^{(1)}$ and show that
$$
\Delta_{n j}^{(1)} \equiv \Delta_{K}^{(1)}+\Delta_{L S}^{(1)}+\Delta_{D}^{(1)}=\frac{m c^{2}(Z \alpha)^{4}}{2 n^{3}}\left[\frac{3}{4 n}-\frac{1}{j+1 / 2}\right]
$$
In Section $8.4$ we will compare this expression to the result of solving the Dirac equation in the presence of the Coulomb potential.

Chai Santi
Chai Santi
Numerade Educator
10:36

Problem 18

These questions are meant to associate numbers with atomic hydrogen phenomena.
a. The red $n=3 \rightarrow 2$ Balmer transition has a wavelength $\lambda \approx 656 \mathrm{~nm}$. Calculate the wavelength difference $\Delta \lambda($ in $\mathrm{nm})$ between the $3 p_{3 / 2} \rightarrow 2 s_{1 / 2}$ and $3 p_{1 / 2} \rightarrow 2 s_{1 / 2}$ transitions due to the spin-orbit interaction. Comment on how you might measure this splitting.
b. How large an electric field 8 is needed so that the Stark splitting in the $n=2$ level is the same as the correction from relativistic kinetic energy between the $2 s$ and $2 p$ levels? How easy or difficult is it to achieve an electric field of this magnitude in the laboratory?
c. The Zeeman effect can be calculated with a "weak" or "strong" magnetic field, depending on the size of the energy shift relative to the spin-orbit splitting. Give examples of a weak and a strong field. How easy or difficult is it to achieve such a magnetic field?

Eduard Sanchez
Eduard Sanchez
Numerade Educator
04:35

Problem 19

Compute the Stark effect for the $2 s_{1 / 2}$ and $2 p_{1 / 2}$ levels of hydrogen for a field $\mathscr{}$ sufficiently weak so that $e \& a_{0}$ is small compared to the fine structure, but take the Lamb shift $\delta(\delta=1057 \mathrm{MHz})$ into account (that is, ignore $2 p_{3 / 2}$ in this calculation). Show that for $e \mathscr{a}_{0} \ll \delta$, the energy shifts are quadratic in $\mathscr{\text { , whereas for }} e \delta a_{0} \gg \delta$ they are linear in $\&$. Briefly discuss the consequences (if any) of time reversal for this problem. This problem is from Gottfried (1966), Problem 7-3.

Zachary Warner
Zachary Warner
Numerade Educator
01:13

Problem 20

Work out the Stark effect to lowest nonvanishing order for the $n=3$ level of the hydrogen atom. Ignoring the spin-orbit force and relativistic correction (Lamb shift), obtain not only the energy shifts to lowest nonvanishing order but also the corresponding zeroth-order eigenket.

Penny Riley
Penny Riley
Numerade Educator
04:50

Problem 21

Suppose the electron had a very small intrinsic electric dipole moment analogous to the spin magnetic moment (that is, $\mu_{e l}$ proportional to $\sigma$ ). Treating the hypothetical $-\mu_{e l} \cdot \mathbf{E}$ interaction as a small perturbation, discuss qualitatively how the energy levels of the Na atom $(Z=11)$ would be altered in the absence of any external electromagnetic field. Are the level shifts first order or second order? State explicitly which states get mixed with each other. Obtain an expression for the energy shift of the lowest level that is affected by the perturbation. Assume throughout that only the valence electron is subjected to the hypothetical interaction.

Suzanne W.
Suzanne W.
Numerade Educator
15:48

Problem 22

Consider a particle bound to a fixed center by a spherically symmetric potential $V(r)$.
a. Prove
$$
|\psi(0)|^{2}=\left(\frac{m}{2 \pi \hbar^{2}}\right)\left\langle\frac{d V}{d r}\right\rangle
$$
for all $s$ states, ground and excited.
b. Check this relation for the ground state of a three-dimensional isotropic oscillator, the hydrogen atom, and so on. (Note: This relation has actually been found to be useful in guessing the form of the potential between a quark and an antiquark. See Moxhay and Rosner, J. Math. Phys., 21 (1980) 1688.)

Mahnoor Amin
Mahnoor Amin
Numerade Educator
08:06

Problem 23

a. Suppose the Hamiltonian of a rigid rotator in a magnetic field perpendicular to the axis is of the form (Merzbacher 1970 , Problem 17-1)
$$
A \mathbf{L}^{2}+B L_{z}+C L_{y}
$$
if terms quadratic in the field are neglected. Assuming $B \gg C$, use perturbation theory to lowest nonvanishing order to get approximate energy eigenvalues.
b. Consider the matrix elements
$$
\begin{gathered}
\left\langle n^{\prime} l^{\prime} m_{l}^{\prime} m_{s}^{\prime}\left|\left(3 z^{2}-r^{2}\right)\right| n l m_{l} m_{s}\right\rangle \\
\left\langle n^{\prime} R^{\prime} m_{l}^{\prime} m_{s}^{\prime}|x y| n l m_{l} m_{s}\right\rangle
\end{gathered}
$$
of a one-electron (for example, alkali) atom. Write the selection rules for $\Delta l, \Delta m l$, and $\Delta m_{s}$. Justify your answer.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
10:39

Problem 24

The $n=2$ state of hydrogen is eightfold degenerate, accounting for both spin and orbital angular momentum. This degeneracy is broken by a perturbation
$$
V=\frac{A}{\hbar^{2}} \mathbf{L} \cdot \mathbf{S}+\frac{B}{\hbar}\left(L_{z}+2 S_{z}\right)
$$
where $\mathbf{L}$ and $\mathbf{S}$ are the orbital and spin angular-momentum operators, $A$ is a constant, and $B$ is proportional (but not equal) to an applied magnetic field in the $z$-direction.
a. Write $V$ in terms of $\mathbf{J}^{2}, \mathbf{L}^{2}, \mathbf{S}^{2}, J_{z}$, and $S_{z}$, where $\mathbf{J}=\mathbf{L}+\mathbf{S}$.
b. Find all nonzero matrix elements of $V$ in the basis $|l, s=1 / 2, j=l \pm 1 / 2, m\rangle$ for the eight $n=2$ states. Hint: Show that the $8 \times 8$ matrix decouples into four $2 \times 2$ matrices, two of which are diagonal.
c. Use degenerate perturbation theory to find the first-order energy shifts $\Delta$. For all eight states, plot $\Delta / A$ as a function of $B / A$. See Figure $5.3 .$ Explain why the resulting spectrum looks qualitatively different for $B / A \ll 1$ and $B / A \gg 1$.

Yaqub Khan
Yaqub Khan
Numerade Educator
13:12

Problem 25

Work out the quadratic Zeeman effect for the ground-state hydrogen atom due to the usually neglected $e^{2} \mathbf{A}^{2} / 2 m_{e} c^{2}$-term in the Hamiltonian taken to first order. Write the energy shift as
$$
\Delta=-\frac{1}{2} \chi \mathbf{B}^{2}
$$
and obtain an expression for diamagnetic susceptibility, $\chi$.

Dr. Rajveer Singh
Dr. Rajveer Singh
Numerade Educator
08:06

Problem 26

(Merzbacher (1970), p. 448, Problem 11.) For the He wave function, use
$$
\psi\left(\mathbf{x}_{1}, \mathbf{x}_{2}\right)=\left(Z_{\mathrm{efl}}^{3} / \pi a_{0}^{3}\right) \exp \left[\frac{-Z_{\mathrm{eff}}\left(r_{1}+r_{2}\right)}{a_{0}}\right]
$$
with $Z_{\mathrm{eff}}=2-\frac{5}{16}$, as obtained by the variational method. The measured value of the diamagnetic susceptibility is $1.88 \times 10^{-6} \mathrm{~cm}^{3} / \mathrm{mole}$.
a. Using the Hamiltonian for an atomic electron in a magnetic field, determine, for a state of zero angular momentum, the energy change to order $B^{2}$ if the system is in a uniform magnetic field represented by the vector potential $\mathbf{A}=\frac{1}{2} \mathbf{B} \times \mathbf{x}$.
b. Defining the atomic diamagnetic susceptibility $\chi$ by $E=-\frac{1}{2} \chi B^{2}$, calculate $\chi$ for a helium atom in the ground state and compare the result with the measured value.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
05:15

Problem 27

Estimate the ground-state energy of a one-dimensional simple harmonic oscillator using
$$
\langle x \mid \tilde{0}\rangle=e^{-\beta|x|}
$$
as a trial function with $\beta$ to be varied.

Keshav Singh
Keshav Singh
Numerade Educator
08:26

Problem 28

Estimate the lowest eigenvalue $(\lambda)$ of the differential equation
$$
\frac{d^{2} \psi}{d x^{2}}+(\lambda-|x|) \psi=0
$$
where $\psi \rightarrow 0$ as $|x| \rightarrow \infty$ using the variational method with
$$
\psi=\left\{\begin{array}{ll}
c(\alpha-|x|) & \text { for }|x|<\alpha \\
0 & \text { for }|x|>\alpha
\end{array} \quad(\alpha \text { to be varied })\right.
$$
as a trial function. (Caution: $d \psi / d x$ is discontinuous at $x=0$.) The exact value of the lowest eigenvalue can be shown to be $1.019$.

Hafiz Shahzaib
Hafiz Shahzaib
Numerade Educator
02:07

Problem 29

Consider a one-dimensional simple harmonic oscillator whose classical angular frequency is $\omega_{0}$. For $t<0$ it is known to be in the ground state. For $t>0$ there is also a time-dependent potential
$$
V(t)=F_{0} x \cos \omega t
$$
where $F_{0}$ is constant in both space and time. Obtain an expression for the expectation value $\langle x\rangle$ as a function of time using time-dependent perturbation theory to lowest nonvanishing order. Is this procedure valid for $\omega \simeq \omega_{0}$ ?

Penny Riley
Penny Riley
Numerade Educator
01:51

Problem 30

A one-dimensional harmonic oscillator is in its ground state for $t<0$. For $t \geq 0$ it is subjected to a time-dependent but spatially uniform force (not potential!) in the $x$-direction,
$$
F(t)=F_{0} e^{-t / \tau}
$$
a. Using time-dependent perturbation theory to first order, obtain the probability of finding the oscillator in its first excited state for $t>0$. Show that the $t \rightarrow \infty$ ( $\tau$ finite) limit of your expression is independent of time. Is this reasonable or surprising?
b. Can we find higher excited states?

Penny Riley
Penny Riley
Numerade Educator
View

Problem 31

Consider a particle bound in a simple harmonic oscillator potential. Initially $(t<0)$, it is in the ground state. At $t=0$ a perturbation of the form
$$
H^{\prime}(x, t)=A x^{2} e^{-t / \tau}
$$
is switched on. Using time-dependent perturbation theory, calculate the probability that, after a sufficiently long time $(t \gg \tau)$, the system will have made a transition to a given excited state. Consider all final states.

Victor Salazar
Victor Salazar
Numerade Educator
20:49

Problem 32

The unperturbed Hamiltonian of a two-state system is represented by
$$
H_{0}=\left(\begin{array}{cc}
E_{1}^{0} & 0 \\
0 & E_{2}^{0}
\end{array}\right)
$$
There is, in addition, a time-dependent perturbation
$$
V(t)=\left(\begin{array}{cc}
0 & \lambda \cos \omega t \\
\lambda \cos \omega t & 0
\end{array}\right) \quad(\lambda \text { real })
$$
a. At $t=0$ the system is known to be in the first state, represented by
$$
\left(\begin{array}{l}
1 \\
0
\end{array}\right)
$$
Using time-dependent perturbation theory and assuming that $E_{1}^{0}-E_{2}^{0}$ is not close to $\pm \hbar \omega$, derive an expression for the probability that the system be found in the second state represented by
$$
\left(\begin{array}{l}
0 \\
1
\end{array}\right)
$$
as a function of $t(t>0)$.
b. Why is this procedure not valid when $E_{1}^{0}-E_{2}^{0}$ is close to $\pm \hbar \omega$ ?

Mahnoor Amin
Mahnoor Amin
Numerade Educator
01:51

Problem 33

A one-dimensional simple harmonic oscillator of angular frequency $\omega$ is acted upon by a spatially uniform but time-dependent force (not potential)
$$
F(t)=\frac{\left(F_{0} \tau / \omega\right)}{\left(\tau^{2}+t^{2}\right)}, \quad-\infty<t<\infty .
$$
At $t=-\infty$, the oscillator is known to be in the ground state. Using the time-dependent perturbation theory to first order, calculate the probability that the oscillator is found in the first excited state at $t=+\infty$.
Challenge for experts: $F(t)$ is so normalized that the impulse
$$
\int F(t) d t
$$
imparted to the oscillator is always the same, that is, independent of $\tau$; yet for $\tau \geqslant$ $1 / \omega$, the probability for excitation is essentially negligible. Is this reasonable?

Penny Riley
Penny Riley
Numerade Educator
09:21

Problem 34

Consider a particle in one dimension moving under the influence of some timeindependent potential. The energy levels and the corresponding eigenfunctions for this problem are assumed to be known. We now subject the particle to a traveling pulse represented by a time-dependent potential,
$$
V(t)=A \delta(x-c t)
$$
a. Suppose at $t=-\infty$ the particle is known to be in the ground state whose energy eigenfunction is $\langle x \mid i\rangle=u_{i}(x)$. Obtain the probability for finding the system in some excited state with energy eigenfunction $\langle x \mid f\rangle=u_{f}(x)$ at $t=+\infty$.
b. Interpret your result in (a) physically by regarding the $\delta$-function pulse as a superposition of harmonic perturbations; recall
$$
\delta(x-c t)=\frac{1}{2 \pi c} \int_{-\infty}^{\infty} d \omega e^{i \omega[(x / c)-t]}
$$
Emphasize the role played by energy conservation, which holds even quantum mechanically as long as the perturbation has been on for a very long time.

Ren Jie Tuieng
Ren Jie Tuieng
Numerade Educator
06:06

Problem 35

A hydrogen atom in its ground state $[(n, l, m)=(1,0,0)]$ is placed between the plates of a capacitor. A time-dependent but spatial uniform electric field (not potential!) is applied as follows:
$\mathbf{E}=\left\{\begin{array}{ll}0 & \text { for } t<0 \\ \mathbf{E}_{0} e^{-t / \tau} & \text { for } t>0\end{array}\left(\mathbf{E}_{0}\right.\right.$ in the positive $z$-direction $)$
Using first-order time-dependent perturbation theory, compute the probability for the atom to be found at $t \gg \tau$ in each of the three $2 p$ states: $(n, l, m)=(2,1, \pm 1$ or 0$)$. Repeat the problem for the $2 s$ state: $(n, l, m)=(2,0,0)$. Consider the limit $\tau \rightarrow \infty$.

Keshav Singh
Keshav Singh
Numerade Educator
07:06

Problem 36

Consider a composite system made up of two spin $\frac{1}{2}$ objects. For $t<0$, the Hamiltonian does not depend on spin and can be taken to be zero by suitably adjusting the energy scale. For $t>0$, the Hamiltonian is given by
$$
H=\left(\frac{4 \Delta}{\hbar^{2}}\right) \mathbf{S}_{1} \cdot \mathbf{S}_{2}
$$
Suppose the system is in $|+-\rangle$ for $t \leq 0 .$ Find, as a function of time, the probability for being found in each of the following states $|++\rangle,|+-\rangle,|-+\rangle$, and $|--\rangle .$
a. By solving the problem exactly.
b. By solving the problem assuming the validity of first-order time-dependent perturbation theory with $H$ as a perturbation switched on at $t=0$. Under what condition does (b) give the correct results?

Andrew Eddins
Andrew Eddins
Emory University
03:21

Problem 37

Consider a two-level system with $E_{1}<E_{2}$. There is a time-dependent potential that connects the two levels as follows:
$$
V_{11}=V_{22}=0, \quad V_{12}=\gamma e^{i \omega t}, \quad V_{21}=\gamma e^{-i \omega t} \quad(\gamma \text { real })
$$
At $t=0$, it is known that only the lower level is populated, that is, $c_{1}(0)=1$, $c_{2}(0)=0$.
a. Find $\left|c_{1}(t)\right|^{2}$ and $\left|c_{2}(t)\right|^{2}$ for $t>0$ by exactly solving the coupled differential equation
$$
i \hbar \dot{c}_{k}=\sum_{n=1}^{2} V_{k n}(t) e^{i \omega_{k n} t} c_{n} \quad(k=1,2)
$$
b. Do the same problem using time-dependent perturbation theory to lowest nonvanishing order. Compare the two approaches for small values of $\gamma$. Treat the following two cases separately: (i) $\omega$ very different from $\omega_{21}$ and (ii) $\omega$ close to $\omega_{21}$.
Answer for (a): (Rabi's formula)
$$
\begin{aligned}
\left|c_{2}(t)\right|^{2} &=\frac{\gamma^{2} / \hbar^{2}}{\gamma^{2} / \hbar^{2}+\left(\omega-\omega_{21}\right)^{2} / 4} \sin ^{2}\left\{\left[\frac{\gamma^{2}}{\hbar^{2}}+\frac{\left(\omega-\omega_{21}\right)^{2}}{4}\right]^{1 / 2} t\right\} \\
\left|c_{1}(t)\right|^{2} &=1-\left|c_{2}(t)\right|^{2}
\end{aligned}
$$

Manish Jain
Manish Jain
Numerade Educator
00:58

Problem 38

Show that the slow-turn-on of perturbation $V \rightarrow V e^{\eta t}$ (see Baym (1969), p. 257) can generate contribution from the second term in (5.295).

Bruce Edelman
Bruce Edelman
Numerade Educator
05:53

Problem 39

a. Consider the positronium problem solved in Chapter 3 , Problem $3.5$. In the presence of a uniform and static magnetic field $B$ along the $z$-axis, the Hamiltonian is given by
$$
H=A \mathbf{S}_{1} \cdot \mathbf{S}_{2}+\left(\frac{e B}{m_{e} c}\right)\left(S_{1 z}-S_{2 z}\right)
$$
Solve this problem to obtain the energy levels of all four states $u$ sing degenerate time-independent perturbation theory (instead of diagonalizing the Hamiltonian matrix). Regard the first and the second terms in the expression for $H$ as $H_{0}$ and $V$, respectively. Compare your results with the exact expressions
$$
\begin{array}{ll}
E=-\left(\frac{\hbar^{2} A}{4}\right)\left[1 \pm 2 \sqrt{1+4\left(\frac{e B}{m_{e} c \hbar A}\right)^{2}}\right] & \text { for }\left\{\begin{array}{l}
\operatorname{singlet} m=0 \\
\operatorname{triplet} m=0
\end{array}\right. \\
E=\frac{\hbar^{2} A}{4} & \text { for triplet } m=\pm 1,
\end{array}
$$
$E=-\left(\frac{\hbar^{2} A}{4}\right)\left[1 \pm 2 \sqrt{1+4\left(\frac{e B}{m_{e} c h A}\right)^{2}}\right] \quad$ for $\left\{\begin{array}{l}\operatorname{singlet} m=0 \\ \operatorname{triplet} m=0\end{array}\right.$ $E=\frac{\hbar^{2} A}{4}$$\quad$ for triplet $m=\pm 1$
where triplet (singlet) $m=0$ stands for the state that becomes a pure triplet (singlet) with $m=0$ as $B \rightarrow 0$.
b. We now attempt to cause transitions (via stimulated emission and absorption) between the two $m=0$ states by introducing an oscillating magnetic field of the "right" frequency. Should we orient the magnetic field along the $z$-axis or along the $x$ - (or $y^{-}$) axis? Justify your choice. (The original static field is assumed to be along the $z$-axis throughout.)
c. Calculate the eigenvectors to first order.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
02:27

Problem 40

Repeat Problem $5.39$ above, but with the atomic hydrogen Hamiltonian
$$
H=A \mathbf{S}_{1} \cdot \mathbf{S}_{2}+\left(\frac{e B}{m_{e} c}\right) \mathbf{S}_{1} \cdot \mathbf{B}
$$
where in the hyperfine term $A \mathbf{S}_{1} \cdot \mathbf{S}_{2}, \mathbf{S}_{1}$ is the electron spin, while $\mathbf{S}_{2}$ is the proton spin. (Note the problem here has less symmetry than that of the positronium case.)

Adriano Chikande
Adriano Chikande
Numerade Educator
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Problem 41

Consider the spontaneous emission of a photon by an excited atom. The process is known to be an $E 1$ transition. Suppose the magnetic quantum number of the atom decreases by one unit. What is the angular distribution of the emitted photon? Also discuss the polarization of the photon with attention to angular-momentum conservation for the whole (atom plus photon) system.

Susan Hallstrom
Susan Hallstrom
Numerade Educator
03:18

Problem 42

Consider an atom made up of an electron and a singly charged $(Z=1)$ triton $\left({ }^{3} \mathrm{H}\right)$. Initially the system is in its ground state $(n=1, l=0)$. Suppose the system undergoes beta decay, in which the nuclear charge suddenly increases by one unit (realistically by emitting an electron and an antineutrino). This means that the tritium nucleus (called a "triton") turns into a helium $(Z=2)$ nucleus of mass $3\left({ }^{3} \mathrm{He}\right)$.
a. Obtain the probability for the system to be found in the ground state of the resulting helium ion.
b. The available energy in tritium beta decay is about $18 \mathrm{keV}$ and the size of the ${ }^{3} \mathrm{He}$ atom is about $1 \AA$. Check that the time scale $T$ for the transformation satisfies the criterion of validity for the sudden approximation.

Farhanul Hasan
Farhanul Hasan
Numerade Educator
00:42

Problem 43

Show that $\boldsymbol{A}_{n}(\mathbf{R})$ defined in (5.234) is a purely real quantity.

Julie Silva
Julie Silva
Numerade Educator
01:28

Problem 44

Consider a neutron in a magnetic field, fixed at an angle $\theta$ with respect to the $\mathbf{z}$-axis, but rotating slowly in the $\phi$ direction. That is, the tip of the magnetic field traces out a circle on the surface of the sphere, at "latitude" $\pi-\theta$. Explicitly calculate the Berry potential $\mathbf{A}$ for the spin-up state from (5.234), take its curl, and determine Berry's phase $\gamma_{+}$. Thus, verify (5.253) for this particular example of a curve $C$. (For hints, see "The adiabatic theorem and Berry's phase" by Holstein, Am.J. Phys., $\mathbf{5 7}(1989)$ 1079.)

Penny Riley
Penny Riley
Numerade Educator
06:17

Problem 45

The ground state of a hydrogen atom $(n=1, l=0)$ is subjected to a time-dependent potential as follows:
$$
V(\mathbf{x}, t)=V_{0} \cos (k z-\omega t)
$$
Using time-dependent perturbation theory, obtain an expression for the transition rate at which the electron is emitted with momentum p. Show, in particular, how you
may compute the angular distribution of the ejected electron (in terms of $\theta$ and $\phi$ defined with respect to the $z$-axis). Discuss briefly the similarities and the differences between this problem and the (more realistic) photoelectric effect. (Note: For the initial wave function see Problem 5.42. If you have a normalization problem, the final wave function may be taken to be
$$
\psi_{f}(\mathbf{x})=\left(\frac{1}{L^{3 / 2}}\right) e^{j \mathbf{p}-\mathbf{x} / \hbar}
$$
with $L$ very large, but you should be able to show that the observable effects are independent of $L$.)

Zachary Warner
Zachary Warner
Numerade Educator
12:14

Problem 46

A particle of mass $m$ constrained to move in one dimension is confined within $0<x<L$ by an infinite-wall potential
$$
\begin{array}{ll}
V=\infty & \text { for } x<0, x>L, \\
V=0 & \text { for } 0 \leq x \leq L .
\end{array}
$$
Obtain an expression for the density of states (that is, the number of states per unit energy interval) for high energies as a function of $E$. (Check your dimension!)

Robert Zaballa
Robert Zaballa
Numerade Educator
01:29

Problem 47

Linearly polarized light of angular frequency $\omega$ is incident on a one-electron "atom" whose wave function can be approximated by the ground state of a three-dimensional isotropic harmonic oscillator of angular frequency $\omega_{0}$. Show that the differential cross section for the ejection of a photoelectron is given by
$$
\begin{aligned}
\frac{d \sigma}{d \Omega}=& \frac{4 \alpha \hbar^{2} k_{f}^{3}}{m^{2} \omega \omega_{0}} \sqrt{\frac{\pi \hbar}{m \omega_{0}}} \exp \left\{-\frac{\hbar}{m \omega_{0}}\left[k_{f}^{2}+\left(\frac{\omega}{c}\right)^{2}\right]\right\} \\
& \times \sin ^{2} \theta \cos ^{2} \phi \exp \left[\left(\frac{2 \hbar k_{f} \omega}{m \omega_{0} c}\right) \cos \theta\right]
\end{aligned}
$$
provided the ejected electron of momentum $\hbar k_{f}$ can be regarded as being in a plane wave state. (The coordinate system used is shown in Figure 5.13.)

Dominador Tan
Dominador Tan
Numerade Educator
08:28

Problem 48

Find the probability $\left|\phi\left(\mathbf{p}^{\prime}\right)\right|^{2} d^{3} p^{\prime}$ of the particular momentum $\mathbf{p}^{\prime}$ for the ground-state hydrogen atom. (This is a nice exercise in three-dimensional Fourier transforms. To perform the angular integration choose the $z$-axis in the direction of $\mathbf{p}$.)

Rahul Nikhar
Rahul Nikhar
Numerade Educator
02:30

Problem 49

Calculate the lifetimes for the $2 p \rightarrow 1 s$ and $3 p \rightarrow 1 s$ transitions in the hydrogen atom. You can find measurements of these lifetimes in Bickel and Goodman, Phys. Rev., $148(1966) 1$.

Ajay Singhal
Ajay Singhal
Numerade Educator
02:36

Problem 50

A hydrogen atom is prepared in the $2 p, m=+1$ state. Find the rate for the $2 p \rightarrow 1 s$ transition as a function of polar angle $\theta$ with respect to the $z$-axis. Describe the polarization of the emitted radiation as a function of $\theta$ and use this to qualitatively explain the intensity pattern.

Mayukh Banik
Mayukh Banik
Numerade Educator
06:34

Problem 51

This problem highlights anomalies in the "exponential" decay of a state. It is inspired by Winter, Phys. Rev., 123 (1961) 1503 , but modern computer applications make it straightforward to directly evaluate the integrals numerically. Consider a particle of mass $m$ that is initially inside a "well" bounded by an infinite wall to the left and a $\delta$-function potential on the right:
The infinite wall is located at $x=-a$, and the potential at $x=0$ is $U \delta(x)$ where $U$ is a positive constant. The figure also shows a plausible "ground state" initial wave function $\Psi(x, t=0)=(2 / a)^{1 / 2} \sin (n \pi x / a)$ with $n=1$.
a. Show that the wave function at all times can be written as
$$
\Psi(x, t)=2 n\left(\frac{2}{a}\right)^{1 / 2} \int_{0}^{\infty} d q \frac{e^{-i T q^{2}} q \sin q[q \sin (l+1) q+f]}{\left(q^{2}-n^{2} \pi^{2}\right)\left(q^{2}+G q \sin 2 q+G^{2} \sin ^{2} q\right)}
$$
where $q \equiv\left[a(2 m E)^{1 / 2}\right] / \hbar$ for a particle with energy $E, T \equiv \hbar t / 2 m a^{2}, l \equiv x / a$, and $G \equiv 2 \mathrm{maU} / \hbar^{2}$ are all dimensionless quantities, and $f=0$ for $-a \leq x \leq 0$ and $f=G \sin q \sin l q$ for $x \geq 0$. This is most easily done by expanding the wave function in energy eigenstates $|E\rangle$, as
$$
\Psi(x, t)=\int_{0}^{\infty} d E \phi_{E}(x) e^{-i E t h}
$$
where $\phi_{E}(x)$ is an energy eigenfunction and $\left\langle E \mid E^{\prime}\right\rangle=\delta\left(E-E^{\prime}\right)$.
b. Write a computer program to (numerically) integrate the probability of finding the particle inside the well. Carry out the integration for a series of values of $T$ between zero and 12 , using the same parameters as Winter, namely $n=1$ and $G=6$. Plotting these probabilities as a function of $T$ should resemble Figure 2 of Winter's paper. Fit the points for $2 \leq T \leq 8$ to an exponential, and compare the decay time to Winter's value of $0.644$.
c. Examine the behavior for $T \geq 8$, and compare to the behavior Winter found for the current at $x=0$. This suggests an experimental measurement. See Norman et al., Phys. Rev. Lett., $\mathbf{6 0}$ (1988) 2246 .

Christopher Provencher
Christopher Provencher
Numerade Educator