# Quantum Mechanics

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Paper 1, Section II, C

commentConsider a quantum mechanical particle of mass $m$ in a one-dimensional stepped potential well $U(x)$ given by:

$U(x)= \begin{cases}\infty & \text { for } x<0 \text { and } x>a \\ 0 & \text { for } 0 \leqslant x \leqslant a / 2 \\ U_{0} & \text { for } a / 2<x \leqslant a\end{cases}$

where $a>0$ and $U_{0} \geqslant 0$ are constants.

(i) Show that all energy levels $E$ of the particle are non-negative. Show that any level $E$ with $0<E<U_{0}$ satisfies

$\frac{1}{k} \tan \frac{k a}{2}=-\frac{1}{l} \tanh \frac{l a}{2}$

where

$k=\sqrt{\frac{2 m E}{\hbar^{2}}}>0 \quad \text { and } \quad l=\sqrt{\frac{2 m\left(U_{0}-E\right)}{\hbar^{2}}}>0$

(ii) Suppose that initially $U_{0}=0$ and the particle is in the ground state of the potential well. $U_{0}$ is then changed to a value $U_{0}>0$ (while the particle's wavefunction stays the same) and the energy of the particle is measured. For $0<E<U_{0}$, give an expression in terms of $E$ for prob $(E)$, the probability that the energy measurement will find the particle having energy $E$. The expression may be left in terms of integrals that you need not evaluate.

Paper 2, Section II, C

comment(a) Write down the expressions for the probability density $\rho$ and associated current density $j$ of a quantum particle in one dimension with wavefunction $\psi(x, t)$. Show that if $\psi$ is a stationary state then the function $j$ is constant.

For the non-normalisable free particle wavefunction $\psi(x, t)=A e^{i k x-i E t / \hbar}$ (where $E$ and $k$ are real constants and $A$ is a complex constant) compute the functions $\rho$ and $j$, and briefly give a physical interpretation of the functions $\psi, \rho$ and $j$ in this case.

(b) A quantum particle of mass $m$ and energy $E>0$ moving in one dimension is incident from the left in the potential $V(x)$ given by

$V(x)=\left\{\begin{array}{cl} -V_{0} & 0 \leqslant x \leqslant a \\ 0 & x<0 \text { or } x>a \end{array}\right.$

where $a$ and $V_{0}$ are positive constants. Write down the form of the wavefunction in the regions $x<0,0 \leqslant x \leqslant a$ and $x>a$.

Suppose now that $V_{0}=3 E$. Show that the probability $T$ of transmission of the particle into the region $x>a$ is given by

$T=\frac{16}{16+9 \sin ^{2}\left(\frac{a \sqrt{8 m E}}{\hbar}\right)}$

Paper 3, Section I, C

commentThe electron in a hydrogen-like atom moves in a spherically symmetric potential $V(r)=-K / r$ where $K$ is a positive constant and $r$ is the radial coordinate of spherical polar coordinates. The two lowest energy spherically symmetric normalised states of the electron are given by

$\chi_{1}(r)=\frac{1}{\sqrt{\pi} a^{3 / 2}} e^{-r / a} \quad \text { and } \quad \chi_{2}(r)=\frac{1}{4 \sqrt{2 \pi} a^{3 / 2}}\left(2-\frac{r}{a}\right) e^{-r / 2 a}$

where $a=\hbar^{2} / m K$ and $m$ is the mass of the electron. For any spherically symmetric function $f(r)$, the Laplacian is given by $\nabla^{2} f=\frac{d^{2} f}{d r^{2}}+\frac{2}{r} \frac{d f}{d r}$.

(i) Suppose that the electron is in the state $\chi(r)=\frac{1}{2} \chi_{1}(r)+\frac{\sqrt{3}}{2} \chi_{2}(r)$ and its energy is measured. Find the expectation value of the result.

(ii) Suppose now that the electron is in state $\chi(r)$ (as above) at time $t=0$. Let $R(t)$ be the expectation value of a measurement of the electron's radial position $r$ at time $t$. Show that the value of $R(t)$ oscillates sinusoidally about a constant level and determine the frequency of the oscillation.

Paper 4, Section I, C

commentLet $\Psi(x, t)$ be the wavefunction for a particle of mass $m$ moving in one dimension in a potential $U(x)$. Show that, with suitable boundary conditions as $x \rightarrow \pm \infty$,

$\frac{d}{d t} \int_{-\infty}^{\infty}|\Psi(x, t)|^{2} d x=0$

Why is this important for the interpretation of quantum mechanics?

Verify the result above by first calculating $|\Psi(x, t)|^{2}$ for the free particle solution

$\Psi(x, t)=C f(t)^{1 / 2} \exp \left(-\frac{1}{2} f(t) x^{2}\right) \quad \text { with } \quad f(t)=\left(\alpha+\frac{i \hbar}{m} t\right)^{-1}$

where $C$ and $\alpha>0$ are real constants, and then considering the resulting integral.

Paper 4, Section II, C

comment(a) Consider the angular momentum operators $\hat{L}_{x}, \hat{L}_{y}, \hat{L}_{z}$ and $\hat{\mathbf{L}}^{2}=\hat{L}_{x}^{2}+\hat{L}_{y}^{2}+\hat{L}_{z}^{2}$ where

$\hat{L}_{z}=\hat{x} \hat{p}_{y}-\hat{y} \hat{p}_{x}, \quad \hat{L}_{x}=\hat{y} \hat{p}_{z}-\hat{z} \hat{p}_{y} \text { and } \hat{L}_{y}=\hat{z} \hat{p}_{x}-\hat{x} \hat{p}_{z} .$

Use the standard commutation relations for these operators to show that

$\hat{L}_{\pm}=\hat{L}_{x} \pm i \hat{L}_{y} \quad \text { obeys } \quad\left[\hat{L}_{z}, \hat{L}_{\pm}\right]=\pm \hbar \hat{L}_{\pm} \quad \text { and } \quad\left[\hat{\mathbf{L}}^{2}, \hat{L}_{\pm}\right]=0$

Deduce that if $\varphi$ is a joint eigenstate of $\hat{L}_{z}$ and $\hat{\mathbf{L}}^{2}$ with angular momentum quantum numbers $m$ and $\ell$ respectively, then $\hat{L}_{\pm} \varphi$ are also joint eigenstates, provided they are non-zero, with quantum numbers $m \pm 1$ and $\ell$.

(b) A harmonic oscillator of mass $M$ in three dimensions has Hamiltonian

$\hat{H}=\frac{1}{2 M}\left(\hat{p}_{x}^{2}+\hat{p}_{y}^{2}+\hat{p}_{z}^{2}\right)+\frac{1}{2} M \omega^{2}\left(\hat{x}^{2}+\hat{y}^{2}+\hat{z}^{2}\right) .$

Find eigenstates of $\hat{H}$ in terms of eigenstates $\psi_{n}$ for an oscillator in one dimension with $n=0,1,2, \ldots$ and eigenvalues $\hbar \omega\left(n+\frac{1}{2}\right)$; hence determine the eigenvalues $E$ of $\hat{H}$.

Verify that the ground state for $\hat{H}$ is a joint eigenstate of $\hat{L}_{z}$ and $\hat{\mathbf{L}}^{2}$ with $\ell=m=0$. At the first excited energy level, find an eigenstate of $\hat{L}_{z}$ with $m=0$ and construct from this two eigenstates of $\hat{L}_{z}$ with $m=\pm 1$.

Why should you expect to find joint eigenstates of $\hat{L}_{z}, \hat{\mathbf{L}}^{2}$ and $\hat{H}$ ?

[ The first two eigenstates for an oscillator in one dimension are $\psi_{0}(x)=$ $C_{0} \exp \left(-M \omega x^{2} / 2 \hbar\right)$ and $\psi_{1}(x)=C_{1} x \exp \left(-M \omega x^{2} / 2 \hbar\right)$, where $C_{0}$ and $C_{1}$ are normalisation constants. ]

Paper 1, Section I,

commentDefine what it means for an operator $Q$ to be hermitian and briefly explain the significance of this definition in quantum mechanics.

Define the uncertainty $(\Delta Q)_{\psi}$ of $Q$ in a state $\psi$. If $P$ is also a hermitian operator, show by considering the state $(Q+i \lambda P) \psi$, where $\lambda$ is a real number, that

$\left\langle Q^{2}\right\rangle_{\psi}\left\langle P^{2}\right\rangle_{\psi} \geqslant \frac{1}{4}\left|\langle i[Q, P]\rangle_{\psi}\right|^{2}$

Hence deduce that

$(\Delta Q)_{\psi}(\Delta P)_{\psi} \geqslant \frac{1}{2}\left|\langle i[Q, P]\rangle_{\psi}\right|$

Give a physical interpretation of this result.

Paper 1, Section II, A

commentConsider a quantum system with Hamiltonian $H$ and wavefunction $\Psi$ obeying the time-dependent Schrödinger equation. Show that if $\Psi$ is a stationary state then $\langle Q\rangle_{\Psi}$ is independent of time, if the observable $Q$ is independent of time.

A particle of mass $m$ is confined to the interval $0 \leqslant x \leqslant a$ by infinite potential barriers, but moves freely otherwise. Let $\Psi(x, t)$ be the normalised wavefunction for the particle at time $t$, with

$\Psi(x, 0)=c_{1} \psi_{1}(x)+c_{2} \psi_{2}(x)$

where

$\psi_{1}(x)=\left(\frac{2}{a}\right)^{1 / 2} \sin \frac{\pi x}{a}, \quad \psi_{2}(x)=\left(\frac{2}{a}\right)^{1 / 2} \sin \frac{2 \pi x}{a}$

and $c_{1}, c_{2}$ are complex constants. If the energy of the particle is measured at time $t$, what are the possible results, and what is the probability for each result to be obtained? Give brief justifications of your answers.

Calculate $\langle\hat{x}\rangle_{\Psi}$ at time $t$ and show that the result oscillates with a frequency $\omega$, to be determined. Show in addition that

$\left|\langle\hat{x}\rangle_{\Psi}-\frac{a}{2}\right| \leqslant \frac{16 a}{9 \pi^{2}} .$

Paper 2, Section II, A

comment(a) The potential $V(x)$ for a particle of mass $m$ in one dimension is such that $V \rightarrow 0$ rapidly as $x \rightarrow \pm \infty$. Let $\psi(x)$ be a wavefunction for the particle satisfying the time-independent Schrödinger equation with energy $E$.

Suppose $\psi$ has the asymptotic behaviour

$\psi(x) \sim A e^{i k x}+B e^{-i k x} \quad(x \rightarrow-\infty), \quad \psi(x) \sim C e^{i k x} \quad(x \rightarrow+\infty)$

where $A, B, C$ are complex coefficients. Explain, in outline, how the probability current $j(x)$ is used in the interpretation of such a solution as a scattering process and how the transmission and reflection probabilities $P_{\mathrm{tr}}$ and $P_{\text {ref }}$ are found.

Now suppose instead that $\psi(x)$ is a bound state solution. Write down the asymptotic behaviour in this case, relating an appropriate parameter to the energy $E$.

(b) Consider the potential

$V(x)=-\frac{\hbar^{2}}{m} \frac{a^{2}}{\cosh ^{2} a x}$

where $a$ is a real, positive constant. Show that

$\psi(x)=N e^{i k x}(a \tanh a x-i k)$

where $N$ is a complex coefficient, is a solution of the time-independent Schrödinger equation for any real $k$ and find the energy $E$. Show that $\psi$ represents a scattering process for which $P_{\text {ref }}=0$, and find $P_{\mathrm{tr}}$ explicitly.

Now let $k=i \lambda$ in the formula for $\psi$ above. Show that this defines a bound state if a certain real positive value of $\lambda$ is chosen and find the energy of this solution.

Paper 1, Section II, B

commentStarting from the time-dependent Schrödinger equation, show that a stationary state $\psi(x)$ of a particle of mass $m$ in a harmonic oscillator potential in one dimension with frequency $\omega$ satisfies

$-\frac{\hbar^{2}}{2 m} \frac{d^{2} \psi}{d x^{2}}+\frac{1}{2} m \omega^{2} x^{2} \psi=E \psi .$

Find a rescaling of variables that leads to the simplified equation

$-\frac{d^{2} \psi}{d y^{2}}+y^{2} \psi=\varepsilon \psi$

Setting $\psi=f(y) e^{-\frac{1}{2} y^{2}}$, find the equation satisfied by $f(y)$.

Assume now that $f$ is a polynomial

$f(y)=y^{N}+a_{N-1} y^{N-1}+a_{N-2} y^{N-2}+\ldots+a_{0}$

Determine the value of $\varepsilon$ and deduce the corresponding energy level $E$ of the harmonic oscillator. Show that if $N$ is even then the stationary state $\psi(x)$ has even parity.

Paper 2, Section II, B

commentLet $x, y, z$ be Cartesian coordinates in $\mathbb{R}^{3}$. The angular momentum operators satisfy the commutation relation

$\left[L_{x}, L_{y}\right]=i \hbar L_{z}$

and its cyclic permutations. Define the total angular momentum operator $\mathbf{L}^{2}$ and show that $\left[L_{z}, \mathbf{L}^{2}\right]=0$. Write down the explicit form of $L_{z}$.

Show that a function of the form $(x+i y)^{m} z^{n} f(r)$, where $r^{2}=x^{2}+y^{2}+z^{2}$, is an eigenfunction of $L_{z}$ and find the eigenvalue. State the analogous result for $(x-i y)^{m} z^{n} f(r)$.

There is an energy level for a particle in a certain spherically symmetric potential well that is 6-fold degenerate. A basis for the (unnormalized) energy eigenstates is of the form

$\left(x^{2}-1\right) f(r),\left(y^{2}-1\right) f(r),\left(z^{2}-1\right) f(r), x y f(r), x z f(r), y z f(r) \text {. }$

Find a new basis that consists of simultaneous eigenstates of $L_{z}$ and $\mathbf{L}^{2}$ and identify their eigenvalues.

[You may quote the range of $L_{z}$ eigenvalues associated with a particular eigenvalue of $\mathbf{L}^{2}$.]

Paper 3, Section $I$, B

commentConsider a quantum mechanical particle moving in two dimensions with Cartesian coordinates $x, y$. Show that, for wavefunctions with suitable decay as $x^{2}+y^{2} \rightarrow \infty$, the operators

$x \quad \text { and } \quad-i \hbar \frac{\partial}{\partial x}$

are Hermitian, and similarly

$y \text { and }-i \hbar \frac{\partial}{\partial y}$

are Hermitian.

Show that if $F$ and $G$ are Hermitian operators, then

$\frac{1}{2}(F G+G F)$

is Hermitian. Deduce that

$L=-i \hbar\left(x \frac{\partial}{\partial y}-y \frac{\partial}{\partial x}\right) \quad \text { and } \quad D=-i \hbar\left(x \frac{\partial}{\partial x}+y \frac{\partial}{\partial y}+1\right)$

are Hermitian. Show that

$[L, D]=0 .$

Paper 3, Section II, B

commentConsider a particle of unit mass in a one-dimensional square well potential

$V(x)=0 \text { for } 0 \leqslant x \leqslant \pi,$

with $V$ infinite outside. Find all the stationary states $\psi_{n}(x)$ and their energies $E_{n}$, and write down the general normalized solution of the time-dependent Schrödinger equation in terms of these.

The particle is initially constrained by a barrier to be in the ground state in the narrower square well potential

$V(x)=0 \quad \text { for } \quad 0 \leqslant x \leqslant \frac{\pi}{2}$

with $V$ infinite outside. The barrier is removed at time $t=0$, and the wavefunction is instantaneously unchanged. Show that the particle is now in a superposition of stationary states of the original potential well, and calculate the probability that an energy measurement will yield the result $E_{n}$.

Paper 4, Section I, B

comment(a) Define the probability density $\rho$ and probability current $j$ for the wavefunction $\Psi(x, t)$ of a particle of mass $m$. Show that

$\frac{\partial \rho}{\partial t}+\frac{\partial j}{\partial x}=0$

and deduce that $j=0$ for a normalizable, stationary state wavefunction. Give an example of a non-normalizable, stationary state wavefunction for which $j$ is non-zero, and calculate the value of $j$.

(b) A particle has the instantaneous, normalized wavefunction

$\Psi(x, 0)=\left(\frac{2 \alpha}{\pi}\right)^{1 / 4} e^{-\alpha x^{2}+i k x}$

where $\alpha$ is positive and $k$ is real. Calculate the expectation value of the momentum for this wavefunction.

Paper 1, Section II, B

commentThe relative motion of a neutron and proton is described by the Schrödinger equation for a single particle of mass $m$ under the influence of the central potential

$V(r)=\left\{\begin{array}{rr} -U & r<a \\ 0 & r>a \end{array}\right.$

where $U$ and $a$ are positive constants. Solve this equation for a spherically symmetric state of the deuteron, which is a bound state of a proton and a neutron, giving the condition on $U$ for this state to exist.

[If $\psi$ is spherically symmetric then $\nabla^{2} \psi=\frac{1}{r} \frac{d^{2}}{d r^{2}}(r \psi)$.]

Paper 2, Section II, B

commentFor an electron in a hydrogen atom, the stationary-state wavefunctions are of the form $\psi(r, \theta, \phi)=R(r) Y_{l m}(\theta, \phi)$, where in suitable units $R$ obeys the radial equation

$\frac{d^{2} R}{d r^{2}}+\frac{2}{r} \frac{d R}{d r}-\frac{l(l+1)}{r^{2}} R+2\left(E+\frac{1}{r}\right) R=0$

Explain briefly how the terms in this equation arise.

This radial equation has bound-state solutions of energy $E=E_{n}$, where $E_{n}=-\frac{1}{2 n^{2}}(n=1,2,3, \ldots)$. Show that when $l=n-1$, there is a solution of the form $R(r)=r^{\alpha} e^{-r / n}$, and determine $\alpha$. Find the expectation value $\langle r\rangle$ in this state.

Determine the total degeneracy of the energy level with energy $E_{n}$.

Paper 3, Section I, B

commentWhat is meant by the statement that an operator is Hermitian?

Consider a particle of mass $m$ in a real potential $V(x)$ in one dimension. Show that the Hamiltonian of the system is Hermitian.

Starting from the time-dependent Schrödinger equation, show that

$\frac{d}{d t}\langle\hat{x}\rangle=\frac{1}{m}\langle\hat{p}\rangle, \quad \frac{d}{d t}\langle\hat{p}\rangle=-\left\langle V^{\prime}(\hat{x})\right\rangle$

where $\hat{p}$ is the momentum operator and $\langle\hat{A}\rangle$ denotes the expectation value of the operator $\hat{A}$.

Paper 3, Section II, B

commentWhat is the physical significance of the expectation value

$\langle Q\rangle=\int \psi^{*}(x) Q \psi(x) d x$

of an observable $Q$ in the normalised state $\psi(x)$ ? Let $P$ and $Q$ be two observables. By considering the norm of $(Q+i \lambda P) \psi$ for real values of $\lambda$, show that

$\left\langle Q^{2}\right\rangle\left\langle P^{2}\right\rangle \geqslant \frac{1}{4}|\langle[Q, P]\rangle|^{2} .$

Deduce the generalised uncertainty relation

$\Delta Q \Delta P \geqslant \frac{1}{2}|\langle[Q, P]\rangle|,$

where the uncertainty $\Delta Q$ in the state $\psi(x)$ is defined by

$(\Delta Q)^{2}=\left\langle(Q-\langle Q\rangle)^{2}\right\rangle$

A particle of mass $m$ moves in one dimension under the influence of the potential $\frac{1}{2} m \omega^{2} x^{2}$. By considering the commutator $[x, p]$, show that every energy eigenvalue $E$ satisfies

$E \geqslant \frac{1}{2} \hbar \omega$

Paper 4, Section I, B

commentA particle moving in one space dimension with wavefunction $\Psi(x, t)$ obeys the timedependent Schrödinger equation. Write down the probability density $\rho$ and current density $j$ in terms of the wavefunction and show that they obey the equation

$\frac{\partial j}{\partial x}+\frac{\partial \rho}{\partial t}=0$

Evaluate $j(x, t)$ in the case that

$\Psi(x, t)=\left(A e^{i k x}+B e^{-i k x}\right) e^{-i E t / \hbar}$

where $E=\hbar^{2} k^{2} / 2 m$, and $A$ and $B$ are constants, which may be complex.

Paper 1, Section II, B

commentConsider the time-independent Schrödinger equation in one dimension for a particle of mass $m$ with potential $V(x)$.

(a) Show that if the potential is an even function then any non-degenerate stationary state has definite parity.

(b) A particle of mass $m$ is subject to the potential $V(x)$ given by

$V(x)=-\lambda(\delta(x-a)+\delta(x+a))$

where $\lambda$ and $a$ are real positive constants and $\delta(x)$ is the Dirac delta function.

Derive the conditions satisfied by the wavefunction $\psi(x)$ around the points $x=\pm a$.

Show (using a graphical method or otherwise) that there is a bound state of even parity for any $\lambda>0$, and that there is an odd parity bound state only if $\lambda>\hbar^{2} /(2 m a)$. [Hint: You may assume without proof that the functions $x \tanh x$ and $x \operatorname{coth} x$ are monotonically increasing for $x>0$.]

Paper 2, Section II, B

comment(a) The potential for the one-dimensional harmonic oscillator is $V(x)=\frac{1}{2} m \omega^{2} x^{2}$. By considering the associated time-independent Schrödinger equation for the wavefunction $\psi(x)$ with substitutions

$\xi=\left(\frac{m \omega}{\hbar}\right)^{1 / 2} x \quad \text { and } \quad \psi(x)=f(\xi) e^{-\xi^{2} / 2}$

show that the allowed energy levels are given by $E_{n}=\left(n+\frac{1}{2}\right) \hbar \omega$ for $n=0,1,2, \ldots$ [You may assume without proof that $f$ must be a polynomial for $\psi$ to be normalisable.]

(b) Consider a particle with charge $q$ and mass $m=1$ subject to the one-dimensional harmonic oscillator potential $U_{0}(x)=x^{2} / 2$. You may assume that the normalised ground state of this potential is

$\psi_{0}(x)=\left(\frac{1}{\pi \hbar}\right)^{1 / 4} e^{-x^{2} /(2 \hbar)}$

The particle is in the stationary state corresponding to $\psi_{0}(x)$ when at time $t=t_{0}$, an electric field of constant strength $E$ is turned on, adding an extra term $U_{1}(x)=-q E x$ to the harmonic potential.

(i) Using the result of part (a) or otherwise, find the energy levels of the new potential.

(ii) Show that the probability of finding the particle in the ground state immediately after $t_{0}$ is given by $e^{-q^{2} E^{2} /(2 \hbar)}$. [You may assume that $\int_{-\infty}^{\infty} e^{-x^{2}+2 A x} d x=$ $\sqrt{\pi} e^{A^{2}}$.]

Paper 3, Section I, B

commentA particle of mass $m$ is confined to a one-dimensional box $0 \leqslant x \leqslant a$. The potential $V(x)$ is zero inside the box and infinite outside.

(a) Find the allowed energies of the particle and the normalised energy eigenstates.

(b) At time $t=0$ the particle has wavefunction $\psi_{0}$ that is uniform in the left half of the box i.e. $\psi_{0}(x)=\sqrt{\frac{2}{a}}$ for $0<x<a / 2$ and $\psi_{0}(x)=0$ for $a / 2<x<a$. Find the probability that a measurement of energy at time $t=0$ will yield a value less than $5 \hbar^{2} \pi^{2} /\left(2 m a^{2}\right)$.

Paper 3, Section II, B

comment(a) Given the position and momentum operators $\hat{x}_{i}=x_{i}$ and $\hat{p}_{i}=-i \hbar \partial / \partial x_{i}$ (for $i=1,2,3)$ in three dimensions, define the angular momentum operators $\hat{L}_{i}$ and the total angular momentum $\hat{L}^{2}$.

Show that $\hat{L}_{3}$ is Hermitian.

(b) Derive the generalised uncertainty relation for the observables $\hat{L}_{3}$ and $\hat{x}_{1}$ in the form

$\Delta_{\psi} \hat{L}_{3} \Delta_{\psi} \hat{x}_{1} \geqslant M$

for any state $\psi$ and a suitable expression $M$ that you should determine. [Hint: It may be useful to consider the operator $\hat{L}_{3}+i \lambda \hat{x}_{1}$.]

(c) Consider a particle with wavefunction

$\psi=K\left(x_{1}+x_{2}+2 x_{3}\right) e^{-\alpha r}$

where $r=\sqrt{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}$ and $K$ and $\alpha$ are real positive constants.

Show that $\psi$ is an eigenstate of total angular momentum $\hat{L}^{2}$ and find the corresponding angular momentum quantum number $l$. Find also the expectation value of a measurement of $\hat{L}_{3}$ on the state $\psi$.

Paper 4, Section I, B

comment(a) Give a physical interpretation of the wavefunction $\phi(x, t)=A e^{i k x} e^{-i E t / \hbar}$ (where $A, k$ and $E$ are real constants).

(b) A particle of mass $m$ and energy $E>0$ is incident from the left on the potential step

$V(x)=\left\{\begin{array}{cl} 0 & \text { for }-\infty<x<a \\ V_{0} & \text { for } a<x<\infty \end{array}\right.$

with $V_{0}>0$.

State the conditions satisfied by a stationary state at the point $x=a$.

Compute the probability that the particle is reflected as a function of $E$, and compare your result with the classical case.

Paper 1, Section II, B

comment(a) A particle of mass $m$ in one space dimension is confined to move in a potential $V(x)$ given by

$V(x)= \begin{cases}0 & \text { for } 0<x<a \\ \infty & \text { for } x<0 \text { or } x>a\end{cases}$

The normalised initial wavefunction of the particle at time $t=0$ is

$\psi_{0}(x)=\frac{4}{\sqrt{5 a}} \sin ^{3}\left(\frac{\pi x}{a}\right)$

(i) Find the expectation value of the energy at time $t=0$.

(ii) Find the wavefunction of the particle at time $t=1$.

[Hint: It may be useful to recall the identity $\sin 3 \theta=3 \sin \theta-4 \sin ^{3} \theta$.]

(b) The right hand wall of the potential is lowered to a finite constant value $U_{0}>0$ giving the new potential:

$U(x)= \begin{cases}0 & \text { for } 0<x<a \\ \infty & \text { for } x<0 \\ U_{0} & \text { for } x>a\end{cases}$

This potential is set up in the laboratory but the value of $U_{0}$ is unknown. The stationary states of the potential are investigated and it is found that there exists exactly one bound state. Show that the value of $U_{0}$ must satisfy

$\frac{\pi^{2} \hbar^{2}}{8 m a^{2}}<U_{0}<\frac{9 \pi^{2} \hbar^{2}}{8 m a^{2}}$

Paper 2, Section II, B

commentThe one dimensional quantum harmonic oscillator has Hamiltonian

$\hat{H}=\frac{1}{2 m} \hat{p}^{2}+\frac{1}{2} m \omega^{2} \hat{x}^{2},$

where $m$ and $\omega$ are real positive constants and $\hat{x}$ and $\hat{p}$ are the standard position and momentum operators satisfying the commutation relation $[\hat{x}, \hat{p}]=i \hbar$. Consider the operators

$\hat{A}=\hat{p}-i m \omega \hat{x} \quad \text { and } \quad \hat{B}=\hat{p}+i m \omega \hat{x} .$

(a) Show that

$\hat{B} \hat{A}=2 m\left(\hat{H}-\frac{1}{2} \hbar \omega\right) \quad \text { and } \quad \hat{A} \hat{B}=2 m\left(\hat{H}+\frac{1}{2} \hbar \omega\right) .$

(b) Suppose that $\phi$ is an eigenfunction of $\hat{H}$ with eigenvalue $E$. Show that $\hat{A} \phi$ is then also an eigenfunction of $\hat{H}$ and that its corresponding eigenvalue is $E-\hbar \omega$.

(c) Show that for any normalisable wavefunctions $\chi$ and $\psi$,

$\int_{-\infty}^{\infty} \chi^{*}(\hat{A} \psi) d x=\int_{-\infty}^{\infty}(\hat{B} \chi)^{*} \psi d x$

[You may assume that the operators $\hat{x}$ and $\hat{p}$ are Hermitian.]

(d) With $\phi$ as in (b), obtain an expression for the norm of $\hat{A} \phi$ in terms of $E$ and the norm of $\phi$. [The squared norm of any wavefunction $\psi$ is $\int_{-\infty}^{\infty}|\psi|^{2} d x$.]

(e) Show that all eigenvalues of $\hat{H}$ are non-negative.

(f) Using the above results, deduce that each eigenvalue $E$ of $\hat{H}$ must be of the form $E=\left(n+\frac{1}{2}\right) \hbar \omega$ for some non-negative integer $n$.

Paper 3, Section I, B

comment(a) Consider a quantum particle moving in one space dimension, in a timeindependent real potential $V(x)$. For a wavefunction $\psi(x, t)$, define the probability density $\rho(x, t)$ and probability current $j(x, t)$ and show that

$\frac{\partial \rho}{\partial t}+\frac{\partial j}{\partial x}=0$

(b) Suppose now that $V(x)=0$ and $\psi(x, t)=\left(e^{i k x}+R e^{-i k x}\right) e^{-i E t / \hbar}$, where $E=\hbar^{2} k^{2} /(2 m), k$ and $m$ are real positive constants, and $R$ is a complex constant. Compute the probability current for this wavefunction. Interpret the terms in $\psi$ and comment on how this relates to the computed expression for the probability current.

Paper 3, Section II, B

commentThe spherically symmetric bound state wavefunctions $\psi(r)$ for the Coulomb potential $V=-e^{2} /\left(4 \pi \epsilon_{0} r\right)$ are normalisable solutions of the equation

$\frac{d^{2} \psi}{d r^{2}}+\frac{2}{r} \frac{d \psi}{d r}+\frac{2 \lambda}{r} \psi=-\frac{2 m E}{\hbar^{2}} \psi$

Here $\lambda=\left(m e^{2}\right) /\left(4 \pi \epsilon_{0} \hbar^{2}\right)$ and $E<0$ is the energy of the state.

(a) By writing the wavefunction as $\psi(r)=f(r) \exp (-K r)$, for a suitable constant $K$ that you should determine, show that there are normalisable wavefunctions $\psi(r)$ only for energies of the form

$E=\frac{-m e^{4}}{32 \pi^{2} \epsilon_{0}^{2} \hbar^{2} N^{2}}$

with $N$ being a positive integer.

(b) The energies in (a) reproduce the predictions of the Bohr model of the hydrogen atom. How do the wavefunctions above compare to the assumptions in the Bohr model?

Paper 4, Section I, B

comment(a) Define the quantum orbital angular momentum operator $\hat{\boldsymbol{L}}=\left(\hat{L}_{1}, \hat{L}_{2}, \hat{L}_{3}\right)$ in three dimensions, in terms of the position and momentum operators.

(b) Show that $\left[\hat{L}_{1}, \hat{L}_{2}\right]=i \hbar \hat{L}_{3}$. [You may assume that the position and momentum operators satisfy the canonical commutation relations.]

(c) Let $\hat{L}^{2}=\hat{L}_{1}^{2}+\hat{L}_{2}^{2}+\hat{L}_{3}^{2}$. Show that $\hat{L}_{1}$ commutes with $\hat{L}^{2}$.

[In this part of the question you may additionally assume without proof the permuted relations $\left[\hat{L}_{2}, \hat{L}_{3}\right]=i \hbar \hat{L}_{1}$ and $\left.\left[\hat{L}_{3}, \hat{L}_{1}\right]=i \hbar \hat{L}_{2} .\right]$

[Hint: It may be useful to consider the expression $[\hat{A}, \hat{B}] \hat{B}+\hat{B}[\hat{A}, \hat{B}]$ for suitable operators $\hat{A}$ and $\hat{B}$.]

(d) Suppose that $\psi_{1}(x, y, z)$ and $\psi_{2}(x, y, z)$ are normalised eigenstates of $\hat{L}_{1}$ with eigenvalues $\hbar$ and $-\hbar$ respectively. Consider the wavefunction

$\psi=\frac{1}{2} \psi_{1} \cos \omega t+\frac{\sqrt{3}}{2} \psi_{2} \sin \omega t$

with $\omega$ being a positive constant. Find the earliest time $t_{0}>0$ such that the expectation value of $\hat{L}_{1}$ in $\psi$ is zero.

Paper 1, Section II, D

commentWrite down expressions for the probability density $\rho(x, t)$ and the probability current $j(x, t)$ for a particle in one dimension with wavefunction $\Psi(x, t)$. If $\Psi(x, t)$ obeys the timedependent Schrödinger equation with a real potential, show that

$\frac{\partial j}{\partial x}+\frac{\partial \rho}{\partial t}=0$

Consider a stationary state, $\Psi(x, t)=\psi(x) e^{-i E t / \hbar}$, with

$\psi(x) \sim \begin{cases}e^{i k_{1} x}+R e^{-i k_{1} x} & x \rightarrow-\infty \\ T e^{i k_{2} x} & x \rightarrow+\infty\end{cases}$

where $E, k_{1}, k_{2}$ are real. Evaluate $j(x, t)$ for this state in the regimes $x \rightarrow+\infty$ and $x \rightarrow-\infty$.

Consider a real potential,

$V(x)=-\alpha \delta(x)+U(x), \quad U(x)= \begin{cases}0 & x<0 \\ V_{0} & x>0\end{cases}$

where $\delta(x)$ is the Dirac delta function, $V_{0}>0$ and $\alpha>0$. Assuming that $\psi(x)$ is continuous at $x=0$, derive an expression for

$\lim _{\epsilon \rightarrow 0}\left[\psi^{\prime}(\epsilon)-\psi^{\prime}(-\epsilon)\right]$

Hence calculate the reflection and transmission probabilities for a particle incident from $x=-\infty$ with energy $E>V_{0} .$

Paper 2, Section II, D

commentA quantum-mechanical harmonic oscillator has Hamiltonian

$\hat{H}=\frac{\hat{p}^{2}}{2}+\frac{1}{2} k^{2} \hat{x}^{2}$

where $k$ is a positive real constant. Show that $\hat{x}=x$ and $\hat{p}=-i \hbar \frac{\partial}{\partial x}$ are Hermitian operators.

The eigenfunctions of $(*)$ can be written as

$\psi_{n}(x)=h_{n}(x \sqrt{k / \hbar}) \exp \left(-\frac{k x^{2}}{2 \hbar}\right),$

where $h_{n}$ is a polynomial of degree $n$ with even (odd) parity for even (odd) $n$ and $n=0,1,2, \ldots$. Show that $\langle\hat{x}\rangle=\langle\hat{p}\rangle=0$ for all of the states $\psi_{n}$.

State the Heisenberg uncertainty principle and verify it for the state $\psi_{0}$ by computing $(\Delta x)$ and $(\Delta p)$. [Hint: You should properly normalise the state.]

The oscillator is in its ground state $\psi_{0}$ when the potential is suddenly changed so that $k \rightarrow 4 k$. If the wavefunction is expanded in terms of the energy eigenfunctions of the new Hamiltonian, $\phi_{n}$, what can be said about the coefficient of $\phi_{n}$ for odd $n$ ? What is the probability that the particle is in the new ground state just after the change?

[Hint: You may assume that if $I_{n}=\int_{-\infty}^{\infty} e^{-a x^{2}} x^{n} d x$ then $I_{0}=\sqrt{\frac{\pi}{a}}$ and $I_{2}=\frac{1}{2 a} \sqrt{\frac{\pi}{a}}$.]

Paper 3, Section $I$, D

commentA quantum-mechanical system has normalised energy eigenstates $\chi_{1}$ and $\chi_{2}$ with non-degenerate energies $E_{1}$ and $E_{2}$ respectively. The observable $A$ has normalised eigenstates,

$\begin{aligned} \phi_{1} &=C\left(\chi_{1}+2 \chi_{2}\right), & & \text { eigenvalue }=a_{1} \\ \phi_{2} &=C\left(2 \chi_{1}-\chi_{2}\right), & & \text { eigenvalue }=a_{2} \end{aligned}$

where $C$ is a positive real constant. Determine $C$.

Initially, at time $t=0$, the state of the system is $\phi_{1}$. Write down an expression for $\psi(t)$, the state of the system with $t \geqslant 0$. What is the probability that a measurement of energy at time $t$ will yield $E_{2}$ ?

For the same initial state, determine the probability that a measurement of $A$ at time $t>0$ will yield $a_{1}$ and the probability that it will yield $a_{2}$.

Paper 3, Section II, D

commentDefine the angular momentum operators $\hat{L}_{i}$ for a particle in three dimensions in terms of the position and momentum operators $\hat{x}_{i}$ and $\hat{p}_{i}=-i \hbar \frac{\partial}{\partial x_{i}}$. Write down an expression for $\left[\hat{L}_{i}, \hat{L}_{j}\right]$ and use this to show that $\left[\hat{L}^{2}, \hat{L}_{i}\right]=0$ where $\hat{L}^{2}=\hat{L}_{x}^{2}+\hat{L}_{y}^{2}+\hat{L}_{z}^{2}$. What is the significance of these two commutation relations?

Let $\psi(x, y, z)$ be both an eigenstate of $\hat{L}_{z}$ with eigenvalue zero and an eigenstate of $\hat{L}^{2}$ with eigenvalue $\hbar^{2} l(l+1)$. Show that $\left(\hat{L}_{x}+i \hat{L}_{y}\right) \psi$ is also an eigenstate of both $\hat{L}_{z}$ and $\hat{L}^{2}$ and determine the corresponding eigenvalues.

Find real constants $A$ and $B$ such that

$\phi(x, y, z)=\left(A z^{2}+B y^{2}-r^{2}\right) e^{-r}, \quad r^{2}=x^{2}+y^{2}+z^{2},$

is an eigenfunction of $\hat{L}_{z}$ with eigenvalue zero and an eigenfunction of $\hat{L}^{2}$ with an eigenvalue which you should determine. [Hint: You might like to show that $\left.\hat{L}_{i} f(r)=0 .\right]$

Paper 4, Section I, D

commentThe radial wavefunction $R(r)$ for an electron in a hydrogen atom satisfies the equation

$-\frac{\hbar^{2}}{2 m r^{2}} \frac{d}{d r}\left(r^{2} \frac{d}{d r} R(r)\right)+\frac{\hbar^{2}}{2 m r^{2}} \ell(\ell+1) R(r)-\frac{e^{2}}{4 \pi \epsilon_{0} r} R(r)=E R(r)$

Briefly explain the origin of each term in this equation.

The wavefunctions for the ground state and the first radially excited state, both with $\ell=0$, can be written as

$\begin{aligned} &R_{1}(r)=N_{1} e^{-\alpha r} \\ &R_{2}(r)=N_{2}\left(1-\frac{1}{2} r \alpha\right) e^{-\frac{1}{2} \alpha r} \end{aligned}$

where $N_{1}$ and $N_{2}$ are normalisation constants. Verify that $R_{1}(r)$ is a solution of $(*)$, determining $\alpha$ and finding the corresponding energy eigenvalue $E_{1}$. Assuming that $R_{2}(r)$ is a solution of $(*)$, compare coefficients of the dominant terms when $r$ is large to determine the corresponding energy eigenvalue $E_{2}$. [You do not need to find $N_{1}$ or $N_{2}$, nor show that $R_{2}$ is a solution of $\left.(*) .\right]$

A hydrogen atom makes a transition from the first radially excited state to the ground state, emitting a photon. What is the angular frequency of the emitted photon?

Paper 1, Section II, A

commentConsider a particle confined in a one-dimensional infinite potential well: $V(x)=\infty$ for $|x| \geqslant a$ and $V(x)=0$ for $|x|<a$. The normalised stationary states are

$\psi_{n}(x)= \begin{cases}\alpha_{n} \sin \left(\frac{\pi n(x+a)}{2 a}\right) & \text { for }|x|<a \\ 0 & \text { for }|x| \geqslant a\end{cases}$

where $n=1,2, \ldots$.

(i) Determine the $\alpha_{n}$ and the stationary states' energies $E_{n}$.

(ii) A state is prepared within this potential well: $\psi(x) \propto x$ for $0<x<a$, but $\psi(x)=0$ for $x \leqslant 0$ or $x \geqslant a$. Find an explicit expansion of $\psi(x)$ in terms of $\psi_{n}(x) .$

(iii) If the energy of the state is then immediately measured, show that the probability that it is greater than $\frac{\hbar^{2} \pi^{2}}{m a^{2}}$ is

$\sum_{n=0}^{4} \frac{b_{n}}{\pi^{n}}$

where the $b_{n}$ are integers which you should find.

(iv) By considering the normalisation condition for $\psi(x)$ in terms of the expansion in $\psi_{n}(x)$, show that

$\frac{\pi^{2}}{3}=\sum_{p=1}^{\infty} \frac{A}{p^{2}}+\frac{B}{(2 p-1)^{2}}\left(1+\frac{C(-1)^{p}}{(2 p-1) \pi}\right)^{2}$

where $A, B$ and $C$ are integers which you should find.

Paper 2, Section II, A

commentFor an electron of mass $m$ in a hydrogen atom, the time-independent Schrödinger equation may be written as

$-\frac{\hbar^{2}}{2 m r^{2}} \frac{\partial}{\partial r}\left(r^{2} \frac{\partial \psi}{\partial r}\right)+\frac{1}{2 m r^{2}} L^{2} \psi-\frac{e^{2}}{4 \pi \epsilon_{0} r} \psi=E \psi$

Consider normalised energy eigenstates of the form

$\psi_{l m}(r, \theta, \phi)=R(r) Y_{l m}(\theta, \phi)$

where $Y_{l m}$ are orbital angular momentum eigenstates:

$L^{2} Y_{l m}=\hbar^{2} l(l+1) Y_{l m}, \quad L_{3} Y_{l m}=\hbar m Y_{l m}$

where $l=1,2, \ldots$ and $m=0, \pm 1, \pm 2, \ldots \pm l$. The $Y_{l m}$ functions are normalised with $\int_{\theta=0}^{\pi} \int_{\phi=0}^{2 \pi}\left|Y_{l m}\right|^{2} \sin \theta d \theta d \phi=1 .$

(i) Write down the resulting equation satisfied by $R(r)$, for fixed $l$. Show that it has solutions of the form

$R(r)=A r^{l} \exp \left(-\frac{r}{a(l+1)}\right)$

where $a$ is a constant which you should determine. Show that

$E=-\frac{e^{2}}{D \pi \epsilon_{0} a}$

where $D$ is an integer which you should find (in terms of $l$ ). Also, show that

$|A|^{2}=\frac{2^{2 l+3}}{a^{F} G !(l+1)^{H}},$

where $F, G$ and $H$ are integers that you should find in terms of $l$.

(ii) Given the radius of the proton $r_{p} \ll a$, show that the probability of the electron being found within the proton is approximately

$\frac{2^{2 l+3}}{C}\left(\frac{r_{p}}{a}\right)^{2 l+3}\left[1+\mathcal{O}\left(\frac{r_{p}}{a}\right)\right]$

finding the integer $C$ in terms of $l$.

[You may assume that $\int_{0}^{\infty} t^{l} e^{-t} d t=l !$.]

Paper 3, Section I, A

The wavefunction of a normalised Gaussian wavepacket for a particle of mass $m$ in one dimension with potential $V(x)=0$ is given by

$\psi(x, t)=B \sqrt{A(t)} \exp \left(\frac{-x^{2} A(t)}{2}\right)$