The Physics of Quantum Mechanics

James Binney, David Skinner

Chapter 6 Composite systems - all with Video Answers

Educators

Chapter Questions

Problem 1

A system AB consists of two non-interacting parts $\mathrm{A}$ and $\mathrm{B}$. The dynamical state of $\mathrm{A}$ is described by $|a\rangle$, and that of $\mathrm{B}$ by $|b\rangle$, so $|a\rangle$ satisfies the TDSE for A and similarly for $|b\rangle .$ What is the ket describing the dynamical state of $\mathrm{AB} ?$ In terms of the Hamiltonians $H_{\mathrm{A}}$ and $H_{\mathrm{B}}$ of the subsystems, write down the TDSE for the evolution of this ket and show that it is automatically satisfied. Do $H_{\mathrm{A}}$ and $H_{\mathrm{B}}$ commute? How is the TDSE changed when the subsystems are coupled by a small dynamical interaction $H_{\text {int }} ?$ If $\mathrm{A}$ and $\mathrm{B}$ are harmonic oscillators, write down $H_{\mathrm{A}}, H_{\mathrm{B}}$. The oscillating particles are connected by a weak spring. Write down the appropriate form of the interaction Hamiltonian $H_{\text {int }} .$ Does $H_{\mathrm{A}}$ commute with $H_{\text {int }} ?$ Explain the physical significance of your answer.

Manish Jain

Numerade Educator

03:49

Problem 2

Explain what is implied by the statement that "the physical state of system A is correlated with the state of system B." Illustrate your answer by considering the momenta of cars on (i) London's circular motorway (the M25) at rush-hour, and (ii) the road over the Nullarbor Plain in southern Australia in the dead of night.
Explain why the states of $A$ and $B$ must be uncorrelated if it is possible to write the state of $\mathrm{AB}$ as a ket $|\mathrm{AB} ; \psi\rangle=\left|\mathrm{A}_{1} \psi_{1}\right\rangle\left|\mathrm{B} ; \psi_{2}\right\rangle$ that is a product of states of $\mathrm{A}$ and $\mathrm{B}$. Given a complete set of states for $\mathrm{A}$, $\{|\mathrm{A} ; i\rangle\}$, and a corresponding complete set of states for $\mathrm{B},\{|\mathrm{B} ; i\rangle\}$, write down an expression for a state of $\mathrm{AB}$ in which $\mathrm{B}$ is possibly correlated with A.

Khoobchandra Agrawal

Numerade Educator

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Problem 3

Given that the state $|\mathrm{AB}\rangle$ of a compound system can be written as a product $|\mathrm{A}\rangle|\mathrm{B}\rangle$ of states of the individual systems, show that when $|\mathrm{AB}\rangle$ is written as $\sum_{i j} c_{i j}|\mathrm{~A} ; i\rangle|\mathrm{B} ; j\rangle$ in terms of arbitrary basis vectors for the subsystems, every column of the matrix $c_{i j}$ is a multiple of the leftmost column.

Victor Salazar

Numerade Educator

23:50

Problem 4

Consider a system of two particles of mass $m$ that each move in one dimension along a given rod. Let $|1 ; x\rangle$ be the state of the first particle when it's at $x$ and $|2 ; y\rangle$ be the state of the second particle when it's at $y$. A complete set of states of the pair of particles is $\{|x y\rangle\}=\{|1 ; x\rangle|2 ; y\rangle\}$. Write down the Hamiltonian of this system given that the particles attract one another with a force that's equal to $C$ times their separation.
Suppose the particles experience an additional potential
$$
V(x, y)=\frac{1}{2} C(x+y)^{2}
$$
Show that the dynamics of the two particles is now identical with the dynamics of a single particle that moves in two dimensions in a particular potential $\Phi(x, y)$, and give the form of $\Phi$.

Yaqub Khan

Numerade Educator

00:56

Problem 5

In $\S 6.1 .4$ we derived Bell's inequality by considering measurements by Alice and Bob on an entangled electron-positron pair. Bob measures the component of spin along an axis that is inclined by angle $\theta$ to that used by Alice. Given the expression
$$
|-, \mathbf{b}\rangle=\cos (\theta / 2) \mathrm{e}^{\mathrm{i} \phi / 2}|-\rangle-\sin (\theta / 2) \mathrm{e}^{-\mathrm{i} \phi / 2}|+\rangle,
$$
for the state of a spin-half particle in which it has spin $-\frac{1}{2}$ along the direction $\mathbf{b}$ with polar angles $(\theta, \phi)$, with $|\pm\rangle$ the states in which there is spin $\pm \frac{1}{2}$ along the $z$-axis, calculate the amplitude $A_{\mathrm{B}}(-\mid \mathrm{A}+)$ that $\mathrm{Bob}$ finds the positron's spin to be $-\frac{1}{2}$ given that Alice has found $+\frac{1}{2}$ for the electron's spin. Hence show that $P_{\mathrm{B}}(-\mid \mathrm{A}+)=\cos ^{2}(\theta / 2)$.

Chai Santi

Numerade Educator

08:02

Problem 6

Show that when the Hadamard operator $U_{\mathrm{H}}$ is applied to every qubit of an $n$-qubit register that is initially in a member $|m\rangle$ of the computational basis, the resulting state is
$$
|\psi\rangle=\frac{1}{2^{n / 2}} \sum_{x=0}^{2^{n}-1} a_{x}|x\rangle
$$
where $a_{x}=1$ for all $x$ if $m=0$, but exactly half the $a_{x}=1$ and the other half the $a_{x}=-1$ for any other choice of $m$. Hence show that
$$
\frac{1}{2^{n / 2}} U_{\mathrm{H}} \sum_{x} a_{x}|x\rangle= \begin{cases}|0\rangle & \text { if all } a_{x}=1 \\ |m\rangle \neq|0\rangle & \text { if half the } a_{x}=1 \text { and the other } a_{x}=-1\end{cases}
$$

Hafiz Shahzaib

Numerade Educator

01:44

Problem 7

Show that the trace of every Hermitian operator is real.

Nick Johnson

Numerade Educator

16:39

Problem 8

Let $\rho$ be the density operator of a two-state system. Explain why $\rho$ can be assumed to have the matrix representation
$$
\rho=\left(\begin{array}{cc}
a & c \\
c^{*} & b
\end{array}\right)
$$
where $a$ and $b$ are real numbers. Let $E_{0}$ and $E_{1}>E_{0}$ be the eigenenergies of this system and $|0\rangle$ and $|1\rangle$ the corresponding stationary states. Show from the equation of motion of $\rho$ that in the energy representation $a$ and $b$ are time-independent while $c(t)=c(0) \mathrm{e}^{\mathrm{i} \omega t}$ with $\omega=\left(E_{1}-E_{0}\right) / \hbar$.
Determine the values of $a, b$ and $c(t)$ for the case that initially the system is in the state $|\psi\rangle=(|0\rangle+|1\rangle) / \sqrt{2}$. Given that the parities of $|0\rangle$ and $|1\rangle$ are even and odd respectively, find the time evolution of the expectation value $\bar{x}$ in terms of the matrix element $\langle 0|x| 1\rangle$. Interpret your result physically.

Sheh Lit Chang

University of Washington

16:39

Problem 9

In this problem we consider an alternative interpretation of the density operator. Any quantum state can be expanded in the energy basis as
$$
|\psi ; \boldsymbol{\phi}\rangle \equiv \sum_{n=1}^{N} \sqrt{p_{n}} \mathrm{e}^{\mathrm{i} \phi_{n}}|n\rangle
$$
where $\phi_{n}$ is real and $p_{n}$ is the probability that a measurement of energy will return $E_{n}$. Suppose we know the values of the $p_{n}$ but not the values of the phases $\phi_{n}$. Then the density operator is
$$
\rho=\int_{0}^{2 \pi} \frac{\mathrm{d}^{N} \boldsymbol{\phi}}{(2 \pi)^{N}}|\psi ; \boldsymbol{\phi}\rangle\langle\psi ; \boldsymbol{\phi}|
$$
Show that this expression reduces to $\sum_{n} p_{n}|n\rangle\langle n| .$ Contrast the physical assumptions made in this derivation of $\rho$ with those made in $\S 6.3$.
Clearly $|\psi ; \boldsymbol{\phi}\rangle$ can be expanded in some other basis $\left\{\left|q_{r}\right\rangle\right\}$ as
$$
|\psi ; \boldsymbol{\phi}\rangle \equiv \sum_{r} \sqrt{P_{r}} \mathrm{e}^{\mathrm{i} \eta_{r}}\left|q_{r}\right\rangle
$$
where $P_{r}$ is the probability of obtaining $q_{r}$ on a measurement of the observable $Q$ and the $\eta_{r}(\boldsymbol{\phi})$ are unknown phases. Why does this second expansion not lead to the erroneous conclusion that $\rho$ is necessarily diagonal in the $\left\{\left|q_{r}\right\rangle\right\}$ representation?

Sheh Lit Chang

University of Washington

03:43

Problem 10

Show that the equation of motion of the density operator $\rho$ is solved by
$$
\rho_{t}=U(t) \rho_{0} U^{\dagger}(t)
$$
where $U(t) \equiv \mathrm{e}^{-i H t / \hbar}$ is the time-evolution operator introduced in $\S 4.3$.

Mehdi Hatefipour

Numerade Educator

06:30

Problem 11

Show that when the density operator takes the form $\rho=|\psi\rangle\langle\psi|$, the expression $\bar{Q}=\operatorname{Tr} Q \rho$ for the expectation value of an observable can be reduced to $\langle\psi|Q| \psi\rangle .$ Explain the physical significance of this result. For the given form of the density operator, show that the equation of motion of $\rho$ yields
$$
|\phi\rangle\langle\psi|=| \psi\rangle\langle\phi| \quad \text { where } \quad|\phi\rangle \equiv \mathrm{i} \hbar \frac{\partial|\psi\rangle}{\partial t}-H|\psi\rangle
$$
Show from this equation that $|\phi\rangle=a|\psi\rangle$, where $a$ is real. Hence determine the time evolution of $|\psi\rangle$ given the at $t=0,|\psi\rangle=|E\rangle$ is an eigenket of $H$. Explain why $\rho$ does not depend on the phase of $|\psi\rangle$ and relate this fact to the presence of $a$ in your solution for $|\psi, t\rangle$.

Eduard Sanchez

Numerade Educator

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Problem 12

The density operator is defined to be $\rho=\sum_{\alpha} p_{\alpha}|\alpha\rangle\langle\alpha|$, where $p_{\alpha}$ is the probability that the system is in the state $\alpha$. Given an arbitrary basis $\{|i\rangle\}$ and the expansions $|\alpha\rangle=\sum_{i} a_{\alpha i}|i\rangle$, calculate the matrix elements $\rho_{i j}=\langle i|\rho| j\rangle$ of $\rho$. Show that the diagonal elements $\rho_{i i}$ are non-negative real numbers and interpret them as probabilities.

Victor Salazar

Numerade Educator

01:37

Problem 13

Consider the density operator $\rho=\sum_{i j} \rho_{i j}|i\rangle\langle j|$ of a system that is in a pure state. Show that every row of the matrix $\rho_{i j}$ is a multiple of the first row and every column is a multiple of the first column. Given that these relations between the rows and columns of a density matrix hold, show that the system is in a pure state. Hint: exploit the real, non-negativity of $\rho_{11}$ established in Problem $6.12$ and the Hermiticity of $\rho$

Nick Johnson

Numerade Educator

02:23

Problem 14

Consider the rate of change of the expectation of the observable $Q$ when the system is in an impure state. This is
$$
\frac{\mathrm{d} \bar{Q}}{\mathrm{~d} t}=\sum_{n} p_{n} \frac{\mathrm{d}}{\mathrm{d} t}\langle n|Q| n\rangle
$$
where $p_{n}$ is the probability that the system is in the state $|n\rangle$. By using Ehrenfest's theorem to evaluate the derivative on the right of $(6.123)$, derive the equation of motion i $\hbar \mathrm{d} \bar{Q} / \mathrm{d} t=\operatorname{Tr}(\rho[Q, H])$.

Sheh Lit Chang

University of Washington

02:24

Problem 15

Find the probability distribution $\left(p_{1}, \ldots, p_{n}\right)$ for $n$ possible outcomes that maximises the Shannon entropy. Hint: use a Lagrange multiplier.

Christopher Stanley

Numerade Educator

08:28

Problem 16

Use Lagrange multipliers $\lambda$ and $\beta$ to extremise the Shannon entropy of the probability distribution $\left\{p_{i}\right\}$ subject to the constraints (i) $\sum_{i} p_{i}=1$ and (ii) $\sum_{i} p_{i} E_{i}=U$. Explain the physical significance of your result.

Andrew Eddins

Emory University

03:18

Problem 17

Explain why if at $t=0$ the density operator of a system is given by the Gibbs distribution, it remains so at later times.

Matt Just

Numerade Educator

06:37

Problem 18

A composite system is formed from uncorrelated subsystem A and subsystem B, both in impure states. The numbers $\left\{p_{A i}\right\}$ are the probabilities of the members of the complete set of states $\{|\mathrm{A} ; i\rangle\}$ for subsystem A, while the numbers $\left\{p_{\mathrm{B} i}\right\}$ are the probabilities of the complete set of states $\{|\mathrm{B} ; i\rangle\}$ for subsystem $\mathrm{B}$. Show that the Shannon entropy of the composite system is the sum of the Shannon entropies of its subsystems. What is the relevance of this result for thermodynamics?

Christopher Nilsen

Numerade Educator

04:34

Problem 19

The $|0\rangle$ state of a qubit has energy 0 , while the $|1\rangle$ state has energy $\epsilon$. Show that when the qubit is in thermodynamic equilibrium at temperature $T=1 /\left(k_{\mathrm{B}} \beta\right)$ the internal energy of the qubit is
$$
U=\frac{\epsilon}{\mathrm{e}^{\beta \epsilon}+1}
$$
Show that when $\beta \epsilon \ll 1, U \simeq \frac{1}{2} \epsilon$, while for $\beta \epsilon \gg 1, U \simeq \epsilon \mathrm{e}^{-\beta \epsilon}$. Interpret these results physically and sketch the specific heat $C=\partial U / \partial T$ as a function of $T$.

Sam Stansfield

Numerade Educator

01:14

Problem 20

Show that the time-evolution of the density operator leaves the Shannon entropy $s=-\operatorname{Tr} \rho \log \rho$ invariant.

Ankur S

Numerade Educator

06:05

Problem 21

Show that the partition function of a harmonic oscillator of natural frequency $\omega$ is
$$
Z_{\mathrm{ho}}=\frac{\mathrm{e}^{-\beta \hbar \omega / 2}}{1-\mathrm{e}^{-\beta \hbar \omega}}
$$
Hence show that when the oscillator is at temperature $T=1 /\left(k_{\mathrm{B}} \beta\right)$ the oscillator's internal energy is
$$
U_{\mathrm{ho}}=\hbar \omega\left(\frac{1}{2}+\frac{1}{\mathrm{e}^{\beta \hbar \omega}-1}\right)
$$
Interpret the factor $\left(\mathrm{e}^{\beta \hbar \omega}-1\right)^{-1}$ physically. Show that the specific heat $C=\partial U / \partial T$ is
$$
C=k_{\mathrm{B}} \frac{\mathrm{e}^{\beta \hbar \omega}}{\left(\mathrm{e}^{\beta \hbar \omega}-1\right)^{2}}(\beta \hbar \omega)^{2}
$$
Show that $\lim _{\mathrm{T} \rightarrow 0} C=0$ and obtain a simple expression for $C$ when $k_{\mathrm{B}} T \gg \hbar \omega$

Ameer Said

Numerade Educator

05:33

Problem 22

A classical ideal monatomic gas has internal energy $U=\frac{3}{2} N k_{\mathrm{B}} T$ and pressure $P=N k_{\mathrm{B}} T / \mathcal{V}$, where $N$ is the number of molecules and $\mathcal{V}$ is the volume they occupy. From these relations, and assuming that the entropy vanishes at zero temperature and volume, show that in general the entropy is
$$
S(T, \mathcal{V})=N k_{\mathrm{B}}\left(\frac{3}{2} \ln T+\ln \mathcal{V}\right)
$$
A removable wall divides a cylinder into equal parts of volume $\mathcal{V}$. Initially the wall is in place and each half contains $N$ molecules of ideal monatomic gas at temperature $T$. The wall is removed. Show that equation $(6.128)$ implies that the entropy of the entire body of fluid increases by $2 \ln 2 N k_{\mathrm{B}}$. Can this result be squared with the principle that $\mathrm{d} S=\mathrm{d} Q / T$, where $\mathrm{d} Q$ is the heat absorbed when the change is made reversibly? What conclusion do you draw from this thought experiment?

Averell Hause

Carnegie Mellon University

01:42

Problem 23

Consider a 'gas' of $M$ non-interacting, monatomic molecules of mass $m$ that move in a one-dimensional potential well $V=0$ for $|x|<$ $a$ and $\infty$ otherwise. Assume that at sufficiently low temperatures all molecules are either in the ground or first-excited states. Show that in this approximation the partition function is given by
$$
\ln Z=-M \beta E_{0}+\mathrm{e}^{-3 \beta E_{0}}-\mathrm{e}^{-3(M+1) \beta E_{0}} \quad \text { where } \quad E_{0} \equiv \frac{\pi^{2} \hbar^{2}}{8 m a^{2}}
$$
Show that for $M$ large the internal energy, pressure and specific heat of this gas are given by
$$
U=E_{0}\left(M+3 \mathrm{e}^{-3 \beta E_{0}}\right) ; P=\frac{2 E_{0}}{a}\left(M+3 \mathrm{e}^{-3 \beta E_{0}}\right) ; C_{V}=\frac{9 E_{0}^{2}}{k_{\mathrm{B}} T^{2}} \mathrm{e}^{-3 \beta E_{0}}
$$
In what respects do these results for a quantum ideal gas differ from the properties of a classical ideal gas? Explain these differences physically.

Penny Riley

Numerade Educator