Chapter 5, Motion in step potentials Video Solutions, The Physics of Quantum Mechanics

Problem 1

A particle is confined by the potential well
$$
V(x)= \begin{cases}0 & \text { for }|x|<a \\ \infty & \text { otherwise }\end{cases}
$$
Explain (a) why we can assume that there is a complete set of stationary states with well-defined parity and (b) why to find the stationary states we solve the TISE subject to the boundary condition $\psi(\pm a)=0$.
Determine the particle's energy spectrum and give the wavefunctions of the first two stationary states.

Suzanne W.

Numerade Educator

02:03

Problem 2

At $t=0$ the particle of Problem $5.1$ has the wavefunction
$$
\psi(x)= \begin{cases}1 / \sqrt{2 a} & \text { for }|x|<a \\ 0 & \text { otherwise }\end{cases}
$$
Find the probabilities that a measurement of its energy will yield: (a) $9 \hbar^{2} \pi^{2} /\left(8 m a^{2}\right) ;$ (b) $16 \hbar^{2} \pi^{2} /\left(8 m a^{2}\right)$

Hafiz Shahzaib

Numerade Educator

02:18

Problem 3

Find the probability distribution of measuring momentum $p$ for the particle described in Problem 5.2. Sketch and comment on your distribution. Hint: express $\langle p \mid x\rangle$ in the position representation.

Chai Santi

Numerade Educator

04:00

Problem 4

Particles move in the potential
$$
V(x)= \begin{cases}0 & \text { for } x<0 \\ V_{0} & \text { for } x>0\end{cases}
$$
Particles of mass $m$ and energy $E>V_{0}$ are incident from $x=-\infty .$ Show that the probability that a particle is reflected is
$$
\left(\frac{k-K}{k+K}\right)^{2}
$$
where $k \equiv \sqrt{2 m E} / \hbar$ and $K \equiv \sqrt{2 m\left(E-V_{0}\right)} / \hbar .$ Show directly from the TISE that the probability of transmission is
$$
\frac{4 k K}{(k+K)^{2}}
$$
and check that the flux of particles moving away from the origin is equal to the incident particle flux.

Keshav Singh

Numerade Educator

12:05

Problem 5

Show that the energies of bound, odd-parity stationary states of the square potential well
$$
V(x)= \begin{cases}0 & \text { for }|x|<a \\ V_{0}>0 & \text { otherwise }\end{cases}
$$
are governed by
$\cot (k a)=-\sqrt{\frac{W^{2}}{(k a)^{2}}-1}$ where $\quad W \equiv \sqrt{\frac{2 m V_{0} a^{2}}{\hbar^{2}}}$ and $k^{2}=2 m E / \hbar^{2}$
$(5.77)$
Show that for a bound odd-parity state to exist, we require $W>\pi / 2$

Mirza Aslam Beig

Numerade Educator

02:54

Problem 6

Show that the correctly normalised wavefunction of a particle trapped by the potential $V(x)=-V_{\delta} \delta(x)$ is $\psi(x)=\sqrt{K} e^{-K|x|}$, where $K=$ $m V_{\delta} / \hbar^{2}$. Show that although this wavefunction makes it certain that a measurement of $x$ will find the particle outside the well where its kinetic energy is negative, the expectation value of its kinetic energy $\left\langle E_{K}\right\rangle=\frac{1}{2} m V_{\delta}^{2} / \hbar^{2}$ is in fact positive. Reconcile this apparent paradox as follows: (i) Show that for a narrow, deep potential well of depth $V_{0}$ and half-width $a$, with $2 V_{0} a=V_{\delta}, k a \simeq W \equiv\left(2 m V_{0} a^{2} / \hbar^{2}\right)^{1 / 2}$, while $K a \simeq W^{2}$. (ii) Hence show that the contribution from inside the well to $\left\langle E_{K}\right\rangle$ is $|\psi(0)|^{2} V_{\delta}$ regardless of the value of $a .$ Explain physically what is happening as we send $a \rightarrow 0$.

Jacob Schulze

Numerade Educator

06:55

Problem 7

Reproduce the plots shown in Figure $5.22$ of the wavefunctions of particles that are scattered by a square barrier and a square potential well. Give physical interpretations of as many features of the plots as you can.

Sanu Kumar

Numerade Educator

12:05

Problem 8

Give an example of a potential in which there is a complete set of bound stationary states of well-defined parity, and an alternative complete set of bound stationary states that are not eigenkets of the parity operator. Hint: modify the potential discussed apropos $\mathrm{NH}_{3}$.

Mirza Aslam Beig

Numerade Educator

06:56

Problem 9

A free particle of energy $E$ approaches a square, one-dimensional potential well of depth $V_{0}$ and width $2 a$. Show that the probability of being reflected by the well vanishes when $K a=n \pi / 2$, where $n$ is an integer and $K=\left(2 m\left(E+V_{0}\right) / \hbar^{2}\right)^{1 / 2}$. Explain this phenomenon in physical terms.

Mirza Aslam Beig

Numerade Educator

01:49

Problem 10

Show that the phase shifts $\phi$ (for the even-parity stationary state) and $\phi^{\prime}$ (for the odd-parity state) that are associated with scattering by a
classically allowed region of potential $V_{0}$ and width $2 a$, satisfy $\tan (k a+\phi)=-(k / K) \cot (K a) \quad$ and $\quad \tan \left(k a+\phi^{\prime}\right)=(k / K) \tan (K a)$ where $k$ and $K$ are, respectively, the wavenumbers at infinity and in the scattering potential. Show that
$$
P_{\mathrm{refl}}=\cos ^{2}\left(\phi^{\prime}-\phi\right)=\frac{(K / k-k / K)^{2} \sin ^{2}(2 K a)}{(K / k+k / K)^{2} \sin ^{2}(2 K a)+4 \cos ^{2}(2 K a)}
$$
Hint: apply the cosine rule for an angle in a triangle in terms of the lengths of the triangle's sides to the top triangle in Figure $5.23$.

Arun Bana

Numerade Educator

14:36

Problem 11

A particle of energy $E$ approaches from $x<0$ a barrier in which the potential energy is $V(x)=V_{\delta} \delta(x)$. Show that the probability of its passing the barrier is
$$
P_{\mathrm{tun}}=\frac{1}{1+(K / 2 k)^{2}} \quad \text { where } \quad k=\sqrt{\frac{2 m E}{\hbar^{2}}}, \quad K=\frac{2 m V_{\delta}}{\hbar^{2}} .
$$

David Morabito

Numerade Educator

07:06

Problem 12

An electron moves along an infinite chain of potential wells. For sufficiently low energies we can assume that the set $\{|n\rangle\}$ is complete, where $|n\rangle$ is the state of definitely being in the $n^{\text {th }}$ well. By analogy with our analysis of the $\mathrm{NH}_{3}$ molecule we assume that for all $n$ the only non-vanishing matrix elements of the Hamiltonian are $\mathcal{E} \equiv\langle n|H| n\rangle$ and $A \equiv\langle n \pm 1|H| n\rangle$. Give physical interpretations of the numbers $A$ and $\mathcal{E}$. Explain why we can write
$$
H=\sum_{n=-\infty}^{\infty} \mathcal{E}|n\rangle\langle n|+A(|n\rangle\langle n+1|+| n+1\rangle\langle n|)
$$
Writing an energy eigenket $|E\rangle=\sum_{n} a_{n}|n\rangle$ show that
$$
a_{m}(E-\mathcal{E})-A\left(a_{m+1}+a_{m-1}\right)=0
$$
Obtain solutions of these equations in which $a_{m} \propto \mathrm{e}^{\mathrm{i} k m}$ and thus find the corresponding energies $E_{k}$. Why is there an upper limit on the values of $k$ that need be considered?
Initially the electron is in the state
$$
|\psi\rangle=\frac{1}{\sqrt{2}}\left(\left|E_{k}\right\rangle+\left|E_{k+\Delta}\right\rangle\right)
$$
where $0<k \ll 1$ and $0<\Delta \ll k$. Describe the electron's subsequent motion in as much detail as you can.

Andrew Eddins

Emory University

00:45

Problem 13

This problem is about the coupling of ammonia molecules to electromagnetic waves in an ammonia maser. Let $|+\rangle$ be the state in which the $\mathrm{N}$ atom lies above the plane of the $\mathrm{H}$ atoms and $|-\rangle$ be the state in which the $\mathrm{N}$ lies below the plane. Then when there is an oscillating electric field $\mathcal{E} \cos \omega t$ directed perpendicular to the plane of the hydrogen atoms, the Hamiltonian in the $|\pm\rangle$ basis becomes
$$
H=\left(\begin{array}{cc}
\bar{E}+q \mathcal{E} s \cos \omega t & -A \\
-A & \bar{E}-q \mathcal{E} s \cos \omega t
\end{array}\right)
$$
Transform this Hamiltonian from the $|\pm\rangle$ basis to the basis provided by the states of well-defined parity $|\mathrm{e}\rangle$ and $|\mathrm{o}\rangle$ (where $|\mathrm{e}\rangle=(|+\rangle+|-\rangle) / \sqrt{2}$, etc). Writing
$$
|\psi\rangle=a_{\mathrm{e}}(t) \mathrm{e}^{-\mathrm{i} E_{e} t / \hbar}|\mathrm{e}\rangle+a_{\mathrm{o}}(t) \mathrm{e}^{-\mathrm{i} E_{0} t / \hbar}|\mathrm{o}\rangle
$$
show that the equations of motion of the expansion coefficients are
$$
\begin{aligned}
&\frac{\mathrm{d} a_{\mathrm{e}}}{\mathrm{d} t}=-\mathrm{i} \Omega a_{\mathrm{o}}(t)\left(\mathrm{e}^{\mathrm{i}\left(\omega-\omega_{0}\right) t}+\mathrm{e}^{-\mathrm{i}\left(\omega+\omega_{0}\right) t}\right) \\
&\frac{\mathrm{d} a_{\mathrm{o}}}{\mathrm{d} t}=-\mathrm{i} \Omega a_{\mathrm{e}}(t)\left(\mathrm{e}^{\mathrm{i}\left(\omega+\omega_{0}\right) t}+\mathrm{e}^{-\mathrm{i}\left(\omega-\omega_{0}\right) t}\right)
\end{aligned}
$$
where $\Omega \equiv q \mathcal{E} s / 2 \hbar$ and $\omega_{0}=\left(E_{o}-E_{e}\right) / \hbar .$ Explain why in the case of a maser the exponentials involving $\omega+\omega_{0}$ can be neglected so the equations of motion become
$$
\frac{\mathrm{d} a_{\mathrm{e}}}{\mathrm{d} t}=-\mathrm{i} \Omega a_{\circ}(t) \mathrm{e}^{\mathrm{i}\left(\omega-\omega_{0}\right) t}, \quad \frac{\mathrm{d} a_{\mathrm{o}}}{\mathrm{d} t}=-\mathrm{i} \Omega a_{\mathrm{e}}(t) \mathrm{e}^{-\mathrm{i}\left(\omega-\omega_{0}\right) t}
$$
Solve the equations by multiplying the first equation by $\mathrm{e}^{-\mathrm{i}\left(\omega-\omega_{0}\right) t}$ and differentiating the result. Explain how the solution describes the decay of a population of molecules that are initially all in the higher energy level. Compare your solution to the result of setting $\omega=\omega_{0}$ in $(5.86)$.

Nikhil Choudhary

Numerade Educator

View

Problem 14

${ }^U$ decays by $\alpha$ emission with a mean lifetime of $6.4$ Gyr. Take the nucleus to have a diameter $\sim 10^{-14} \mathrm{~m}$ and suppose that the $\alpha$ particle has been bouncing around within it at speed $\sim c / 3$. Modelling the potential barrier that confines the $\alpha$ particle to be a square one of height $V_{0}$ and width $2 a$, give an order-of-magnitude estimate of $W=\left(2 m V_{0} a^{2} / \hbar^{2}\right)^{1 / 2}$ Given that the energy released by the decay is $\sim 4 \mathrm{MeV}$ and the atomic number of uranium is $Z=92$, estimate the width of the barrier through which the $\alpha$ particle has to tunnel. Hence give a very rough estimate of the barrier's typical height. Outline numerical work that would lead to an improved estimate of the structure of the barrier.

Lainey Roebuck

Numerade Educator

12:18

Problem 15

Particles of mass $m$ and momentum $\hbar k$ at $x<-a$ move in the potential
$$
V(x)=V_{0} \begin{cases}0 & \text { for } x<-a \\ \frac{1}{2}[1+\sin (\pi x / 2 a)] & \text { for }|x|<a \\ 1 & \text { for } x>a\end{cases}
$$
where $V_{0}<\hbar^{2} k^{2} / 2 m$. Numerically reproduce the reflection probabilities plotted in Figure $5.20$ as follows. Let $\psi_{i} \equiv \psi\left(x_{j}\right)$ be the value of the wavefunction at $x_{j}=j \Delta$, where $\Delta$ is a small increment in the $x$ coordinate. From the TISE show that
$$
\psi_{j} \simeq\left(2-\Delta^{2} k^{2}\right) \psi_{j+1}-\psi_{j+2}
$$
where $k \equiv \sqrt{2 m(E-V)} / \hbar$. Determine $\psi_{j}$ at the two grid points with the largest values of $x$ from a suitable boundary condition, and use the recurrence relation (5.88) to determine $\psi_{j}$ at all other grid points. By matching the values of $\psi$ at the points with the smallest values of $x$ to a sum of sinusoidal waves, determine the probabilities required for the figure. Be sure to check the accuracy of your code when $V_{0}=0$, and in the general case explicitly check that your results are consistent with equal fluxes of particles towards and away from the origin.

Equation (12.40) gives an analytical approximation for $\psi$ in the case that there is negligible reflection. Compute this approximate form of $\psi$ and compare it with vour numerical results for larger values of $a$.

Robert Zaballa

Numerade Educator

14:01

Problem 16

In this problem we obtain an analytic estimate of the energy difference between the even- and odd-parity states of a double square well. Show that for large $\theta$, $\operatorname{coth} \theta-\tanh \theta \simeq 4 \mathrm{e}^{-2 \theta}$. Next letting $\delta k$ be the difference between the $k$ values that solve
$$
\tan [r \pi-k(b-a)] \sqrt{\frac{W^{2}}{(k a)^{2}}-1}= \begin{cases}\operatorname{coth}\left(\sqrt{W^{2}-(k a)^{2}}\right) & \text { even parity } \\ \tanh \left(\sqrt{W^{2}-(k a)^{2}}\right) & \text { odd parity }\end{cases}
$$
where
$$
W \equiv \sqrt{\frac{2 m V_{0} a^{2}}{\hbar^{2}}}
$$
for given $r$ in the odd- and even-parity cases, deduce that
$$
\begin{array}{r}
\left\{\left[\left(\frac{W^{2}}{(k a)^{2}}-1\right)^{1 / 2}+\left(\frac{W^{2}}{(k a)^{2}}-1\right)^{-1 / 2}\right](b-a)+\frac{1}{k}\left(1-\frac{(k a)^{2}}{W^{2}}\right)^{-1}\right\} \delta k \\
\simeq-4 \exp \left[-2 \sqrt{W^{2}-(k a)^{2}}\right]
\end{array}
$$
Hence show that when $W \gg 1$ the fractional difference between the energies of the ground and first excited states is
$$
\frac{\delta E}{E} \simeq \frac{-8 a}{W(b-a)} \mathrm{e}^{-2 W \sqrt{1-E / V_{0}}}
$$

Blake Lee

Numerade Educator

02:32

Problem 17

We consider the scattering of free particles of mass $m$ that move in one dimension in the potential $V(x)=-W \delta(x)$, with $W>0$. (a) For a well of finite depth $V_{0}$ and width $2 a$ the condition on the phases $\phi$ and
$\phi^{\prime}$ of the even- and odd-parity wavefunctions $\psi \propto \sin (k x+\phi)$, etc., for free particles are
$$
\tan (k a+\phi)=-\frac{k}{K} \cot (K a), \quad \tan \left(k a+\phi^{\prime}\right)=-\frac{k}{K} \tan (K a)
$$
Show that in the limit $a \rightarrow 0, V_{0}=W / 2 a \rightarrow \infty$ we have $\tan \phi \rightarrow$ $-\hbar^{2} k / m W$ and $\phi^{\prime} \rightarrow 0$. Hence obtain the scattering cross-section by the $\delta$-function potential
$$
\sigma=\frac{2}{1+(\hbar k / m W)^{2}}
$$
(b) Re-derive the equation above for $\phi$ by requiring that $\psi=\sin (k|x|+\phi)$ satisfy the TISE. Convince yourself that $\psi=\sin (k x)$ is also consistent with the TISE.

Andreas Papavassiliou

Numerade Educator