# Part II, 2003

### Jump to course

B2.10

comment(a) For which polynomials $f(x)$ of degree $d>0$ does the equation $y^{2}=f(x)$ define a smooth affine curve?

(b) Now let $C$ be the completion of the curve defined in (a) to a projective curve. For which polynomials $f(x)$ of degree $d>0$ is $C$ a smooth projective curve?

(c) Suppose that $C$, defined in (b), is a smooth projective curve. Consider a map $p: C \rightarrow \mathbb{P}^{1}$, given by $p(x, y)=x$. Find the degree and the ramification points of $p$.

B3.10

comment(a) Let $X \subseteq \mathbb{A}^{n}$ be an affine algebraic variety. Define the tangent space $T_{p} X$ for $p \in X$. Show that the set

$\left\{p \in X \mid \operatorname{dim} T_{p} X \geqslant d\right\}$

is closed, for every $d \geqslant 0$.

(b) Let $C$ be an irreducible projective curve, $p \in C$, and $f: C \backslash\{p\} \rightarrow \mathbb{P}^{n}$ a rational map. Show, carefully quoting any theorems that you use, that if $C$ is smooth at $p$ then $f$ extends to a regular map at $p$.

B4.9

commentLet $X$ be a smooth curve of genus 0 over an algebraically closed field $k$. Show that $k(X)=k\left(\mathbb{P}^{1}\right) .$

Now let $C$ be a plane projective curve defined by an irreducible homogeneous cubic polynomial.

(a) Show that if $C$ is smooth then $C$ is not isomorphic to $\mathbb{P}^{1}$. Standard results on the canonical class may be assumed without proof, provided these are clearly stated.

(b) Show that if $C$ has a singularity then there exists a non-constant morphism from $\mathbb{P}^{1}$ to $C$.

B2.8

commentDefine the fundamental group of a topological space and explain briefly why a continuous map gives rise to a homomorphism between fundamental groups.

Let $X$ be a subspace of the Euclidean space $\mathbb{R}^{3}$ which contains all of the points $(x, y, 0)$ with $(x, y) \neq(0,0)$, and which does not contain any of the points $(0,0, z)$. Show that $X$ has an infinite fundamental group.

B3.7

commentDefine a covering map. Prove that any covering map induces an injective homomorphisms of fundamental groups.

Show that there is a non-trivial covering map of the real projective plane. Explain how to use this to find the fundamental group of the real projective plane.

B4.5

commentState the Mayer-Vietoris theorem. You should give the definition of all the homomorphisms involved.

Compute the homology groups of the union of the 2 -sphere with the line segment from the North pole to the South pole.

A2.10

comment(i) Consider a network with node set $N$ and set of directed arcs $A$ equipped with functions $d^{+}: A \rightarrow \mathbb{Z}$ and $d^{-}: A \rightarrow \mathbb{Z}$ with $d^{-} \leqslant d^{+}$. Given $u: N \rightarrow \mathbb{R}$ we define the differential $\Delta u: A \rightarrow \mathbb{R}$ by $\Delta u(j)=u\left(i^{\prime}\right)-u(i)$ for $j=\left(i, i^{\prime}\right) \in A$. We say that $\Delta u$ is a feasible differential if

$d^{-}(j) \leqslant \Delta u(j) \leqslant d^{+}(j) \text { for all } j \in A$

Write down a necessary and sufficient condition on $d^{+}, d^{-}$for the existence of a feasible differential and prove its necessity.

Assuming Minty's Lemma, describe an algorithm to construct a feasible differential and outline how this algorithm establishes the sufficiency of the condition you have given.

(ii) Let $E \subseteq S \times T$, where $S, T$ are finite sets. A matching in $E$ is a subset $M \subseteq E$ such that, for all $s, s^{\prime} \in S$ and $t, t^{\prime} \in T$,

$\begin{array}{lll} (s, t),\left(s^{\prime}, t\right) \in M & \text { implies } & s=s^{\prime} \\ (s, t),\left(s, t^{\prime}\right) \in M & \text { implies } & t=t^{\prime} \end{array}$

A matching $M$ is maximal if for any other matching $M^{\prime}$ with $M \subseteq M^{\prime}$ we must have $M=M^{\prime}$. Formulate the problem of finding a maximal matching in $E$ in terms of an optimal distribution problem on a suitably defined network, and hence in terms of a standard linear optimization problem.

[You may assume that the optimal distribution subject to integer constraints is integervalued.]

A3.10

comment(i) Consider the problem

$\begin{aligned} \operatorname{minimize} & f(x) \\ \text { subject to } & h(x)=b, \quad x \in X, \end{aligned}$

where $f: \mathbb{R}^{n} \rightarrow \mathbb{R}, h: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}, X \subseteq \mathbb{R}^{n}$ and $b \in \mathbb{R}^{m}$. State and prove the Lagrangian sufficiency theorem.

In each of the following cases, where $n=2, m=1$ and $X=\{(x, y): x, y \geqslant 0\}$, determine whether the Lagrangian sufficiency theorem can be applied to solve the problem:

(ii) Consider the problem in $\mathbb{R}^{n}$

$\begin{aligned} \operatorname{minimize} & \frac{1}{2} x^{T} Q x+c^{T} x \\ \text { subject to } & A x=b \end{aligned}$

where $Q$ is a positive-definite symmetric $n \times n$ matrix, $A$ is an $m \times n$ matrix, $c \in \mathbb{R}^{n}$ and $b \in \mathbb{R}^{m}$. Explain how to reduce this problem to the solution of simultaneous linear equations.

Consider now the problem

$\begin{aligned} \operatorname{minimize} & \frac{1}{2} x^{T} Q x+c^{T} x \\ \text { subject to } & A x \geqslant b \end{aligned}$

Describe the active set method for its solution.

Consider the problem

$\begin{aligned} \operatorname{minimize} &(x-a)^{2}+(y-b)^{2}+x y \\ \text { subject to } & 0 \leqslant x \leqslant 1 \text { and } 0 \leqslant y \leqslant 1 \end{aligned}$

where $a, b \in \mathbb{R}$. Draw a diagram partitioning the $(a, b)$-plane into regions according to which constraints are active at the minimum.

$\begin{aligned} & \text { (a) } \quad f(x, y)=-x, \quad h(x, y)=x^{2}+y^{2}, \quad b=1 \text {; } \\ & \text { (b) } \quad f(x, y)=e^{-x y}, \quad h(x)=x, \quad b=0 \text {. } \end{aligned}$

A4.11

commentDefine the optimal distribution problem. State what it means for a circuit $P$ to be flow-augmenting, and what it means for $P$ to be unbalanced. State the optimality theorem for flows. Describe the simplex-on-a-graph algorithm, giving a brief justification of its stopping rules.

Consider the problem of finding, in the network shown below, a minimum-cost flow from $s$ to $t$ of value 2 . Here the circled numbers are the upper arc capacities, the lower arc capacities all being zero, and the uncircled numbers are costs. Apply the simplex-on-agraph algorithm to solve this problem, taking as initial flow the superposition of a unit flow along the path $s,(s, a), a,(a, t), t$ and a unit flow along the path $s,(s, a), a,(a, b), b,(b, t), t$.

Part II 2003

B1.23

commentDefine the differential cross section $\frac{d \sigma}{d \Omega}$. Show how it may be related to a scattering amplitude $f$ defined in terms of the behaviour of a wave function $\psi$ satisfying suitable boundary conditions as $r=|\mathbf{r}| \rightarrow \infty$.

For a particle scattering off a potential $V(r)$ show how $f(\theta)$, where $\theta$ is the scattering angle, may be expanded, for energy $E=\hbar^{2} k^{2} / 2 m$, as

$f(\theta)=\sum_{\ell=0}^{\infty} f_{\ell}(k) P_{\ell}(\cos \theta),$

and find $f_{\ell}(k)$ in terms of the phase shift $\delta_{\ell}(k)$. Obtain the optical theorem relating $\sigma_{\text {total }}$ and $f(0)$.

Suppose that

$e^{2 i \delta_{1}}=\frac{E-E_{0}-\frac{1}{2} i \Gamma}{E-E_{0}+\frac{1}{2} i \Gamma}$

Why for $E \approx E_{0}$ may $f_{1}(k)$ be dominant, and what is the expected behaviour of $\frac{d \sigma}{d \Omega}$ for $E \approx E_{0}$ ?

[For large $r$

$e^{i k r \cos \theta} \sim \frac{1}{2 i k r} \sum_{\ell=0}^{\infty}(2 \ell+1)\left((-1)^{\ell+1} e^{-i k r}+e^{i k r}\right) P_{\ell}(\cos \theta)$

Legendre polynomials satisfy

$\left.\int_{-1}^{1} P_{\ell}(t) P_{\ell^{\prime}}(t) d t=\frac{2}{2 \ell+1} \delta_{\ell \ell^{\prime}} \cdot\right]$

B2.22

commentThe Hamiltonian $H_{0}$ for a single electron atom has energy eigenstates $\left|\psi_{n}\right\rangle$ with energy eigenvalues $E_{n}$. There is an interaction with an electromagnetic wave of the form

$H_{1}=-e \mathbf{r} \cdot \boldsymbol{\epsilon} \cos (\mathbf{k} \cdot \mathbf{r}-\omega t), \quad \omega=|\mathbf{k}| c,$

where $\boldsymbol{\epsilon}$ is the polarisation vector. At $t=0$ the atom is in the state $\left|\psi_{0}\right\rangle$. Find a formula for the probability amplitude, to first order in $e$, to find the atom in the state $\left|\psi_{1}\right\rangle$ at time $t$. If the atom has a size $a$ and $|\mathbf{k}| a \ll 1$ what are the selection rules which are relevant? For $t$ large, under what circumstances will the transition rate be approximately constant?

[You may use the result

$\left.\int_{-\infty}^{\infty} \frac{\sin ^{2} \lambda t}{\lambda^{2}} d \lambda=\pi|t| . \quad\right]$

B3.23

commentConsider the two Hamiltonians

$\begin{aligned} &H_{1}=\frac{\mathbf{p}^{2}}{2 m}+V(|\mathbf{r}|), \\ &H_{2}=\frac{\mathbf{p}^{2}}{2 m}+\sum_{n_{i} \in \mathbb{Z}} V\left(\left|\mathbf{r}-n_{1} \mathbf{a}_{1}-n_{2} \mathbf{a}_{2}-n_{3} \mathbf{a}_{3}\right|\right), \end{aligned}$

where $\mathbf{a}_{i}$ are three linearly independent vectors. For each of the Hamiltonians $H=H_{1}$ and $H=H_{2}$, what are the symmetries of $H$ and what unitary operators $U$ are there such that $U H U^{-1}=H$ ?

For $\mathrm{H}_{2}$ derive Bloch's theorem. Suppose that $H_{1}$ has energy eigenfunction $\psi_{0}(\mathbf{r})$ with energy $E_{0}$ where $\psi_{0}(\mathbf{r}) \sim N e^{-K r}$ for large $r=|\mathbf{r}|$. Assume that $K\left|\mathbf{a}_{i}\right| \gg 1$ for each $i$. In a suitable approximation derive the energy eigenvalues for $H_{2}$ when $E \approx E_{0}$. Verify that the energy eigenfunctions and energy eigenvalues satisfy Bloch's theorem. What happens if $K\left|\mathbf{a}_{i}\right| \rightarrow \infty$ ?

B4.24

commentAtoms of mass $m$ in an infinite one-dimensional periodic array, with interatomic spacing $a$, have perturbed positions $x_{n}=n a+y_{n}$, for integer $n$. The potential between neighbouring atoms is

$\frac{1}{2} \lambda\left(x_{n+1}-x_{n}-a\right)^{2}$

for positive constant $\lambda$. Write down the Lagrangian for the variables $y_{n}$. Find the frequency $\omega(k)$ of a normal mode of wavenumber $k$. Define the Brillouin zone and explain why $k$ may be restricted to lie within it.

Assume now that the array is periodically-identified, so that there are effectively only $N$ atoms in the array and the atomic displacements $y_{n}$ satisfy the periodic boundary conditions $y_{n+N}=y_{n}$. Determine the allowed values of $k$ within the Brillouin zone. Show, for allowed wavenumbers $k$ and $k^{\prime}$, that

$\sum_{n=0}^{N-1} e^{i n\left(k-k^{\prime}\right) a}=N \delta_{k, k^{\prime}}$

By writing $y_{n}$ as

$y_{n}=\frac{1}{\sqrt{N}} \sum_{k} q_{k} e^{i n k a}$

where the sum is over allowed values of $k$, find the Lagrangian for the variables $q_{k}$, and hence the Hamiltonian $H$ as a function of $q_{k}$ and the conjugate momenta $p_{k}$. Show that the Hamiltonian operator $\hat{H}$ of the quantum theory can be written in the form

$\hat{H}=E_{0}+\sum_{k} \hbar \omega(k) a_{k}^{\dagger} a_{k}$

where $E_{0}$ is a constant and $a_{k}, a_{k}^{\dagger}$ are harmonic oscillator annihilation and creation operators. What is the physical interpretation of $a_{k}$ and $a_{k}^{\dagger}$ ? How does this show that phonons have quantized energies?

B2.13

commentLet $S_{k}$ be the sum of $k$ independent exponential random variables of rate $k \mu$. Compute the moment generating function of $S_{k}$.

Consider, for each fixed $k$ and for $0<\lambda<\mu$, an $M / G / 1$ queue with arrival rate $\lambda$ and with service times distributed as $S_{k}$. Assume that the queue is empty at time 0 and write $T_{k}$ for the earliest time at which a customer departs leaving the queue empty. Show that, as $k \rightarrow \infty, T_{k}$ converges in distribution to a random variable $T$ whose moment generating function $M_{T}(\theta)$ satisfies

$\log \left(1-\frac{\theta}{\lambda}\right)+\log M_{T}(\theta)=\left(\frac{\theta-\lambda}{\mu}\right)\left(1-M_{T}(\theta)\right)$

Hence obtain the mean value of $T$.

For what service-time distribution would the empty-to-empty time correspond exactly to $T$ ?

B3.13

commentState the product theorem for Poisson random measures.

Consider a system of $n$ queues, each with infinitely many servers, in which, for $i=1, \ldots, n-1$, customers leaving the $i$ th queue immediately arrive at the $(i+1)$ th queue. Arrivals to the first queue form a Poisson process of rate $\lambda$. Service times at the $i$ th queue are all independent with distribution $F$, and independent of service times at other queues, for all $i$. Assume that initially the system is empty and write $V_{i}(t)$ for the number of customers at queue $i$ at time $t \geqslant 0$. Show that $V_{1}(t), \ldots, V_{n}(t)$ are independent Poisson random variables.

In the case $F(t)=1-e^{-\mu t}$ show that

$\mathbb{E}\left(V_{i}(t)\right)=\frac{\lambda}{\mu} \mathbb{P}\left(N_{t} \geqslant i\right), \quad t \geqslant 0, \quad i=1, \ldots, n,$

where $\left(N_{t}\right)_{t \geqslant 0}$ is a Poisson process of rate $\mu$.

Suppose now that arrivals to the first queue stop at time $T$. Determine the mean number of customers at the $i$ th queue at each time $t \geqslant T$.

B4.12

commentExplain what is meant by a renewal process and by a renewal-reward process.

State and prove the law of large numbers for renewal-reward processes.

A component used in a manufacturing process has a maximum lifetime of 2 years and is equally likely to fail at any time during that period. If the component fails whilst in use, it is replaced immediately by a similar component, at a cost of $£ 1000$. The factory owner may alternatively replace the component before failure, at a time of his choosing, at a cost of $£ 200$. What should the factory owner do?

A1.10

comment(i) We work over the field of two elements. Define what is meant by a linear code of length $n$. What is meant by a generator matrix for a linear code?

Define what is meant by a parity check code of length $n$. Show that a code is linear if and only if it is a parity check code.

Give the original Hamming code in terms of parity checks and then find a generator matrix for it.

[You may use results from the theory of vector spaces provided that you quote them correctly.]

(ii) Suppose that $1 / 4>\delta>0$ and let $\alpha(n, n \delta)$ be the largest information rate of any binary error correcting code of length $n$ which can correct $[n \delta]$ errors.

Show that

$1-H(2 \delta) \leqslant \liminf _{n \rightarrow \infty} \alpha(n, n \delta) \leqslant 1-H(\delta)$

where

$H(\eta)=-\eta \log _{2} \eta-(1-\eta) \log _{2}(1-\eta)$

[You may assume any form of Stirling's theorem provided that you quote it correctly.]

A2.9

comment(i) Answer the following questions briefly but clearly.

(a) How does coding theory apply when the error rate $p>1 / 2$ ?

(b) Give an example of a code which is not a linear code.

(c) Give an example of a linear code which is not a cyclic code.

(d) Give an example of a general feedback register with output $k_{j}$, and initial fill $\left(k_{0}, k_{1}, \ldots, k_{N}\right)$, such that

$\left(k_{n}, k_{n+1}, \ldots, k_{n+N}\right) \neq\left(k_{0}, k_{1}, \ldots, k_{N}\right)$

for all $n \geqslant 1$.

(e) Explain why the original Hamming code can not always correct two errors.

(ii) Describe the Rabin-Williams scheme for coding a message $x$ as $x^{2}$ modulo a certain $N$. Show that, if $N$ is chosen appropriately, breaking this code is equivalent to factorising the product of two primes.

B1.5

commentLet $G$ be a graph of order $n \geqslant 4$. Prove that if $G$ has $t_{2}(n)+1$ edges then it contains two triangles with a common edge. Here, $t_{2}(n)=\left\lfloor n^{2} / 4\right\rfloor$ is the Turán number.

Suppose instead that $G$ has exactly one triangle. Show that $G$ has at most $t_{2}(n-1)+2$ edges, and that this number can be attained.

B2.5

commentProve Ramsey's theorem in its usual infinite form, namely, that if $\mathbb{N}^{(r)}$ is finitely coloured then there is an infinite subset $M \subset \mathbb{N}$ such that $M^{(r)}$ is monochromatic.

Now let the graph $\mathbb{N}^{(2)}$ be coloured with an infinite number of colours in such a way that there is no infinite $M \subset \mathbb{N}$ with $M^{(2)}$ monochromatic. By considering a suitable 2-colouring of the set $\mathbb{N}^{(4)}$ of 4 -sets, show that there is an infinite $M \subset \mathbb{N}$ with the property that any two edges of $M^{(2)}$ of the form $a d, b c$ with $a<b<c<d$ have different colours.

By considering two further 2-colourings of $\mathbb{N}^{(4)}$, show that there is an infinite $M \subset \mathbb{N}$ such that any two non-incident edges of $M^{(2)}$ have different colours.

B4.1

commentWrite an essay on the Kruskal-Katona theorem. As well as stating the theorem and giving a detailed sketch of a proof, you should describe some further results that may be derived from it.

A1.13

comment(i) Suppose $Y_{i}, 1 \leqslant i \leqslant n$, are independent binomial observations, with $Y_{i} \sim B i\left(t_{i}, \pi_{i}\right)$, $1 \leqslant i \leqslant n$, where $t_{1}, \ldots, t_{n}$ are known, and we wish to fit the model

$\omega: \log \frac{\pi_{i}}{1-\pi_{i}}=\mu+\beta^{T} x_{i} \quad \text { for each } i$

where $x_{1}, \ldots, x_{n}$ are given covariates, each of dimension $p$. Let $\hat{\mu}, \hat{\beta}$ be the maximum likelihood estimators of $\mu, \beta$. Derive equations for $\hat{\mu}, \hat{\beta}$ and state without proof the form of the approximate distribution of $\hat{\beta}$.

(ii) In 1975 , data were collected on the 3-year survival status of patients suffering from a type of cancer, yielding the following table

\begin{tabular}{ccrr} & & \multicolumn{2}{c}{ survive? } \ age in years & malignant & yes & no \ under 50 & no & 77 & 10 \ under 50 & yes & 51 & 13 \ $50-69$ & no & 51 & 11 \ $50-69$ & yes & 38 & 20 \ $70+$ & no & 7 & 3 \ $70+$ & yes & 6 & 3 \end{tabular}

Here the second column represents whether the initial tumour was not malignant or was malignant.

Let $Y_{i j}$ be the number surviving, for age group $i$ and malignancy status $j$, for $i=1,2,3$ and $j=1,2$, and let $t_{i j}$ be the corresponding total number. Thus $Y_{11}=77$, $t_{11}=87$. Assume $Y_{i j} \sim B i\left(t_{i j}, \pi_{i j}\right), 1 \leqslant i \leqslant 3,1 \leqslant j \leqslant 2$. The results from fitting the model

$\log \left(\pi_{i j} /\left(1-\pi_{i j}\right)\right)=\mu+\alpha_{i}+\beta_{j}$

with $\alpha_{1}=0, \beta_{1}=0$ give $\hat{\beta}_{2}=-0.7328(\mathrm{se}=0.2985)$, and deviance $=0.4941$. What do you conclude?

Why do we take $\alpha_{1}=0, \beta_{1}=0$ in the model?

What "residuals" should you compute, and to which distribution would you refer them?

A2.12

comment(i) Suppose $Y_{1}, \ldots, Y_{n}$ are independent Poisson variables, and

$\mathbb{E}\left(Y_{i}\right)=\mu_{i}, \quad \log \mu_{i}=\alpha+\beta t_{i}, \quad \text { for } \quad i=1, \ldots, n,$

where $\alpha, \beta$ are two unknown parameters, and $t_{1}, \ldots, t_{n}$ are given covariates, each of dimension 1. Find equations for $\hat{\alpha}, \hat{\beta}$, the maximum likelihood estimators of $\alpha, \beta$, and show how an estimate of $\operatorname{var}(\hat{\beta})$ may be derived, quoting any standard theorems you may need.

(ii) By 31 December 2001, the number of new vCJD patients, classified by reported calendar year of onset, were

$8,10,11,14,17,29,23$

for the years

$1994, \ldots, 2000 \text { respectively }$

Discuss carefully the (slightly edited) $R$ output for these data given below, quoting any standard theorems you may need.

year

year

[1] 1994199519961997199819992000

$>$ tot

[1] $\begin{array}{lllllll}8 & 10 & 11 & 14 & 17 & 29 & 23\end{array}$

first.glm - glm(tot year, family = poisson)

$>\operatorname{summary}$ (first.glm)

Call:

glm(formula $=$ tot year, family $=$ poisson $)$

Coefficients

Estimate Std. Error z value $\operatorname{Pr}(>|z|)$

(Intercept) $-407.8128599 .35366-4.1054 .05 \mathrm{e}-05$

year $\quad 0.20556 \quad 0.04973 \quad 4.1333 .57 e-05$

(Dispersion parameter for poisson family taken to be 1)

Null deviance: $20.7753$ on 6 degrees of freedom

Residual deviance: $2.7931$ on 5 degrees of freedom

Number of Fisher Scoring iterations: 3

Part II 2003

A4.14

commentThe nave height $x$, and the nave length $y$ for 16 Gothic-style cathedrals and 9 Romanesque-style cathedrals, all in England, have been recorded, and the corresponding $R$ output (slightly edited) is given below.

You may assume that $x, y$ are in suitable units, and that "style" has been set up as a factor with levels 1,2 corresponding to Gothic, Romanesque respectively.

(a) Explain carefully, with suitable graph(s) if necessary, the results of this analysis.

(b) Using the general model $Y=X \beta+\epsilon$ (in the conventional notation) explain carefully the theory needed for (a).

[Standard theorems need not be proved.]

B1.8

commentState the Implicit Function Theorem and outline how it produces submanifolds of Euclidean spaces.

Show that the unitary group $U(n) \subset G L(n, \mathbb{C})$ is a smooth manifold and find its dimension.

Identify the tangent space to $U(n)$ at the identity matrix as a subspace of the space of $n \times n$ complex matrices.

B2.7

commentLet $M$ and $N$ be smooth manifolds. If $\pi: M \times N \rightarrow M$ is the projection onto the first factor and $\pi^{*}$ is the map in cohomology induced by the pull-back map on differential forms, show that $\pi^{*}\left(H^{k}(M)\right)$ is a direct summand of $H^{k}(M \times N)$ for each $k \geqslant 0$.

Taking $H^{k}(M)$ to be zero for $k<0$ and $k>\operatorname{dim} M$, show that for $n \geqslant 1$ and all $k$

$H^{k}\left(M \times S^{n}\right) \cong H^{k}(M) \oplus H^{k-n}(M)$

[You might like to use induction in n.]

B4.4

commentDefine the 'pull-back' homomorphism of differential forms determined by the smooth map $f: M \rightarrow N$ and state its main properties.

If $\theta: W \rightarrow V$ is a diffeomorphism between open subsets of $\mathbb{R}^{m}$ with coordinates $x_{i}$ on $V$ and $y_{j}$ on $W$ and the $m$-form $\omega$ is equal to $f d x_{1} \wedge \ldots \wedge d x_{m}$ on $V$, state and prove the expression for $\theta^{*}(\omega)$ as a multiple of $d y_{1} \wedge \ldots \wedge d y_{m}$.

Define the integral of an $m$-form $\omega$ over an oriented $m$-manifold $M$ and prove that it is well-defined.

Show that the inclusion map $f: N \hookrightarrow M$, of an oriented $n$-submanifold $N$ (without boundary) into $M$, determines an element $\nu$ of $H_{n}(M) \cong \operatorname{Hom}\left(H^{n}(M), \mathbb{R}\right)$. If $M=N \times P$ and $f(x)=(x, p)$, for $x \in N$ and $p$ fixed in $P$, what is the relation between $\nu$ and $\pi^{*}\left(\left[\omega_{N}\right]\right)$, where $\left[\omega_{N}\right]$ is the fundamental cohomology class of $N$ and $\pi$ is the projection onto the first factor?

B1.17

commentConsider the one-dimensional map $f: \mathbb{R} \rightarrow \mathbb{R}$, where $f(x)=\mu x^{2}(1-x)$ with $\mu$ a real parameter. Find the range of values of $\mu$ for which the open interval $(0,1)$ is mapped into itself and contains at least one fixed point. Describe the bifurcation at $\mu=4$ and find the parameter value for which there is a period-doubling bifurcation. Determine whether the fixed point is an attractor at this bifurcation point.

B3.17

commentLet $f: I \rightarrow I$ be a continuous one-dimensional map of the interval $I \subset \mathbb{R}$. Explain what is meant by saying (a) that the map $f$ is topologically transitive, and (b) that the map $f$ has a horseshoe.

Consider the tent map defined on the interval $[0,1]$ by

$f(x)= \begin{cases}\mu x & 0 \leqslant x<\frac{1}{2} \\ \mu(1-x) & \frac{1}{2} \leqslant x \leqslant 1\end{cases}$

for $1<\mu \leqslant 2$. Show that if $\mu>\sqrt{2}$ then this map is topologically transitive, and also that $f^{2}$ has a horseshoe.

B4.17

commentLet $f: S^{1} \rightarrow S^{1}$ be an orientation-preserving invertible map of the circle onto itself, with a lift $F: \mathbb{R} \rightarrow \mathbb{R}$. Define the rotation numbers $\rho_{0}(F)$ and $\rho(f)$.

Suppose that $\rho_{0}(F)=p / q$, where $p$ and $q$ are coprime integers. Prove that the map $f$ has periodic points of least period $q$, and no periodic points with any least period not equal to $q$.

Now suppose that $\rho_{0}(F)$ is irrational. Explain the distinction between wandering and non-wandering points under $f$. Let $\Omega(x)$ be the set of limit points of the sequence $\left\{x, f(x), f^{2}(x), \ldots\right\}$. Prove

(a) that the set $\Omega(x)=\Omega$ is independent of $x$ and is the smallest closed, non-empty, $f$-invariant subset of $S^{1}$;

(b) that $\Omega$ is the set of non-wandering points of $S^{1}$;

(c) that $\Omega$ is either the whole of $S^{1}$ or a Cantor set in $S^{1}$.

A1.6

comment(i) State and prove Dulac's Criterion for the non-existence of periodic orbits in $\mathbb{R}^{2}$. Hence show (choosing a weighting factor of the form $x^{\alpha} y^{\beta}$ ) that there are no periodic orbits of the equations

$\dot{x}=x\left(2-6 x^{2}-5 y^{2}\right), \quad \dot{y}=y\left(-3+10 x^{2}+3 y^{2}\right)$

(ii) State the Poincaré-Bendixson Theorem. A model of a chemical reaction (the Brusselator) is defined by the second order system

$\dot{x}=a-x(1+b)+x^{2} y, \quad \dot{y}=b x-x^{2} y$

where $a, b$ are positive parameters. Show that there is a unique fixed point. Show that, for a suitable choice of $p>0$, trajectories enter the closed region bounded by $x=p, y=b / p$, $x+y=a+b / p$ and $y=0$. Deduce that when $b>1+a^{2}$, the system has a periodic orbit.

A2.6 B2.4

comment(i) What is a Liapunov function?

Consider the second order ODE

$\dot{x}=y, \quad \dot{y}=-y-\sin ^{3} x$

By finding a suitable Liapunov function of the form $V(x, y)=f(x)+g(y)$, where $f$ and $g$ are to be determined, show that the origin is asymptotically stable. Using your form of $V$, find the greatest value of $y_{0}$ such that a trajectory through $\left(0, y_{0}\right)$ is guaranteed to tend to the origin as $t \rightarrow \infty$.

[Any theorems you use need not be proved but should be clearly stated.]

(ii) Explain the use of the stroboscopic method for investigating the dynamics of equations of the form $\ddot{x}+x=\epsilon f(x, \dot{x}, t)$, when $|\epsilon| \ll 1$. In particular, for $x=R \cos (t+\theta)$, $\dot{x}=-R \sin (t+\theta)$ derive the equations, correct to order $\epsilon$,

$\dot{R}=-\epsilon\langle f \sin (t+\theta)\rangle, \quad R \dot{\theta}=-\epsilon\langle f \cos (t+\theta)\rangle$

where the brackets denote an average over the period of the unperturbed oscillator.

Find the form of the right hand sides of these equations explicitly when $f=$ $\Gamma x^{2} \cos t-3 q x$, where $\Gamma>0, q \neq 0$. Show that apart from the origin there is another fixed point of $(*)$, and determine its stability. Sketch the trajectories in $(R, \theta)$ space in the case $q>0$. What do you deduce about the dynamics of the full equation?

[You may assume that $\left\langle\cos ^{2} t\right\rangle=\frac{1}{2},\left\langle\cos ^{4} t\right\rangle=\frac{3}{8},\left\langle\cos ^{2} t \sin ^{2} t\right\rangle=\frac{1}{8}$.]

A3.6 B3.4

comment(i) Define the Poincaré index of a curve $\mathcal{C}$ for a vector field $\mathbf{f}(\mathbf{x}), \mathbf{x} \in \mathbb{R}^{2}$. Explain why the index is uniquely given by the sum of the indices for small curves around each fixed point within $\mathcal{C}$. Write down the indices for a saddle point and for a focus (spiral) or node, and show that the index of a periodic solution of $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ has index unity.

A particular system has a periodic orbit containing five fixed points, and two further periodic orbits. Sketch the possible arrangements of these orbits, assuming there are no degeneracies.

(ii) A dynamical system in $\mathbb{R}^{2}$ depending on a parameter $\mu$ has a homoclinic orbit when $\mu=\mu_{0}$. Explain how to determine the stability of this orbit, and sketch the different behaviours for $\mu<\mu_{0}$ and $\mu>\mu_{0}$ in the case that the orbit is stable.

Now consider the system

$\dot{x}=y, \quad \dot{y}=x-x^{2}+y(\alpha+\beta x)$

where $\alpha, \beta$ are constants. Show that the origin is a saddle point, and that if there is an orbit homoclinic to the origin then $\alpha, \beta$ are related by

$\oint y^{2}(\alpha+\beta x) d t=0$

where the integral is taken round the orbit. Evaluate this integral for small $\alpha, \beta$ by approximating $y$ by its form when $\alpha=\beta=0$. Hence give conditions on (small) $\alpha, \beta$ that lead to a stable homoclinic orbit at the origin. [Note that $y d t=d x$.]

A4.6

commentExplain what is meant by a steady-state bifurcation of a fixed point $\mathbf{x}_{0}(\mu)$ of an ODE $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x}, \mu)$, in $\mathbb{R}^{n}$, where $\mu$ is a real parameter. Give examples for $n=1$ of equations exhibiting saddle-node, transcritical and pitchfork bifurcations.

Consider the system in $\mathbb{R}^{2}$, with $\mu>0$,

$\dot{x}=x\left(1-y-4 x^{2}\right), \quad \dot{y}=y\left(\mu-y-x^{2}\right) .$

Show that the fixed point $(0, \mu)$ has a bifurcation when $\mu=1$, while the fixed points $\left(\pm \frac{1}{2}, 0\right)$ have a bifurcation when $\mu=\frac{1}{4}$. By finding the first approximation to the extended centre manifold, construct the normal form at the bifurcation point in each case, and determine the respective bifurcation types. Deduce that for $\mu$ just greater than $\frac{1}{4}$, and for $\mu$ just less than 1 , there is a stable pair of "mixed-mode" solutions with $x^{2}>0, y>0$.

B1.21

commentA particle of charge $q$ and mass $m$ moves non-relativistically with 4 -velocity $u^{a}(t)$ along a trajectory $x^{a}(t)$. Its electromagnetic field is determined by the Liénard-Wiechert potential

$A^{a}\left(\mathbf{x}^{\prime}, t^{\prime}\right)=\frac{q}{4 \pi \epsilon_{0}} \frac{u^{a}(t)}{u_{b}(t)\left(x^{\prime}-x(t)\right)^{b}}$

where $t^{\prime}=t+\left|\mathbf{x}-\mathbf{x}^{\prime}\right|$ and $\mathbf{x}$ denotes the spatial part of the 4 -vector $x^{a}$.

Derive a formula for the Poynting vector at very large distances from the particle. Hence deduce Larmor's formula for the rate of loss of energy due to electromagnetic radiation by the particle.

A particle moves in the $(x, y)$ plane in a constant magnetic field $\mathbf{B}=(0,0, B)$. Initially it has kinetic energy $E_{0}$; derive a formula for the kinetic energy of this particle as a function of time.

B2.20

commentA plane electromagnetic wave of frequency $\omega$ and wavevector $\mathbf{k}$ has an electromagnetic potential given by

$A^{a}=A \epsilon^{a} e^{i(\mathbf{k} \cdot \mathbf{x}-\omega t)}$

where $A$ is the amplitude of the wave and $\epsilon^{a}$ is the polarization vector. Explain carefully why there are two independent polarization states for such a wave, and why $|\mathbf{k}|^{2}=\omega^{2}$.

A wave travels in the positive $z$-direction with polarization vector $\epsilon^{a}=(0,1, i, 0)$. It is incident at $z=0$ on a plane surface which conducts perfectly in the $x$-direction, but not at all in the $y$-direction. Find an expression for the electromagnetic potential of the radiation that is reflected from this surface.

B4.21

commentDescribe the physical meaning of the various components of the stress-energy tensor $T^{a b}$ of the electromagnetic field.

Suppose that one is given an electric field $\mathbf{E}(\mathbf{x})$ and a magnetic field $\mathbf{B}(\mathbf{x})$. Show that the angular momentum about the origin of these fields is

$\mathbf{J}=\frac{1}{\mu_{0}} \int \mathbf{x} \times(\mathbf{E} \times \mathbf{B}) d^{3} \mathbf{x}$

where the integral is taken over all space.

A point electric charge $Q$ is at the origin, and has electric field

$\mathbf{E}=\frac{Q}{4 \pi \epsilon_{0}} \frac{\mathbf{x}}{|\mathbf{x}|^{3}}$

A point magnetic monopole of strength $P$ is at $\mathbf{y}$ and has magnetic field

$\mathbf{B}=\frac{\mu_{0} P}{4 \pi} \frac{\mathbf{x}-\mathbf{y}}{|\mathbf{x}-\mathbf{y}|^{3}}$

Find the component, along the axis between the electric charge and the magnetic monopole, of the angular momentum of the electromagnetic field about the origin.

[Hint: You may find it helpful to express both $\mathbf{E}$ and $\mathbf{B}$ as gradients of scalar potentials.]

A $1 . 5 \quad$ B $1 . 4 \quad$

comment(i) Using Maxwell's equations as they apply to magnetostatics, show that the magnetic field $\mathbf{B}$ can be described in terms of a vector potential $\mathbf{A}$ on which the condition $\nabla \cdot \mathbf{A}=0$ may be imposed. Hence derive an expression, valid at any point in space, for the vector potential due to a steady current distribution of density $\mathbf{j}$ that is non-zero only within a finite domain.

(ii) Verify that the vector potential $\mathbf{A}$ that you found in Part (i) satisfies $\nabla \cdot \mathbf{A}=0$, and use it to obtain the Biot-Savart law expression for $\mathbf{B}$. What is the corresponding result for a steady surface current distribution of density $\mathbf{s}$ ?

In cylindrical polar coordinates $(\rho, \phi, z)$ (oriented so that $\mathbf{e}_{\rho} \times \mathbf{e}_{\phi}=\mathbf{e}_{z}$ ) a surface current

$\mathbf{s}=s(\rho) \mathbf{e}_{\phi}$

flows in the plane $z=0$. Given that

$s(\rho)= \begin{cases}4 I\left(1+\frac{a^{2}}{\rho^{2}}\right)^{\frac{1}{2}} & a \leqslant \rho \leqslant 3 a \\ 0 & \text { otherwise }\end{cases}$

show that the magnetic field at the point $\mathbf{r}=a \mathbf{e}_{z}$ has $z$-component

$B_{z}=\mu_{0} I \log 5 .$

State, with justification, the full result for $\mathbf{B}$ at the point $\mathbf{r}=a \mathbf{e}_{z}$.

A2 $. 5 \quad$

comment(i) A plane electromagnetic wave has electric and magnetic fields

$\mathbf{E}=\mathbf{E}_{0} e^{i(\mathbf{k} \cdot \mathbf{r}-\omega t)}, \quad \mathbf{B}=\mathbf{B}_{0} e^{i(\mathbf{k} \cdot \mathbf{r}-\omega t)}$

for constant vectors $\mathbf{E}_{0}, \mathbf{B}_{0}$, constant positive angular frequency $\omega$ and constant wavevector $\mathbf{k}$. Write down the vacuum Maxwell equations and show that they imply

$\mathbf{k} \cdot \mathbf{E}_{0}=0, \quad \mathbf{k} \cdot \mathbf{B}_{0}=0, \quad \omega \mathbf{B}_{0}=\mathbf{k} \times \mathbf{E}_{0}$

Show also that $|\mathbf{k}|=\omega / c$, where $c$ is the speed of light.

(ii) State the boundary conditions on $\mathbf{E}$ and $\mathbf{B}$ at the surface $S$ of a perfect conductor. Let $\sigma$ be the surface charge density and s the surface current density on $S$. How are $\sigma$ and $\mathbf{s}$ related to $\mathbf{E}$ and $\mathbf{B}$ ?

A plane electromagnetic wave is incident from the half-space $x<0$ upon the surface $x=0$ of a perfectly conducting medium in $x>0$. Given that the electric and magnetic fields of the incident wave take the form $(*)$ with

$\mathbf{k}=k(\cos \theta, \sin \theta, 0) \quad(0<\theta<\pi / 2)$

and

$\mathbf{E}_{0}=\lambda(-\sin \theta, \cos \theta, 0),$

find $\mathbf{B}_{0}$.

Reflection of the incident wave at $x=0$ produces a reflected wave with electric field

$\mathbf{E}_{0}^{\prime} e^{i\left(\mathbf{k}^{\prime} \cdot \mathbf{r}-\omega t\right)}$

with

$\mathbf{k}^{\prime}=k(-\cos \theta, \sin \theta, 0)$

By considering the boundary conditions at $x=0$ on the total electric field, show that

$\mathbf{E}_{0}^{\prime}=-\lambda(\sin \theta, \cos \theta, 0)$

Show further that the electric charge density on the surface $x=0$ takes the form

$\sigma=\sigma_{0} e^{i k(y \sin \theta-c t)}$

for a constant $\sigma_{0}$ that you should determine. Find the magnetic field of the reflected wave and hence the surface current density $\mathbf{s}$ on the surface $x=0$.

A3.5 B3.3

comment(i) Given the electric field (in cartesian components)

$\mathbf{E}(\mathbf{r}, t)=\left(0, x / t^{2}, 0\right)$

use the Maxwell equation

$\nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}$

to find $\mathbf{B}$ subject to the boundary condition that $|\mathbf{B}| \rightarrow 0$ as $t \rightarrow \infty$.

Let $S$ be the planar rectangular surface in the $x y$-plane with corners at

$(0,0,0), \quad(L, 0,0), \quad(L, a, 0), \quad(0, a, 0)$

where $a$ is a constant and $L=L(t)$ is some function of time. The magnetic flux through $S$ is given by the surface integral

$\Phi=\int_{S} \mathbf{B} \cdot d \mathbf{S}$

Compute $\Phi$ as a function of $t$.

Let $\mathcal{C}$ be the closed rectangular curve that bounds the surface $S$, taken anticlockwise in the $x y$-plane, and let $\mathbf{v}$ be its velocity (which depends, in this case, on the segment of $\mathcal{C}$ being considered). Compute the line integral

$\oint_{\mathcal{C}}(\mathbf{E}+\mathbf{v} \times \mathbf{B}) \cdot d \mathbf{r}$

Hence verify that

$\oint_{\mathcal{C}}(\mathbf{E}+\mathbf{v} \times \mathbf{B}) \cdot d \mathbf{r}=-\frac{d \Phi}{d t}$

(ii) A surface $S$ is bounded by a time-dependent closed curve $\mathcal{C}(t)$ such that in time $\delta t$ it sweeps out a volume $\delta V$. By considering the volume integral

$\int_{\delta V} \nabla \cdot \mathbf{B} d \tau$

and using the divergence theorem, show that the Maxwell equation $\nabla \cdot \mathbf{B}=0$ implies that

$\frac{d \Phi}{d t}=\int_{S} \frac{\partial \mathbf{B}}{\partial t} \cdot d \mathbf{S}-\oint_{\mathcal{C}}(\mathbf{v} \times \mathbf{B}) \cdot d \mathbf{r}$

where $\Phi$ is the magnetic flux through $S$ as given in Part (i). Hence show, using (1) and Stokes' theorem, that (2) is a consequence of Maxwell's equations.

A4.5

commentLet $\mathbf{E}(\mathbf{r})$ be the electric field due to a continuous static charge distribution $\rho(\mathbf{r})$ for which $|\mathbf{E}| \rightarrow 0$ as $|\mathbf{r}| \rightarrow \infty$. Starting from consideration of a finite system of point charges, deduce that the electrostatic energy of the charge distribution $\rho$ is

$W=\frac{1}{2} \varepsilon_{0} \int|\mathbf{E}|^{2} d \tau$

where the volume integral is taken over all space.

A sheet of perfectly conducting material in the form of a surface $S$, with unit normal $\mathbf{n}$, carries a surface charge density $\sigma$. Let $E_{\pm}=\mathbf{n} \cdot \mathbf{E}_{\pm}$denote the normal components of the electric field $\mathbf{E}$ on either side of $S$. Show that

$\frac{1}{\varepsilon_{0}} \sigma=E_{+}-E_{-} .$

Three concentric spherical shells of perfectly conducting material have radii $a, b, c$ with $a<b<c$. The innermost and outermost shells are held at zero electric potential. The other shell is held at potential $V$. Find the potentials $\phi_{1}(r)$ in $a<r<b$ and $\phi_{2}(r)$ in $b<r<c$. Compute the surface charge density $\sigma$ on the shell of radius $b$. Use the formula $(*)$ to compute the electrostatic energy of the system.

B1.25

commentConsider a two-dimensional horizontal vortex sheet of strength $U$ at height $h$ above a horizontal rigid boundary at $y=0$, so that the inviscid fluid velocity is

$\mathbf{u}= \begin{cases}(U, 0) & 0<y<h \\ (0,0) & y>h\end{cases}$

Examine the temporal linear instabililty of the sheet and determine the relevant dispersion relationship.

For what wavelengths is the sheet unstable?

Evaluate the temporal growth rate and the wave propagation speed in the limit of both short and long waves. Comment briefly on the significance of your results.

B2 24

commentA plate is drawn vertically out of a bath and the resultant liquid drains off the plate as a thin film. Using lubrication theory, show that the governing equation for the thickness of the film, $h(x, t)$ is

$\frac{\partial h}{\partial t}+\left(\frac{g h^{2}}{\nu}\right) \frac{\partial h}{\partial x}=0$

where $t$ is time and $x$ is the distance down the plate measured from the top.

Verify that

$h(x, t)=F\left(x-\frac{g h^{2}}{\nu} t\right)$

satisfies $(*)$ and identify the function $F(x)$. Using this relationship or otherwise, determine an explicit expression for the thickness of the film assuming that it was initially of uniform thickness $h_{0}$.

B3.24

commentA steady two-dimensional jet is generated in an infinite, incompressible fluid of density $\rho$ and kinematic viscosity $\nu$ by a point source of momentum with momentum flux in the $x$ direction $F$ per unit length located at the origin.

Using boundary layer theory, analyse the motion in the jet and show that the $x$-component of the velocity is given by

$u=U(x) f^{\prime}(\eta)$

where

$\eta=y / \delta(x), \quad \delta(x)=\left(\rho \nu^{2} x^{2} / F\right)^{1 / 3} \text { and } U(x)=\left(F^{2} / \rho^{2} \nu x\right)^{1 / 3}$

Show that $f$ satisfies the differential equation

$f^{\prime \prime \prime}+\frac{1}{3}\left(f f^{\prime \prime}+f^{\prime^{2}}\right)=0 .$

Write down the appropriate boundary conditions for this equation. [You need not solve the equation.]

B4.26

commentShow that the complex potential in the complex $\zeta$ plane,

$w=(U-i V) \zeta+(U+i V) \frac{c^{2}}{\zeta}-\frac{i \kappa}{2 \pi} \log \zeta$

describes irrotational, inviscid flow past the rigid cylinder $|\zeta|=c$, placed in a uniform flow $(U, V)$ with circulation $\kappa$.

Show that the transformation

$z=\zeta+\frac{c^{2}}{\zeta}$

maps the circle $|\zeta|=c$ in the $\zeta$ plane onto the flat plate airfoil $-2 c<x<2 c, y=0$ in the $z$ plane $(z=x+i y)$. Hence, write down an expression for the complex potential, $w_{p}$, of uniform flow past the flat plate, with circulation $\kappa$. Indicate very briefly how the value of $\kappa$ might be chosen to yield a physical solution.

Calculate the turning moment, $M$, exerted on the flat plate by the flow.

(You are given that

$M=-\frac{1}{2} \rho \operatorname{Re}\left\{\oint\left[\frac{\left(\frac{d w}{d \zeta}\right)^{2}}{\frac{d z}{d \zeta}}\right] z(\zeta) d \zeta\right\} \text {, }$

where $\rho$ is the fluid density and the integral is to be completed around a contour enclosing the circle $|\zeta|=c$ ).

A2.13 B2.21

comment(i) Define the Heisenberg picture of quantum mechanics in relation to the Schrödinger picture. Explain how the two pictures provide equivalent descriptions of observable results.

Derive the equation of motion for an operator in the Heisenberg picture.

(ii) For a particle moving in one dimension, the Hamiltonian is

$\hat{H}=\frac{\hat{p}^{2}}{2 m}+V(\hat{x}),$

where $\hat{x}$ and $\hat{p}$ are the position and momentum operators, and the state vector is $|\Psi\rangle$.

Eigenstates of $\hat{x}$ and $\hat{p}$ satisfy

$\langle x \mid p\rangle=\left(\frac{1}{2 \pi \hbar}\right)^{1 / 2} e^{i p x / \hbar}, \quad\left\langle x \mid x^{\prime}\right\rangle=\delta\left(x-x^{\prime}\right), \quad\left\langle p \mid p^{\prime}\right\rangle=\delta\left(p-p^{\prime}\right) .$

Use standard methods in the Dirac formalism to show that

$\begin{aligned} &\left\langle x|\hat{p}| x^{\prime}\right\rangle=-i \hbar \frac{\partial}{\partial x} \delta\left(x-x^{\prime}\right) \\ &\left\langle p|\hat{x}| p^{\prime}\right\rangle=i \hbar \frac{\partial}{\partial p} \delta\left(p-p^{\prime}\right) \end{aligned}$

Calculate $\left\langle x|\hat{H}| x^{\prime}\right\rangle$ and express $\langle x|\hat{p}| \Psi\rangle,\langle x|\hat{H}| \Psi\rangle$ in terms of the position space wave function $\Psi(x)$.

Compute the momentum space Hamiltonian for the harmonic oscillator with potential $V(\hat{x})=\frac{1}{2} m \omega^{2} \hat{x}^{2}$.

A3.13 B3.21

comment(i) What are the commutation relations satisfied by the components of an angular momentum vector $\mathbf{J}$ ? State the possible eigenvalues of the component $J_{3}$ when $\mathbf{J}^{2}$ has eigenvalue $j(j+1) \hbar^{2}$.

Describe how the Pauli matrices

$\sigma_{1}=\left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right), \quad \sigma_{2}=\left(\begin{array}{cc} 0 & -i \\ i & 0 \end{array}\right), \quad \sigma_{3}=\left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right)$

are used to construct the components of the angular momentum vector $\mathbf{S}$ for a spin $\frac{1}{2}$ system. Show that they obey the required commutation relations.

Show that $S_{1}, S_{2}$ and $S_{3}$ each have eigenvalues $\pm \frac{1}{2} \hbar$. Verify that $\mathbf{S}^{2}$ has eigenvalue $\frac{3}{4} \hbar^{2} .$

(ii) Let $\mathbf{J}$ and $|j m\rangle$ denote the standard operators and state vectors of angular momentum theory. Assume units where $\hbar=1$. Consider the operator

$U(\theta)=e^{-i \theta J_{2}}$

Show that

$\begin{aligned} &U(\theta) J_{1} U(\theta)^{-1}=\cos \theta J_{1}-\sin \theta J_{3} \\ &U(\theta) J_{3} U(\theta)^{-1}=\sin \theta J_{1}+\cos \theta J_{3} \end{aligned}$

Show that the state vectors $U\left(\frac{\pi}{2}\right)|j m\rangle$ are eigenvectors of $J_{1}$. Suppose that $J_{1}$ is measured for a system in the state $|j m\rangle$; show that the probability that the result is $m^{\prime}$ equals

$\left|\left\langle j m^{\prime}\left|e^{i \frac{\pi}{2} J_{2}}\right| j m\right\rangle\right|^{2}$

Consider the case $j=m=\frac{1}{2}$. Evaluate the probability that the measurement of $J_{1}$ will result in $m^{\prime}=-\frac{1}{2}$.

A4.15 B4.22

commentDiscuss the quantum mechanics of the one-dimensional harmonic oscillator using creation and annihilation operators, showing how the energy levels are calculated.

A quantum mechanical system consists of two interacting harmonic oscillators and has the Hamiltonian

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

For $\lambda=0$, what are the degeneracies of the three lowest energy levels? For $\lambda \neq 0$ compute, to lowest non-trivial order in perturbation theory, the energies of the ground state and first excited state.

[Standard results for perturbation theory may be stated without proof.]

A1.3

(i) Let $T: H_{1} \rightarrow H_{2}$ be a continuous linear map between two Hilbert spaces $H_{1}, H_{2}$. Define the adjoint $T^{*}$ of $T$. Explain what it means to say that $T$ is Hermitian or unitary.

Let $\phi: \mathbb{R} \rightarrow \mathbb{C}$ be a bounded continuous function. Show that the map

$T: L^{2}(\mathbb{R}) \rightarrow L^{2}(\mathbb{R})$

with $T f(x)=\phi(x) f(x+1)$ is a continuous linear map and find its adjoint. When is $T$ Hermitian? When is it unitary?

(ii) Let $C$ be a closed, non-empty, convex subset of a real Hilbert space $H$. Show that there exists a unique point $x_{o} \in C$