Part II, 2005

Algebraic Topology

• 1.II.21H

  Algebraic Topology | Part II, 2005

  (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.]

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• 2.II.21H

  Algebraic Topology | Part II, 2005

  State 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 $T$.

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• 3.II.20H

  Algebraic Topology | Part II, 2005

  Let $X$ 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.]

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• 4.II.21H

  Algebraic Topology | Part II, 2005

  Let $X$ 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 $H_{*} C \rightarrow H_{*} X$.

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Applications of Quantum Mechanics

• 1.II.33B

  Applications of Quantum Mechanics | Part II, 2005

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

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• 2.II.33B

  Applications of Quantum Mechanics | Part II, 2005

  Describe 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.

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• 3.II.33B

  Applications of Quantum Mechanics | Part II, 2005

  Let $\{\mathbf{l}\}$ 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

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

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• 4.II.33B

  Applications of Quantum Mechanics | Part II, 2005

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

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Applied Probability

• 1.II.26I

  Applied Probability | Part II, 2005

  A 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.

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• 2.II.26I

  Applied Probability | Part II, 2005

  What 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 $s \rightarrow \infty$,

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

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• 3.II.25I

  Applied Probability | Part II, 2005

  Consider 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.]

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• 4.II.26I

  Applied Probability | Part II, 2005

  A 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 $(*)$.

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Asymptotic Methods

• 1.II $. 30 \mathrm{~A}$

  Asymptotic Methods | Part II, 2005

  Explain 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.

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• 3.II $. 30$

  Asymptotic Methods | Part II, 2005

  Explain, 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?

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• 4.II $. 31 \mathrm{~A}$

  Asymptotic Methods | Part II, 2005

  Consider 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

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

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Classical Dynamics

• 1.I.9C

  Classical Dynamics | Part II, 2005

  A 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.

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• 1.II.15C

  Classical Dynamics | Part II, 2005

  (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.

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• 2.I.9C

  Classical Dynamics | Part II, 2005

  A 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.

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• 3.I.9C

  Classical Dynamics | Part II, 2005

  Define 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 $\left\{L_{a}, L_{b}\right\}$.

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• 3.II.15C

  Classical Dynamics | Part II, 2005

  (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.

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• 4.I.9C

  Classical Dynamics | Part II, 2005

  Define 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) $Q=q^{\alpha} \cos (\beta p), P=q^{\alpha} \sin (\beta p)$.

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Coding and Cryptography

• 1.I.4J

  Coding and Cryptography | Part II, 2005

  Briefly 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.

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• 2.I.4J

  Coding and Cryptography | Part II, 2005

  What 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

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

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• 2.II.12J

  Coding and Cryptography | Part II, 2005

  What 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.

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• 3.I.4J

  Coding and Cryptography | Part II, 2005

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

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• 3.II.12J

  Coding and Cryptography | Part II, 2005

  Define 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?

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• 4.I.4J

  Coding and Cryptography | Part II, 2005

  What 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) $p>1 / 2$ ?

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Cosmology

• 1.I.10D

  Cosmology | Part II, 2005

  (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.

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• 2.I.10D

  Cosmology | Part II, 2005

  (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.

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• 2.II.15D

  Cosmology | Part II, 2005

  (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

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

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• 3.I.10D

  Cosmology | Part II, 2005

  (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?

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• 4.I.10D

  Cosmology | Part II, 2005

  The 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?

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• 4.II.15D

  Cosmology | Part II, 2005

  For 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.

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Differential Geometry

• 1.II.24H

  Differential Geometry | Part II, 2005

  Let $f: X \rightarrow Y$ 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.

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• 2.II.24H

  Differential Geometry | Part II, 2005

  State 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.

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• 3.II.23H

  Differential Geometry | Part II, 2005

  (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.

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• 4.II.24H

  Differential Geometry | Part II, 2005

  (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 ${\phi }_1,{\phi }_2$