# Part II, 2005

### Jump to course

1.II.21H

comment(i) Show that if $E \rightarrow T$ is a covering map for the torus $T=S^{1} \times S^{1}$, then $E$ is homeomorphic to one of the following: the plane $\mathbb{R}^{2}$, the cylinder $\mathbb{R} \times S^{1}$, or the torus $T$.

(ii) Show that any continuous map from a sphere $S^{n}(n \geqslant 2)$ to the torus $T$ is homotopic to a constant map.

[General theorems from the course may be used without proof, provided that they are clearly stated.]

2.II.21H

commentState the Van Kampen Theorem. Use this theorem and the fact that $\pi_{1} S^{1}=\mathbb{Z}$ to compute the fundamental groups of the torus $T=S^{1} \times S^{1}$, the punctured torus $T \backslash\{p\}$, for some point $p \in T$, and the connected sum $T \# T$ of two copies of $T$.

3.II.20H

commentLet $X$ be a space that is triangulable as a simplicial complex with no $n$-simplices. Show that any continuous map from $X$ to $S^{n}$ is homotopic to a constant map.

[General theorems from the course may be used without proof, provided they are clearly stated.]

4.II.21H

commentLet $X$ be a simplicial complex. Suppose $X=B \cup C$ for subcomplexes $B$ and $C$, and let $A=B \cap C$. Show that the inclusion of $A$ in $B$ induces an isomorphism $H_{*} A \rightarrow H_{*} B$ if and only if the inclusion of $C$ in $X$ induces an isomorphism $H_{*} C \rightarrow H_{*} X$.

1.II.33B

commentA beam of particles is incident on a central potential $V(r)(r=|\mathbf{x}|)$ that vanishes for $r>R$. Define the differential cross-section $d \sigma / d \Omega$.

Given that each incoming particle has momentum $\hbar \mathbf{k}$, explain the relevance of solutions to the time-independent Schrödinger equation with the asymptotic form

$\psi(\mathbf{x}) \sim e^{i \mathbf{k} \cdot \mathbf{x}}+f(\hat{\mathbf{x}}) \frac{e^{i k r}}{r}$

as $r \rightarrow \infty$, where $k=|\mathbf{k}|$ and $\hat{\mathbf{x}}=\mathbf{x} / r$. Write down a formula that determines $d \sigma / d \Omega$ in this case.

Write down the time-independent Schrödinger equation for a particle of mass $m$ and energy $E=\frac{\hbar^{2} k^{2}}{2 m}$ in a central potential $V(r)$, and show that it allows a solution of the form

$\psi(\mathbf{x})=e^{i \mathbf{k} \cdot \mathbf{x}}-\frac{m}{2 \pi \hbar^{2}} \int d^{3} x^{\prime} \frac{e^{i k\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|} V\left(r^{\prime}\right) \psi\left(\mathbf{x}^{\prime}\right) .$

Show that this is consistent with $(*)$ and deduce an expression for $f(\hat{\mathbf{x}})$. Obtain the Born approximation for $f(\hat{\mathbf{x}})$, and show that $f(\hat{\mathbf{x}})=F(k \hat{\mathbf{x}}-\mathbf{k})$, where

$F(\mathbf{q})=-\frac{m}{2 \pi \hbar^{2}} \int d^{3} x e^{-i \mathbf{q} \cdot \mathbf{x}} V(r)$

Under what conditions is the Born approximation valid?

Obtain a formula for $f(\hat{\mathbf{x}})$ in terms of the scattering angle $\theta$ in the case that

$V(r)=K \frac{e^{-\mu r}}{r},$

for constants $K$ and $\mu$. Hence show that $f(\hat{\mathbf{x}})$ is independent of $\hbar$ in the limit $\mu \rightarrow 0$, when expressed in terms of $\theta$ and the energy $E$.

[You may assume that $\left.\left(\nabla^{2}+k^{2}\right)\left(\frac{e^{i k r}}{r}\right)=-4 \pi \delta^{3}(\mathbf{x}) .\right]$

2.II.33B

commentDescribe briefly the variational approach to the determination of an approximate ground state energy $E_{0}$ of a Hamiltonian $H$.

Let $\left|\psi_{1}\right\rangle$ and $\left|\psi_{2}\right\rangle$ be two states, and consider the trial state

$|\psi\rangle=a_{1}\left|\psi_{1}\right\rangle+a_{2}\left|\psi_{2}\right\rangle$

for real constants $a_{1}$ and $a_{2}$. Given that

$\begin{aligned} \left\langle\psi_{1} \mid \psi_{1}\right\rangle &=\left\langle\psi_{2} \mid \psi_{2}\right\rangle=1, &\left\langle\psi_{2} \mid \psi_{1}\right\rangle=\left\langle\psi_{1} \mid \psi_{2}\right\rangle=s, \\ \left\langle\psi_{1}|H| \psi_{1}\right\rangle &=\left\langle\psi_{2}|H| \psi_{2}\right\rangle=\mathcal{E}, &\left\langle\psi_{2}|H| \psi_{1}\right\rangle=\left\langle\psi_{1}|H| \psi_{2}\right\rangle=\epsilon, \end{aligned}$

and that $\epsilon<s \mathcal{E}$, obtain an upper bound on $E_{0}$ in terms of $\mathcal{E}, \epsilon$ and $s$.

The normalized ground-state wavefunction of the Hamiltonian

$H_{1}=\frac{p^{2}}{2 m}-K \delta(x), \quad K>0,$

$\psi_{1}(x)=\sqrt{\lambda} e^{-\lambda|x|}, \quad \lambda=\frac{m K}{\hbar^{2}} .$

Verify that the ground state energy of $H_{1}$ is

$E_{B} \equiv\left\langle\psi_{1}|H| \psi_{1}\right\rangle=-\frac{1}{2} K \lambda .$

Now consider the Hamiltonian

$H=\frac{p^{2}}{2 m}-K \delta(x)-K \delta(x-R)$

and let $E_{0}(R)$ be its ground-state energy as a function of $R$. Assuming that

$\psi_{2}(x)=\sqrt{\lambda} e^{-\lambda|x-R|},$

use $(*)$ to compute $s, \mathcal{E}$ and $\epsilon$ for $\psi_{1}$ and $\psi_{2}$ as given. Hence show that

$E_{0}(R) \leqslant E_{B}\left[1+2 \frac{e^{-\lambda R}\left(1+e^{-\lambda R}\right)}{1+(1+\lambda R) e^{-\lambda R}}\right]$

Why should you expect this inequality to become an approximate equality for sufficiently large $R$ ? Describe briefly how this is relevant to molecular binding.

3.II.33B

commentLet $\{\mathbf{l}\}$ be the set of lattice vectors of some lattice. Define the reciprocal lattice. What is meant by a Bravais lattice?

Let $\mathbf{i}, \mathbf{j}, \mathbf{k}$ be mutually orthogonal unit vectors. A crystal has identical atoms at positions given by the vectors

$\begin{array}{ll} a\left[n_{1} \mathbf{i}+n_{2} \mathbf{j}+n_{3} \mathbf{k}\right], & a\left[\left(n_{1}+\frac{1}{2}\right) \mathbf{i}+\left(n_{2}+\frac{1}{2}\right) \mathbf{j}+n_{3} \mathbf{k}\right] \\ a\left[\left(n_{1}+\frac{1}{2}\right) \mathbf{i}+\mathbf{j}+\left(n_{3}+\frac{1}{2}\right) \mathbf{k}\right], & a\left[n_{1} \mathbf{i}+\left(n_{2}+\frac{1}{2}\right) \mathbf{j}+\left(n_{3}+\frac{1}{2}\right) \mathbf{k}\right] \end{array}$

where $\left(n_{1}, n_{2}, n_{3}\right)$ are arbitrary integers and $a$ is a constant. Show that these vectors define a Bravais lattice with basis vectors

$\mathbf{a}_{1}=a \frac{1}{2}(\mathbf{j}+\mathbf{k}), \quad \mathbf{a}_{2}=a \frac{1}{2}(\mathbf{i}+\mathbf{k}), \quad \mathbf{a}_{3}=a \frac{1}{2}(\mathbf{i}+\mathbf{j})$

Verify that a basis for the reciprocal lattice is

$\mathbf{b}_{1}=\frac{2 \pi}{a}(\mathbf{j}+\mathbf{k}-\mathbf{i}), \quad \mathbf{b}_{2}=\frac{2 \pi}{a}(\mathbf{i}+\mathbf{k}-\mathbf{j}), \quad \mathbf{b}_{3}=\frac{2 \pi}{a}(\mathbf{i}+\mathbf{j}-\mathbf{k})$

In Bragg scattering, an incoming plane wave of wave-vector $\mathbf{k}$ is scattered to an outgoing wave of wave-vector $\mathbf{k}^{\prime}$. Explain why $\mathbf{k}^{\prime}=\mathbf{k}+\mathbf{g}$ for some reciprocal lattice vector g. Given that $\theta$ is the scattering angle, show that

$\sin \frac{1}{2} \theta=\frac{|\mathbf{g}|}{2|\mathbf{k}|} .$

For the above lattice, explain why you would expect scattering through angles $\theta_{1}$ and $\theta_{2}$ such that

$\frac{\sin \frac{1}{2} \theta_{1}}{\sin \frac{1}{2} \theta_{2}}=\frac{\sqrt{3}}{2}$

4.II.33B

commentA semiconductor has a valence energy band with energies $E \leqslant 0$ and density of states $g_{v}(E)$, and a conduction energy band with energies $E \geqslant E_{g}$ and density of states $g_{c}(E)$. Assume that $g_{v}(E) \sim A_{v}(-E)^{\frac{1}{2}}$ as $E \rightarrow 0$, and that $g_{c}(E) \sim A_{c}\left(E-E_{g}\right)^{\frac{1}{2}}$ as $E \rightarrow E_{g}$. At zero temperature all states in the valence band are occupied and the conduction band is empty. Let $p$ be the number of holes in the valence band and $n$ the number of electrons in the conduction band at temperature $T$. Under suitable approximations derive the result

$p n=N_{v} N_{c} e^{-E_{g} / k T}$

where

$N_{v}=\frac{1}{2} \sqrt{\pi} A_{v}(k T)^{\frac{3}{2}}, \quad N_{c}=\frac{1}{2} \sqrt{\pi} A_{c}(k T)^{\frac{3}{2}} .$

Briefly describe how a semiconductor may conduct electricity but with a conductivity that is strongly temperature dependent.

Describe how doping of the semiconductor leads to $p \neq n$. A $p n$ junction is formed between an $n$-type semiconductor, with $N_{d}$ donor atoms, and a $p$-type semiconductor, with $N_{a}$ acceptor atoms. Show that there is a potential difference $V_{n p}=\Delta E /|e|$ across the junction, where $e$ is the electron charge, and

$\Delta E=E_{g}-k T \ln \frac{N_{v} N_{c}}{N_{d} N_{a}} .$

Two semiconductors, one $p$-type and one $n$-type, are joined to make a closed circuit with two $p n$ junctions. Explain why a current will flow around the circuit if the junctions are at different temperatures.

[The Fermi-Dirac distribution function at temperature $T$ and chemical potential $\mu$ is $\frac{g(E)}{e^{(E-\mu) / k T}+1}$, where $g(E)$ is the number of states with energy $E$.

Note that $\int_{0}^{\infty} x^{\frac{1}{2}} e^{-x} d x=\frac{1}{2} \sqrt{\pi}$.]

1.II.26I

commentA cell has been placed in a biological solution at time $t=0$. After an exponential time of rate $\mu$, it is divided, producing $k$ cells with probability $p_{k}, k=0,1, \ldots$, with the mean value $\rho=\sum_{k=1}^{\infty} k p_{k}(k=0$ means that the cell dies $)$. The same mechanism is applied to each of the living cells, independently.

(a) Let $M_{t}$ be the number of living cells in the solution by time $t>0$. Prove that $\mathbb{E} M_{t}=\exp [t \mu(\rho-1)]$. [You may use without proof, if you wish, the fact that, if a positive function $a(t)$ satisfies $a(t+s)=a(t) a(s)$ for $t, s \geqslant 0$ and is differentiable at zero, then $a(t)=e^{\alpha t}, t \geqslant 0$, for some $\left.\alpha .\right]$

Let $\phi_{t}(s)=\mathbb{E} s^{M_{t}}$ be the probability generating function of $M_{t}$. Prove that it satisfies the following differential equation

$\frac{\mathrm{d}}{\mathrm{d} t} \phi_{t}(s)=\mu\left(-\phi_{t}(s)+\sum_{k=0}^{\infty} p_{k}\left[\phi_{t}(s)\right]^{k}\right), \quad \text { with } \quad \phi_{0}(s)=s$

(b) Now consider the case where each cell is divided in two cells $\left(p_{2}=1\right)$. Let $N_{t}=M_{t}-1$ be the number of cells produced in the solution by time $t$.

Calculate the distribution of $N_{t}$. Is $\left(N_{t}\right)$ an inhomogeneous Poisson process? If so, what is its rate $\lambda(t)$ ? Justify your answer.

2.II.26I

commentWhat does it mean to say that $\left(X_{t}\right)$ is a renewal process?

Let $\left(X_{t}\right)$ be a renewal process with holding times $S_{1}, S_{2}, \ldots$ and let $s>0$. For $n \geqslant 1$, set $T_{n}=S_{X_{s}+n}$. Show that

$\mathbb{P}\left(T_{n}>t\right) \geqslant \mathbb{P}\left(S_{n}>t\right), \quad t \geqslant 0,$

for all $n$, with equality if $n \geqslant 2$.

Consider now the case where $S_{1}, S_{2}, \ldots$ are exponential random variables. Show that

$\mathbb{P}\left(T_{1}>t\right)>\mathbb{P}\left(S_{1}>t\right), \quad t>0$

and that, as $s \rightarrow \infty$,

$\mathbb{P}\left(T_{1}>t\right) \rightarrow \mathbb{P}\left(S_{1}+S_{2}>t\right), \quad t \geqslant 0$

3.II.25I

commentConsider an $\mathrm{M} / \mathrm{G} / r / 0$ loss system with arrival rate $\lambda$ and service-time distribution $F$. Thus, arrivals form a Poisson process of rate $\lambda$, service times are independent with common distribution $F$, there are $r$ servers and there is no space for waiting. Use Little's Lemma to obtain a relation between the long-run average occupancy $L$ and the stationary probability $\pi$ that the system is full.

Cafe-Bar Duo has 23 serving tables. Each table can be occupied either by one person or two. Customers arrive either singly or in a pair; if a table is empty they are seated and served immediately, otherwise, they leave. The times between arrivals are independent exponential random variables of mean $20 / 3$. Each arrival is twice as likely to be a single person as a pair. A single customer stays for an exponential time of mean 20 , whereas a pair stays for an exponential time of mean 30 ; all these times are independent of each other and of the process of arrivals. The value of orders taken at each table is a constant multiple $2 / 5$ of the time that it is occupied.

Express the long-run rate of revenue of the cafe as a function of the probability $\pi$ that an arriving customer or pair of customers finds the cafe full.

By imagining a cafe with infinitely many tables, show that $\pi \leqslant \mathbb{P}(N \geqslant 23)$ where $N$ is a Poisson random variable of parameter $7 / 2$. Deduce that $\pi$ is very small. [Credit will be given for any useful numerical estimate, an upper bound of $10^{-3}$ being sufficient for full credit.]

4.II.26I

commentA particle performs a continuous-time nearest neighbour random walk on a regular triangular lattice inside an angle $\pi / 3$, starting from the corner. See the diagram below. The jump rates are $1 / 3$ from the corner and $1 / 6$ in each of the six directions if the particle is inside the angle. However, if the particle is on the edge of the angle, the rate is $1 / 3$ along the edge away from the corner and $1 / 6$ to each of three other neighbouring sites in the angle. See the diagram below, where a typical trajectory is also shown.

The particle position at time $t \geqslant 0$ is determined by its vertical level $V_{t}$ and its horizontal position $G_{t}$. For $k \geqslant 0$, if $V_{t}=k$ then $G_{t}=0, \ldots, k$. Here $1, \ldots, k-1$ are positions inside, and 0 and $k$ positions on the edge of the angle, at vertical level $k$.

Let $J_{1}^{V}, J_{2}^{V}, \ldots$ be the times of subsequent jumps of process $\left(V_{t}\right)$ and consider the embedded discrete-time Markov chains

$Y_{n}^{\text {in }}=\left(\widehat{G}_{n}^{\text {in }}, \widehat{V}_{n}\right) \text { and } Y_{n}^{\text {out }}=\left(\widehat{G}_{n}^{\text {out }}, \widehat{V}_{n}\right)$

where $\widehat{V}_{n}$ is the vertical level immediately after time $J_{n}^{V}, \widehat{G}_{n}^{\text {in }}$ is the horizontal position immediately after time $J_{n}^{V}$, and $\widehat{G}_{n}^{\text {out }}$ is the horizontal position immediately before time $J_{n+1}^{V}$. (a) Assume that $\left(\widehat{V}_{n}\right)$ is a Markov chain with transition probabilities

$\mathbb{P}\left(\widehat{V}_{n}=k+1 \mid \widehat{V}_{n-1}=k\right)=\frac{k+2}{2(k+1)}, \mathbb{P}\left(\widehat{V}_{n}=k-1 \mid \widehat{V}_{n-1}=k\right)=\frac{k}{2(k+1)},$

and that $\left(V_{t}\right)$ is a continuous-time Markov chain with rates

$q_{k k-1}=\frac{k}{3(k+1)}, \quad q_{k k}=-\frac{2}{3}, \quad q_{k k+1}=\frac{k+2}{3(k+1)}$

[You will be asked to justify these assumptions in part (b) of the question.] Determine whether the chains $\left(\widehat{V}_{n}\right)$ and $\left(V_{t}\right)$ are transient, positive recurrent or null recurrent.

(b) Now assume that, conditional on $\widehat{V}_{n}=k$ and previously passed vertical levels, the horizontal positions $\widehat{G}_{n}^{\text {in }}$ and $\widehat{G}_{n}^{\text {out }}$ are uniformly distributed on $\{0, \ldots, k\}$. In other words, for all attainable values $k, k_{n-1}, \ldots, k_{1}$ and for all $i=0, \ldots, k$,

$\begin{aligned} &\mathbb{P}\left(\widehat{G}_{n}^{\text {in }}=i \mid \widehat{V}_{n}=k, \widehat{V}_{n-1}=k_{n-1}, \ldots, \widehat{V}_{1}=k_{1}, \widehat{V}_{0}=0\right) \\ &\quad=\mathbb{P}\left(\widehat{G}_{n}^{\text {out }}=i \mid \widehat{V}_{n}=k, \widehat{V}_{n-1}=k_{n-1}, \ldots, \widehat{V}_{1}=k_{1}, \widehat{V}_{0}=0\right)=\frac{1}{k+1} \end{aligned}$

Deduce that $\left(\widehat{V}_{n}\right)$ and $\left(V_{t}\right)$ are indeed Markov chains with transition probabilities and rates as in (a).

(c) Finally, prove property $(*)$.

1.II $. 30 \mathrm{~A}$

commentExplain what is meant by an asymptotic power series about $x=a$ for a real function $f(x)$ of a real variable. Show that a convergent power series is also asymptotic.

Show further that an asymptotic power series is unique (assuming that it exists).

Let the function $f(t)$ be defined for $t \geqslant 0$ by

$f(t)=\frac{1}{\pi^{1 / 2}} \int_{0}^{\infty} \frac{e^{-x}}{x^{1 / 2}(1+2 x t)} d x$

By suitably expanding the denominator of the integrand, or otherwise, show that, as $t \rightarrow 0_{+}$,

$f(t) \sim \sum_{k=0}^{\infty}(-1)^{k} 1.3 \ldots(2 k-1) t^{k}$

and that the error, when the series is stopped after $n$ terms, does not exceed the absolute value of the $(n+1)$ th term of the series.

3.II $. 30$

commentExplain, without proof, how to obtain an asymptotic expansion, as $x \rightarrow \infty$, of

$I(x)=\int_{0}^{\infty} e^{-x t} f(t) d t$

if it is known that $f(t)$ possesses an asymptotic power series as $t \rightarrow 0$.

Indicate the modification required to obtain an asymptotic expansion, under suitable conditions, of

$\int_{-\infty}^{\infty} e^{-x t^{2}} f(t) d t$

Find an asymptotic expansion as $z \rightarrow \infty$ of the function defined by

$I(z)=\int_{-\infty}^{\infty} \frac{e^{-t^{2}}}{(z-t)} d t \quad(\operatorname{Im}(z)<0)$

and its analytic continuation to $\operatorname{Im}(z) \geqslant 0$. Where are the Stokes lines, that is, the critical lines separating the Stokes regions?

4.II $. 31 \mathrm{~A}$

commentConsider the differential equation

$\frac{d^{2} w}{d x^{2}}=q(x) w$

where $q(x) \geqslant 0$ in an interval $(a, \infty)$. Given a solution $w(x)$ and a further smooth function $\xi(x)$, define

$W(x)=\left[\xi^{\prime}(x)\right]^{1 / 2} w(x) .$

Show that, when $\xi$ is regarded as the independent variable, the function $W(\xi)$ obeys the differential equation

$\frac{d^{2} W}{d \xi^{2}}=\left\{\dot{x}^{2} q(x)+\dot{x}^{1 / 2} \frac{d^{2}}{d \xi^{2}}\left[\dot{x}^{-1 / 2}\right]\right\} W$

where $\dot{x}$ denotes $d x / d \xi$.

Taking the choice

$\xi(x)=\int q^{1 / 2}(x) d x$

show that equation $(*)$ becomes

$\frac{d^{2} W}{d \xi^{2}}=(1+\phi) W$

where

$\phi=-\frac{1}{q^{3 / 4}} \frac{d^{2}}{d x^{2}}\left(\frac{1}{q^{1 / 4}}\right)$

In the case that $\phi$ is negligible, deduce the Liouville-Green approximate solutions

$w_{\pm}=q^{-1 / 4} \exp \left(\pm \int q^{1 / 2} d x\right)$

Consider the Whittaker equation

$\frac{d^{2} w}{d x^{2}}=\left[\frac{1}{4}+\frac{s(s-1)}{x^{2}}\right] w$

where $s$ is a real constant. Show that the Liouville-Green approximation suggests the existence of solutions $w_{A, B}(x)$ with asymptotic behaviour of the form

$w_{A} \sim \exp (x / 2)\left(1+\sum_{n=1}^{\infty} a_{n} x^{-n}\right), \quad w_{B} \sim \exp (-x / 2)\left(1+\sum_{n=1}^{\infty} b_{n} x^{-n}\right)$

as $x \rightarrow \infty$.

Given that these asymptotic series may be differentiated term-by-term, show that

$a_{n}=\frac{(-1)^{n}}{n !}(s-n)(s-n+1) \ldots(s+n-1) \text {. }$

1.I.9C

commentA particle of mass $m_{1}$ is constrained to move on a circle of radius $r_{1}$, centre $x=y=0$ in a horizontal plane $z=0$. A second particle of mass $m_{2}$ moves on a circle of radius $r_{2}$, centre $x=y=0$ in a horizontal plane $z=c$. The two particles are connected by a spring whose potential energy is

$V=\frac{1}{2} \omega^{2} d^{2}$

where $d$ is the distance between the particles. How many degrees of freedom are there? Identify suitable generalized coordinates and write down the Lagrangian of the system in terms of them.

1.II.15C

comment(i) The action for a system with generalized coordinates $\left(q_{a}\right)$ is given by

$S=\int_{t_{1}}^{t_{2}} L\left(q_{a}, \dot{q}_{b}\right) d t$

Derive Lagrange's equations from the principle of least action by considering all paths with fixed endpoints, $\delta q_{a}\left(t_{1}\right)=\delta q_{a}\left(t_{2}\right)=0$.

(ii) A pendulum consists of a point mass $m$ at the end of a light rod of length $l$. The pivot of the pendulum is attached to a mass $M$ which is free to slide without friction along a horizontal rail. Choose as generalized coordinates the position $x$ of the pivot and the angle $\theta$ that the pendulum makes with the vertical.

Write down the Lagrangian and derive the equations of motion.

Find the frequency of small oscillations around the stable equilibrium.

Now suppose that a force acts on the pivot causing it to travel with constant acceleration in the $x$-direction. Find the equilibrium angle $\theta$ of the pendulum.

2.I.9C

commentA rigid body has principal moments of inertia $I_{1}, I_{2}$ and $I_{3}$ and is moving under the action of no forces with angular velocity components $\left(\omega_{1}, \omega_{2}, \omega_{3}\right)$. Its motion is described by Euler's equations

$\begin{aligned} &I_{1} \dot{\omega}_{1}-\left(I_{2}-I_{3}\right) \omega_{2} \omega_{3}=0 \\ &I_{2} \dot{\omega}_{2}-\left(I_{3}-I_{1}\right) \omega_{3} \omega_{1}=0 \\ &I_{3} \dot{\omega}_{3}-\left(I_{1}-I_{2}\right) \omega_{1} \omega_{2}=0 \end{aligned}$

Are the components of the angular momentum to be evaluated in the body frame or the space frame?

Now suppose that an asymmetric body is moving with constant angular velocity $(\Omega, 0,0)$. Show that this motion is stable if and only if $I_{1}$ is the largest or smallest principal moment.

3.I.9C

commentDefine the Poisson bracket $\{f, g\}$ between two functions $f\left(q_{a}, p_{a}\right)$ and $g\left(q_{a}, p_{a}\right)$ on phase space. If $f\left(q_{a}, p_{a}\right)$ has no explicit time dependence, and there is a Hamiltonian $H$, show that Hamilton's equations imply

$\frac{d f}{d t}=\{f, H\} .$

A particle with position vector $\mathbf{x}$ and momentum $\mathbf{p}$ has angular momentum $\mathbf{L}=\mathbf{x} \times \mathbf{p}$. Compute $\left\{p_{a}, L_{b}\right\}$ and $\left\{L_{a}, L_{b}\right\}$.

3.II.15C

comment(i) A point mass $m$ with position $q$ and momentum $p$ undergoes one-dimensional periodic motion. Define the action variable $I$ in terms of $q$ and $p$. Prove that an orbit of energy $E$ has period

$T=2 \pi \frac{d I}{d E} .$

(ii) Such a system has Hamiltonian

$H(q, p)=\frac{p^{2}+q^{2}}{\mu^{2}-q^{2}}$

where $\mu$ is a positive constant and $|q|<\mu$ during the motion. Sketch the orbits in phase space both for energies $E \gg 1$ and $E \ll 1$. Show that the action variable $I$ is given in terms of the energy $E$ by

$I=\frac{\mu^{2}}{2} \frac{E}{\sqrt{E+1}} .$

Hence show that for $E \gg 1$ the period of the orbit is $T \approx \frac{1}{2} \pi \mu^{3} / p_{0}$, where $p_{0}$ is the greatest value of the momentum during the orbit.

4.I.9C

commentDefine a canonical transformation for a one-dimensional system with coordinates $(q, p) \rightarrow(Q, P)$. Show that if the transformation is canonical then $\{Q, P\}=1$.

Find the values of constants $\alpha$ and $\beta$ such that the following transformations are canonical: (i) $Q=p q^{\beta}, P=\alpha q^{-1}$. (ii) $Q=q^{\alpha} \cos (\beta p), P=q^{\alpha} \sin (\beta p)$.

1.I.4J

commentBriefly describe the methods of Shannon-Fano and Huffman for economical coding. Illustrate both methods by finding decipherable binary codings in the case where messages $\mu_{1}, \ldots, \mu_{5}$ are emitted with probabilities $0.45,0.25,0.2,0.05,0.05$. Compute the expected word length in each case.

2.I.4J

commentWhat is a linear binary code? What is the weight $w(C)$ of a linear binary code $C ?$ Define the bar product $C_{1} \mid C_{2}$ of two binary linear codes $C_{1}$ and $C_{2}$, stating the conditions that $C_{1}$ and $C_{2}$ must satisfy. Under these conditions show that

$w\left(C_{1} \mid C_{2}\right) \geqslant \min \left(2 w\left(C_{1}\right), w\left(C_{2}\right)\right)$

2.II.12J

commentWhat does it means to say that $f: \mathbb{F}_{2}^{d} \rightarrow \mathbb{F}_{2}^{d}$ is a linear feedback shift register? Let $\left(x_{n}\right)_{n \geqslant 0}$ be a stream produced by such a register. Show that there exist $N, M$ with $N+M \leqslant 2^{d}-1$ such that $x_{r+N}=x_{r}$ for all $r \geqslant M$.

Explain and justify the Berlekamp-Massey method for 'breaking' a cipher stream arising from a linear feedback register of unknown length.

Let $x_{n}, y_{n}, z_{n}$ be three streams produced by linear feedback registers. Set

$\begin{aligned} &k_{n}=x_{n} \text { if } y_{n}=z_{n} \\ &k_{n}=y_{n} \text { if } y_{n} \neq z_{n} \end{aligned}$

Show that $k_{n}$ is also a stream produced by a linear feedback register. Sketch proofs of any theorems that you use.

3.I.4J

commentBriefly explain how and why a signature scheme is used. Describe the el Gamal scheme.

3.II.12J

commentDefine a cyclic code. Define the generator and check polynomials of a cyclic code and show that they exist.

Show that Hamming's original code is a cyclic code with check polynomial $X^{4}+X^{2}+X+1$. What is its generator polynomial? Does Hamming's original code contain a subcode equivalent to its dual?

4.I.4J

commentWhat does it mean to transmit reliably at rate $r$ through a binary symmetric channel (BSC) with error probability $p$ ? Assuming Shannon's second coding theorem, compute the supremum of all possible reliable transmission rates of a BSC. What happens if (i) $p$ is very small, (ii) $p=1 / 2$, or (iii) $p>1 / 2$ ?

1.I.10D

comment(a) Around $t \approx 1 \mathrm{~s}$ after the big bang $(k T \approx 1 \mathrm{MeV})$, neutrons and protons are kept in equilibrium by weak interactions such as

$\tag{*} n+\nu_{e} \leftrightarrow p+e^{-}$

Show that, in equilibrium, the neutron-to-proton ratio is given by

$\frac{n_{n}}{n_{p}} \approx e^{-Q / k T}$

where $Q=\left(m_{n}-m_{p}\right) c^{2}=1.29 \mathrm{MeV}$ corresponds to the mass difference between the neutron and the proton. Explain briefly why we can neglect the difference $\mu_{n}-\mu_{p}$ in the chemical potentials.

(b) The ratio of the weak interaction rate $\Gamma_{W} \propto T^{5}$ which maintains (*) to the Hubble expansion rate $H \propto T^{2}$ is given by

$\tag{†} \frac{\Gamma_{W}}{H} \approx\left(\frac{k T}{0.8 \mathrm{MeV}}\right)^{3}$

Explain why the neutron-to-proton ratio effectively "freezes out" once $k T<0.8 \mathrm{MeV}$, except for some slow neutron decay. Also explain why almost all neutrons are subsequently captured in ${ }^{4} \mathrm{He}$; estimate the value of the relative mass density $Y_{^{4} \mathrm{He}}=\rho_{^{4}\mathrm{He}} / \rho_{\mathrm{B}}$ (with $\rho_{\mathrm{B}}=\rho_{n}+\rho_{p}$ ) given a final ratio $n_{n} / n_{p} \approx 1 / 8$.

(c) Suppose instead that the weak interaction rate were very much weaker than that described by equation $(†)$. Describe the effect on the relative helium density $Y_{^{4} \mathrm{He}}$. Briefly discuss the wider implications of this primordial helium-to-hydrogen ratio on stellar lifetimes and life on earth.

2.I.10D

comment(a) A spherically symmetric star obeys the pressure-support equation

$\frac{d P}{d r}=-\frac{G m \rho}{r^{2}}$

where $P(r)$ is the pressure at a distance $r$ from the centre, $\rho(r)$ is the density, and the mass $m(r)$ is defined through the relation $d m / d r=4 \pi r^{2} \rho(r)$. Multiply $(*)$ by $4 \pi r^{3}$ and integrate over the total volume $V$ of the star to derive the virial theorem

$\langle P\rangle V=-\frac{1}{3} E_{\text {grav }}$

where $\langle P\rangle$ is the average pressure and $E_{\text {grav }}$ is the total gravitational potential energy.

(b) Consider a white dwarf supported by electron Fermi degeneracy pressure $P \approx h^{2} n^{5 / 3} / m_{\mathrm{e}}$, where $m_{\mathrm{e}}$ is the electron mass and $n$ is the number density. Assume a uniform density $\rho(r)=m_{\mathrm{p}} n(r) \approx m_{\mathrm{p}}\langle n\rangle$, so the total mass of the star is given by $M=(4 \pi / 3)\langle n\rangle m_{\mathrm{p}} R^{3}$ where $R$ is the star radius and $m_{\mathrm{p}}$ is the proton mass. Show that the total energy of the white dwarf can be written in the form

$E_{\mathrm{total}}=E_{\mathrm{kin}}+E_{\mathrm{grav}}=\frac{\alpha}{R^{2}}-\frac{\beta}{R}$

where $\alpha, \beta$ are positive constants which you should determine. [You may assume that for an ideal gas $E_{\mathrm{kin}}=\frac{3}{2}\langle P\rangle V$.] Use this expression to explain briefly why a white dwarf is stable.

2.II.15D

comment(a) Consider a homogeneous and isotropic universe with scale factor $a(t)$ and filled with mass density $\rho(t)$. Show how the conservation of kinetic energy plus gravitational potential energy for a test particle on the edge of a spherical region in this universe can be used to derive the Friedmann equation

$\tag{*} \left(\frac{\dot{a}}{a}\right)^{2}+\frac{k c^{2}}{a^{2}}=\frac{8 \pi G}{3} \rho$

where $k$ is a constant. State clearly any assumptions you have made.

(b) Now suppose that the universe was filled throughout its history with radiation with equation of state $P=\rho c^{2} / 3$. Using the fluid conservation equation and the definition of the relative density $\Omega$, show that the density of this radiation can be expressed as

$\rho=\frac{3 H_{0}^{2}}{8 \pi G} \frac{\Omega_{0}}{a^{4}},$

where $H_{0}$ is the Hubble parameter today and $\Omega_{0}$ is the relative density today $\left(t=t_{0}\right)$ and $a_{0} \equiv a\left(t_{0}\right)=1$ is assumed. Show also that $k c^{2}=H_{0}^{2}\left(\Omega_{0}-1\right)$ and hence rewrite the Friedmann equation $(*)$ as

$\tag{†} \left(\frac{\dot{a}}{a}\right)^{2}=H_{0}^{2} \Omega_{0}\left(\frac{1}{a^{4}}-\frac{\beta}{a^{2}}\right)$

where $\beta \equiv\left(\Omega_{0}-1\right) / \Omega_{0}$.

(c) Now consider a closed model with $k>0$ (or $\Omega>1)$. Rewrite ( $\dagger$ ) using the new time variable $\tau$ defined by

$\frac{d t}{d \tau}=a$

Hence, or otherwise, solve $(†)$ to find the parametric solution

$a(\tau)=\frac{1}{\sqrt{\beta}}(\sin \alpha \tau), \quad t(\tau)=\frac{1}{\alpha \sqrt{\beta}}(1-\cos \alpha \tau),$

where $\alpha \equiv H_{0} \sqrt{\left(\Omega_{0}-1\right)} . \quad$ Recall that $\left.\int d x / \sqrt{1-x^{2}}=\sin ^{-1} x .\right]$

Using the solution for $a(\tau)$, find the value of the new time variable $\tau=\tau_{0}$ today and hence deduce that the age of the universe in this model is

$t_{0}=H_{0}^{-1} \frac{\sqrt{\Omega_{0}}-1}{\Omega_{0}-1}$

3.I.10D

comment(a) Define and discuss the concept of the cosmological horizon and the Hubble radius for a homogeneous isotropic universe. Illustrate your discussion with the specific examples of the Einstein-de Sitter universe $\left(a \propto t^{2 / 3}\right.$ for $\left.t>0\right)$ and a de Sitter universe $\left(a \propto e^{H t}\right.$ with $H$ constant, $t>-\infty)$.

(b) Explain the horizon problem for a decelerating universe in which $a(t) \propto t^{\alpha}$ with $\alpha<1$. How can inflation cure the horizon problem?

(c) Consider a Tolman (radiation-filled) universe $\left(a(t) \propto t^{1 / 2}\right.$ ) beginning at $t_{\mathrm{r}} \sim$ $10^{-35} \mathrm{~s}$ and lasting until today at $t_{0} \approx 10^{17} \mathrm{~s}$. Estimate the horizon size today $d_{H}\left(t_{0}\right)$ and project this lengthscale backwards in time to show that it had a physical size of about 1 metre at $t \approx t_{\mathrm{r}}$.

Prior to $t \approx t_{\mathrm{r}}$, assume an inflationary (de Sitter) epoch with constant Hubble parameter $H$ given by its value at $t \approx t_{\mathrm{r}}$ for the Tolman universe. How much expansion during inflation is required for the observable universe today to have begun inside one Hubble radius?

4.I.10D

commentThe linearised equation for the growth of a density fluctuation $\delta_{k}$ in a homogeneous and isotropic universe is

$\frac{d^{2} \delta_{k}}{d t^{2}}+2 \frac{\dot{a}}{a} \frac{d \delta_{k}}{d t}-\left(4 \pi G \rho_{\mathrm{m}}-\frac{v_{s}^{2} k^{2}}{a^{2}}\right) \delta_{k}=0,$

where $\rho_{\mathrm{m}}$ is the non-relativistic matter density, $k$ is the comoving wavenumber and $v_{s}$ is the sound speed $\left(v_{s}^{2} \equiv d P / d \rho\right)$.

(a) Define the Jeans length $\lambda_{\mathrm{J}}$ and discuss its significance for perturbation growth.

(b) Consider an Einstein-de Sitter universe with $a(t)=\left(t / t_{0}\right)^{2 / 3}$ filled with pressure-free matter $(P=0)$. Show that the perturbation equation $(*)$ can be re-expressed as

$\ddot{\delta}_{k}+\frac{4}{3 t} \dot{\delta}_{k}-\frac{2}{3 t^{2}} \delta_{k}=0 .$

By seeking power law solutions, find the growing and decaying modes of this equation.

(c) Qualitatively describe the evolution of non-relativistic matter perturbations $(k>a H)$ in the radiation era, $a(t) \propto t^{1 / 2}$, when $\rho_{\mathrm{r}} \gg \rho_{\mathrm{m}}$. What feature in the power spectrum is associated with the matter-radiation transition?

4.II.15D

commentFor an ideal gas of bosons, the average occupation number can be expressed as

$\bar{n}_{k}=\frac{g_{k}}{e^{\left(E_{k}-\mu\right) / k T}-1},$

where $g_{k}$ has been included to account for the degeneracy of the energy level $E_{k}$. In the approximation in which a discrete set of energies $E_{k}$ is replaced with a continuous set with momentum $p$, the density of one-particle states with momentum in the range $p$ to $p+d p$ is $g(p) d p$. Explain briefly why

$g(p) \propto p^{2} V,$

where $V$ is the volume of the gas. Using this formula with equation $(*)$, obtain an expression for the total energy density $\epsilon=E / V$ of an ultra-relativistic gas of bosons at zero chemical potential as an integral over $p$. Hence show that

$\epsilon \propto T^{\alpha}$

where $\alpha$ is a number you should find. Why does this formula apply to photons?

Prior to a time $t \sim 100,000$ years, the universe was filled with a gas of photons and non-relativistic free electrons and protons. Subsequently, at around $t \sim 400,000$ years, the protons and electrons began combining to form neutral hydrogen,

$p+e^{-} \leftrightarrow H+\gamma$

Deduce Saha's equation for this recombination process stating clearly the steps required:

$\frac{n_{\mathrm{e}}^{2}}{n_{\mathrm{H}}}=\left(\frac{2 \pi m_{\mathrm{e}} k T}{h^{2}}\right)^{3 / 2} \exp (-I / k T)$

where $I$ is the ionization energy of hydrogen. [Note that the equilibrium number density of a non-relativistic species $\left(k T \ll m c^{2}\right)$ is given by $n=g_{s}\left(\frac{2 \pi m k T}{h^{2}}\right)^{3 / 2} \exp \left[\left(\mu-m c^{2}\right) / k T\right]$, while the photon number density is $n_{\gamma}=16 \pi \zeta(3)\left(\frac{k T}{h c}\right)^{3}$, where $\left.\zeta(3) \approx 1.20 \ldots\right]$

Consider now the fractional ionization $X_{\mathrm{e}}=n_{\mathrm{e}} / n_{\mathrm{B}}$, where $n_{B}=n_{\mathrm{p}}+n_{\mathrm{H}}=\eta n_{\gamma}$ is the baryon number of the universe and $\eta$ is the baryon-to-photon ratio. Find an expression for the ratio

$\left(1-X_{\mathrm{e}}\right) / X_{\mathrm{e}}^{2}$

in terms only of $k T$ and constants such as $\eta$ and $I$. One might expect neutral hydrogen to form at a temperature given by $k T \approx I \approx 13 \mathrm{eV}$, but instead in our universe it forms at the much lower temperature $k T \approx 0.3 \mathrm{eV}$. Briefly explain why.

1.II.24H

commentLet $f: X \rightarrow Y$ be a smooth map between manifolds without boundary.

(i) Define what is meant by a critical point, critical value and regular value of $f$.

(ii) Show that if $y$ is a regular value of $f$ and $\operatorname{dim} X \geqslant \operatorname{dim} Y$, then the set $f^{-1}(y)$ is a submanifold of $X$ with $\operatorname{dim} f^{-1}(y)=\operatorname{dim} X-\operatorname{dim} Y$.

[You may assume the inverse function theorem.]

(iii) Let $S L(n, \mathbb{R})$ be the group of all $n \times n$ real matrices with determinant 1. Prove that $S L(n, \mathbb{R})$ is a submanifold of the set of all $n \times n$ real matrices. Find the tangent space to $S L(n, \mathbb{R})$ at the identity matrix.

2.II.24H

commentState the isoperimetric inequality in the plane.

Let $S \subset \mathbb{R}^{3}$ be a surface. Let $p \in S$ and let $S_{r}(p)$ be a geodesic circle of centre $p$ and radius $r$ ( $r$ small). Let $L$ be the length of $S_{r}(p)$ and $A$ be the area of the region bounded by $S_{r}(p)$. Prove that

$4 \pi A-L^{2}=\pi^{2} r^{4} K(p)+\varepsilon(r),$

where $K(p)$ is the Gaussian curvature of $S$ at $p$ and

$\lim _{r \rightarrow 0} \frac{\varepsilon(r)}{r^{4}}=0 .$

When $K(p)>0$ and $r$ is small, compare this briefly with the isoperimetric inequality in the plane.

3.II.23H

comment(i) Define geodesic curvature and state the Gauss-Bonnet theorem.

(ii) Let $\alpha: I \rightarrow \mathbb{R}^{3}$ be a closed regular curve parametrized by arc-length, and assume that $\alpha$ has non-zero curvature everywhere. Let $n: I \rightarrow S^{2} \subset \mathbb{R}^{3}$ be the curve given by the normal vector $n(s)$ to $\alpha(s)$. Let $\bar{s}$ be the arc-length of the curve $n$ on $S^{2}$. Show that the geodesic curvature $k_{g}$ of $n$ is given by

$k_{g}=-\frac{d}{d s} \tan ^{-1}(\tau / k) \frac{d s}{d \bar{s}},$

where $k$ and $\tau$ are the curvature and torsion of $\alpha$.

(iii) Suppose now that $n(s)$ is a simple curve (i.e. it has no self-intersections). Show that $n(I)$ divides $S^{2}$ into two regions of equal area.

4.II.24H

(i) Define what is meant by an isothermal parametrization. Let $\phi: U \rightarrow \mathbb{R}^{3}$ be an isothermal parametrization. Prove that

$\phi_{u u}+\phi_{v v}=2 \lambda^{2} \mathbf{H}$

where $\mathbf{H}$ is the mean curvature vector and $\lambda^{2}=\left\langle\phi_{u}, \phi_{u}\right\rangle$.

Define what it means for $\phi$ to be minimal, and deduce that $\phi$ is minimal if and only if $\Delta \phi=0$.

[You may assume that the mean curvature $H$ can be written as

$\left.H=\frac{e G-2 f F+g E}{2\left(E G-F^{2}\right)} .\right]$

(ii) Write $\phi(u, v)=(x(u, v), y(u, v), z(u, v))$. Consider the complex valued functions

$\varphi_{1}=x_{u}-i x_{v}, \quad \varphi_{2}=y_{u}-i y_{v}, \quad \varphi_{3}=z_{u}-i z_{v}$

Show that $\phi$ is isothermal if and only if $\varphi_{1}^{2}+\varphi_{2}^{2}+\varphi_{3}^{2} \equiv 0$.

Suppose now that $\phi$ is isothermal. Prove that $\phi$ is minimal if and only if $\varphi_{1}, \varphi_{2}$ and $\varphi_{3}$ are holomorphic functions.

(iii) Consider the immersion $\phi: \mathbb{R}^{2} \rightarrow \mathbb{R}^{3}$ given by

$\phi(u, v)=\left(u-u^{3} / 3+u v^{2},-v+v^{3} / 3-u^{2} v, u^{2}-v^{2}\right)$

Find ${\phi}_{1},{\phi}_{2}$