• # Paper 1, Section II, B

(a) Discuss the variational principle that allows one to derive an upper bound on the energy $E_{0}$ of the ground state for a particle in one dimension subject to a potential $V(x)$.

If $V(x)=V(-x)$, how could you adapt the variational principle to derive an upper bound on the energy $E_{1}$ of the first excited state?

(b) Consider a particle of mass $2 m=\hbar^{2}$ (in certain units) subject to a potential

$V(x)=-V_{0} e^{-x^{2}} \quad \text { with } \quad V_{0}>0$

(i) Using the trial wavefunction

$\psi(x)=e^{-\frac{1}{2} x^{2} a}$

with $a>0$, derive the upper bound $E_{0} \leqslant E(a)$, where

$E(a)=\frac{1}{2} a-V_{0} \frac{\sqrt{a}}{\sqrt{1+a}}$

(ii) Find the zero of $E(a)$ in $a>0$ and show that any extremum must obey

$(1+a)^{3}=\frac{V_{0}^{2}}{a} .$

(iii) By sketching $E(a)$ or otherwise, deduce that there must always be a minimum in $a>0$. Hence deduce the existence of a bound state.

(iv) Working perturbatively in $0, show that

$-V_{0}

[Hint: You may use that $\int_{-\infty}^{\infty} e^{-b x^{2}} d x=\sqrt{\frac{\pi}{b}}$ for $\left.b>0 .\right]$

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• # Paper 2, Section II, 36B

(a) The $s$-wave solution $\psi_{0}$ for the scattering problem of a particle of mass $m$ and momentum $\hbar k$ has the asymptotic form

$\psi_{0}(r) \sim \frac{A}{r}[\sin (k r)+g(k) \cos (k r)]$

Define the phase shift $\delta_{0}$ and verify that $\tan \delta_{0}=g(k)$.

(b) Define the scattering amplitude $f$. For a spherically symmetric potential of finite range, starting from $\sigma_{T}=\int|f|^{2} d \Omega$, derive the expression

$\sigma_{T}=\frac{4 \pi}{k^{2}} \sum_{l=0}^{\infty}(2 l+1) \sin ^{2} \delta_{l}$

giving the cross-section $\sigma_{T}$ in terms of the phase shifts $\delta_{l}$ of the partial waves.

(c) For $g(k)=-k / K$ with $K>0$, show that a bound state exists and compute its energy. Neglecting the contributions from partial waves with $l>0$, show that

$\sigma_{T} \approx \frac{4 \pi}{K^{2}+k^{2}}$

(d) For $g(k)=\gamma /\left(K_{0}-k\right)$ with $K_{0}>0, \gamma>0$ compute the $s$-wave contribution to $\sigma_{T}$. Working to leading order in $\gamma \ll K_{0}$, show that $\sigma_{T}$ has a local maximum at $k=K_{0}$. Interpret this fact in terms of a resonance and compute its energy and decay width.

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• # Paper 3, Section II, 34B

(a) In three dimensions, define a Bravais lattice $\Lambda$ and its reciprocal lattice $\Lambda^{*}$.

A particle is subject to a potential $V(\mathbf{x})$ with $V(\mathbf{x})=V(\mathbf{x}+\mathbf{r})$ for $\mathbf{x} \in \mathbb{R}^{3}$ and $\mathbf{r} \in \Lambda$. State and prove Bloch's theorem and specify how the Brillouin zone is related to the reciprocal lattice.

(b) A body-centred cubic lattice $\Lambda_{B C C}$ consists of the union of the points of a cubic lattice $\Lambda_{1}$ and all the points $\Lambda_{2}$ at the centre of each cube:

\begin{aligned} \Lambda_{B C C} & \equiv \Lambda_{1} \cup \Lambda_{2}, \\ \Lambda_{1} & \equiv\left\{\mathbf{r} \in \mathbb{R}^{3}: \mathbf{r}=n_{1} \hat{\mathbf{i}}+n_{2} \hat{\mathbf{j}}+n_{3} \hat{\mathbf{k}}, \text { with } n_{1,2,3} \in \mathbb{Z}\right\}, \\ \Lambda_{2} & \equiv\left\{\mathbf{r} \in \mathbb{R}^{3}: \mathbf{r}=\frac{1}{2}(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}})+\mathbf{r}^{\prime}, \text { with } \mathbf{r}^{\prime} \in \Lambda_{1}\right\}, \end{aligned}

where $\hat{\mathbf{i}}, \hat{\mathbf{j}}$ and $\hat{\mathbf{k}}$ are unit vectors parallel to the Cartesian coordinates in $\mathbb{R}^{3}$. Show that $\Lambda_{B C C}$ is a Bravais lattice and determine the primitive vectors $\mathbf{a}_{1}, \mathbf{a}_{2}$ and $\mathbf{a}_{3}$.

Find the reciprocal lattice $\Lambda_{B C C}^{*} .$ Briefly explain what sort of lattice it is.

$\left[\right.$ Hint: The matrix $M=\frac{1}{2}\left(\begin{array}{ccc}-1 & 1 & 1 \\ 1 & -1 & 1 \\ 1 & 1 & -1\end{array}\right)$ has inverse $M^{-1}=\left(\begin{array}{lll}0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0\end{array}\right)$.

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• # Paper 4, Section II, B

(a) Consider the nearly free electron model in one dimension with mass $m$ and periodic potential $V(x)=\lambda U(x)$ with $0<\lambda \ll 1$ and

$U(x)=\sum_{l=-\infty}^{\infty} U_{l} \exp \left(\frac{2 \pi i}{a} l x\right)$

Ignoring degeneracies, the energy spectrum of Bloch states with wavenumber $k$ is

$E(k)=E_{0}(k)+\lambda\langle k|U| k\rangle+\lambda^{2} \sum_{k^{\prime} \neq k} \frac{\left\langle k|U| k^{\prime}\right\rangle\left\langle k^{\prime}|U| k\right\rangle}{E_{0}(k)-E_{0}\left(k^{\prime}\right)}+\mathcal{O}\left(\lambda^{3}\right)$

where $\{|k\rangle\}$ are normalized eigenstates of the free Hamiltonian with wavenumber $k$. What is $E_{0}$ in this formula?

If we impose periodic boundary conditions on the wavefunctions, $\psi(x)=\psi(x+L)$ with $L=N a$ and $N$ a positive integer, what are the allowed values of $k$ and $k^{\prime}$ ? Determine $\left\langle k|U| k^{\prime}\right\rangle$ for these allowed values.

(b) State when the above expression for $E(k)$ ceases to be a good approximation and explain why. Quoting any result you need from degenerate perturbation theory, calculate to $\mathcal{O}(\lambda)$ the location and width of the band gaps.

(c) Determine the allowed energy bands for each of the potentials

\begin{aligned} &\text { (i) } V(x)=2 \lambda \cos \left(\frac{2 \pi x}{a}\right) \text {, } \\ &\text { (ii) } V(x)=\lambda a \sum_{n=-\infty}^{\infty} \delta(x-n a) \text {. } \end{aligned}

(d) Briefly discuss a macroscopic physical consequence of the existence of energy bands.

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• # Paper 1, Section II, C

Consider the quantum mechanical scattering of a particle of mass $m$ in one dimension off a parity-symmetric potential, $V(x)=V(-x)$. State the constraints imposed by parity, unitarity and their combination on the components of the $S$-matrix in the parity basis,

$S=\left(\begin{array}{cc} S_{++} & S_{+-} \\ S_{-+} & S_{--} \end{array}\right)$

For the specific potential

$V=\hbar^{2} \frac{U_{0}}{2 m}\left[\delta_{D}(x+a)+\delta_{D}(x-a)\right]$

show that

$S_{--}=e^{-i 2 k a}\left[\frac{\left(2 k-i U_{0}\right) e^{i k a}+i U_{0} e^{-i k a}}{\left(2 k+i U_{0}\right) e^{-i k a}-i U_{0} e^{i k a}}\right]$

For $U_{0}<0$, derive the condition for the existence of an odd-parity bound state. For $U_{0}>0$ and to leading order in $U_{0} a \gg 1$, show that an odd-parity resonance exists and discuss how it evolves in time.

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• # Paper 2, Section II, $35 \mathrm{C}$

a) Consider a particle moving in one dimension subject to a periodic potential, $V(x)=V(x+a)$. Define the Brillouin zone. State and prove Bloch's theorem.

b) Consider now the following periodic potential

$V=V_{0}(\cos (x)-\cos (2 x))$

with positive constant $V_{0}$.

i) For very small $V_{0}$, use the nearly-free electron model to compute explicitly the lowest-energy band gap to leading order in degenerate perturbation theory.

ii) For very large $V_{0}$, the electron is localised very close to a minimum of the potential. Estimate the two lowest energies for such localised eigenstates and use the tight-binding model to estimate the lowest-energy band gap.

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• # Paper 3, Section II, C

(a) For the quantum scattering of a beam of particles in three dimensions off a spherically symmetric potential $V(r)$ that vanishes at large $r$, discuss the boundary conditions satisfied by the wavefunction $\psi$ and define the scattering amplitude $f(\theta)$. Assuming the asymptotic form

$\psi=\sum_{l=0}^{\infty} \frac{2 l+1}{2 i k}\left[(-1)^{l+1} \frac{e^{-i k r}}{r}+\left(1+2 i f_{l}\right) \frac{e^{i k r}}{r}\right] P_{l}(\cos \theta),$

state the constraints on $f_{l}$ imposed by the unitarity of the $S$-matrix and define the phase shifts $\delta_{l}$.

(b) For $V_{0}>0$, consider the specific potential

$V(r)=\left\{\begin{array}{lc} \infty, & r \leqslant a \\ -V_{0}, & a2 a \end{array}\right.$

(i) Show that the s-wave phase shift $\delta_{0}$ obeys

$\tan \left(\delta_{0}\right)=\frac{k \cos (2 k a)-\kappa \cot (\kappa a) \sin (2 k a)}{k \sin (2 k a)+\kappa \cot (\kappa a) \cos (2 k a)},$

where $\kappa^{2}=k^{2}+2 m V_{0} / \hbar^{2}$.

(ii) Compute the scattering length $a_{s}$ and find for which values of $\kappa$ it diverges. Discuss briefly the physical interpretation of the divergences. [Hint: you may find this trigonometric identity useful

$\left.\tan (A+B)=\frac{\tan A+\tan B}{1-\tan A \tan B} .\right]$

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• # Paper 4, Section II, $34 C$

(a) For a particle of charge $q$ moving in an electromagnetic field with vector potential $\boldsymbol{A}$ and scalar potential $\phi$, write down the classical Hamiltonian and the equations of motion.

(b) Consider the vector and scalar potentials

$\boldsymbol{A}=\frac{B}{2}(-y, x, 0), \quad \phi=0$

(i) Solve the equations of motion. Define and compute the cyclotron frequency $\omega_{B}$.

(ii) Write down the quantum Hamiltonian of the system in terms of the angular momentum operator

$L_{z}=x p_{y}-y p_{x}$

Show that the states

$\psi(x, y)=f(x+i y) e^{-\left(x^{2}+y^{2}\right) q B / 4 \hbar}$

for any function $f$, are energy eigenstates and compute their energy. Define Landau levels and discuss this result in relation to them.

(iii) Show that for $f(w)=w^{M}$, the wavefunctions in ( $\dagger$ ) are eigenstates of angular momentum and compute the corresponding eigenvalue. These wavefunctions peak in a ring around the origin. Estimate its radius. Using these two facts or otherwise, estimate the degeneracy of Landau levels.

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• # Paper 1, Section II, B

A particle of mass $m$ and charge $q$ moving in a uniform magnetic field $\mathbf{B}=\nabla \times \mathbf{A}=$ $(0,0, B)$ and electric field $\mathbf{E}=-\nabla \phi$ is described by the Hamiltonian

$H=\frac{1}{2 m}|\mathbf{p}-q \mathbf{A}|^{2}+q \phi$

where $\mathbf{p}$ is the canonical momentum.

[ In the following you may use without proof any results concerning the spectrum of the harmonic oscillator as long as they are stated clearly.]

(a) Let $\mathbf{E}=\mathbf{0}$. Choose a gauge which preserves translational symmetry in the $y$ direction. Determine the spectrum of the system, restricted to states with $p_{z}=0$. The system is further restricted to lie in a rectangle of area $A=L_{x} L_{y}$, with sides of length $L_{x}$ and $L_{y}$ parallel to the $x$ - and $y$-axes respectively. Assuming periodic boundary conditions in the $y$-direction, estimate the degeneracy of each Landau level.

(b) Consider the introduction of an additional electric field $\mathbf{E}=(\mathcal{E}, 0,0)$. Choosing a suitable gauge (with the same choice of vector potential $\mathbf{A}$ as in part (a)), write down the resulting Hamiltonian. Find the energy spectrum for a particle on $\mathbb{R}^{3}$ again restricted to states with $p_{z}=0$.

Define the group velocity of the electron and show that its $y$-component is given by $v_{y}=-\mathcal{E} / B$.

When the system is further restricted to a rectangle of area $A$ as above, show that the previous degeneracy of the Landau levels is lifted and determine the resulting energy gap $\Delta E$ between the ground-state and the first excited state.

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• # Paper 2, Section II, B

Give an account of the variational principle for establishing an upper bound on the ground state energy of a Hamiltonian $H$.

A particle of mass $m$ moves in one dimension and experiences the potential $V=A|x|^{n}$ with $n$ an integer. Use a variational argument to prove the virial theorem,

$2\langle T\rangle_{0}=n\langle V\rangle_{0}$

where $\langle\cdot\rangle_{0}$ denotes the expectation value in the true ground state. Deduce that there is no normalisable ground state for $n \leqslant-3$.

For the case $n=1$, use the ansatz $\psi(x) \propto e^{-\alpha^{2} x^{2}}$ to find an estimate for the energy of the ground state.

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• # Paper 3, Section II, B

A Hamiltonian $H$ is invariant under the discrete translational symmetry of a Bravais lattice $\Lambda$. This means that there exists a unitary translation operator $T_{\mathbf{r}}$ such that $\left[H, T_{\mathbf{r}}\right]=0$ for all $\mathbf{r} \in \Lambda$. State and prove Bloch's theorem for $H$.

Consider the two-dimensional Bravais lattice $\Lambda$ defined by the basis vectors

$\mathbf{a}_{1}=\frac{a}{2}(\sqrt{3}, 1), \quad \mathbf{a}_{2}=\frac{a}{2}(\sqrt{3},-1)$

Find basis vectors $\mathbf{b}_{1}$ and $\mathbf{b}_{2}$ for the reciprocal lattice. Sketch the Brillouin zone. Explain why the Brillouin zone has only two physically distinct corners. Show that the positions of these corners may be taken to be $\mathbf{K}=\frac{1}{3}\left(2 \mathbf{b}_{1}+\mathbf{b}_{2}\right)$ and $\mathbf{K}^{\prime}=\frac{1}{3}\left(\mathbf{b}_{1}+2 \mathbf{b}_{2}\right)$.

The dynamics of a single electron moving on the lattice $\Lambda$ is described by a tightbinding model with Hamiltonian

$H=\sum_{\mathbf{r} \in \Lambda}\left[E_{0}|\mathbf{r}\rangle\langle\mathbf{r}|-\lambda\left(|\mathbf{r}\rangle\left\langle\mathbf{r}+\mathbf{a}_{1}|+| \mathbf{r}\right\rangle\left\langle\mathbf{r}+\mathbf{a}_{2}|+| \mathbf{r}+\mathbf{a}_{1}\right\rangle\left\langle\mathbf{r}|+| \mathbf{r}+\mathbf{a}_{2}\right\rangle\langle\mathbf{r}|\right)\right]$

where $E_{0}$ and $\lambda$ are real parameters. What is the energy spectrum as a function of the wave vector $\mathbf{k}$ in the Brillouin zone? How does the energy vary along the boundary of the Brillouin zone between $\mathbf{K}$ and $\mathbf{K}^{\prime}$ ? What is the width of the band?

In a real material, each site of the lattice $\Lambda$ contains an atom with a certain valency. Explain how the conducting properties of the material depend on the valency.

Suppose now that there is a second band, with minimum $E=E_{0}+\Delta$. For what values of $\Delta$ and the valency is the material an insulator?

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• # Paper 4, Section II, B

(a) A classical beam of particles scatters off a spherically symmetric potential $V(r)$. Draw a diagram to illustrate the differential cross-section $d \sigma / d \Omega$, and use this to derive an expression for $d \sigma / d \Omega$ in terms of the impact parameter $b$ and the scattering angle $\theta$.

A quantum beam of particles of mass $m$ and momentum $p=\hbar k$ is incident along the $z$-axis and scatters off a spherically symmetric potential $V(r)$. Write down the asymptotic form of the wavefunction $\psi$ in terms of the scattering amplitude $f(\theta)$. By considering the probability current $\mathbf{J}=-i(\hbar / 2 m)\left(\psi^{\star} \nabla \psi-\left(\nabla \psi^{\star}\right) \psi\right)$, derive an expression for the differential cross-section $d \sigma / d \Omega$ in terms of $f(\theta)$.

(b) The solution $\psi(\mathbf{r})$ of the radial Schrödinger equation for a particle of mass $m$ and wave number $k$ moving in a spherically symmetric potential $V(r)$ has the asymptotic form

$\psi(\mathbf{r}) \sim \sum_{l=0}^{\infty}\left[A_{l}(k) \frac{\sin \left(k r-\frac{l \pi}{2}\right)}{k r}-B_{l}(k) \frac{\cos \left(k r-\frac{l \pi}{2}\right)}{k r}\right] P_{l}(\cos \theta)$

valid for $k r \gg 1$, where $A_{l}(k)$ and $B_{l}(k)$ are constants and $P_{l}$ denotes the $l$ th Legendre polynomial. Define the S-matrix element $S_{l}$ and the corresponding phase shift $\delta_{l}$ for the partial wave of angular momentum $l$, in terms of $A_{l}(k)$ and $B_{l}(k)$. Define also the scattering length $a_{s}$ for the potential $V$.

Outside some core region, $r>r_{0}$, the Schrödinger equation for some such potential is solved by the s-wave (i.e. $l=0$ ) wavefunction $\psi(\mathbf{r})=\psi(r)$ with,

$\psi(r)=\frac{e^{-i k r}}{r}+\frac{k+i \lambda \tanh (\lambda r)}{k-i \lambda} \frac{e^{i k r}}{r}$

where $\lambda>0$ is a constant. Extract the S-matrix element $S_{0}$, the phase shift $\delta_{0}$ and the scattering length $a_{s}$. Deduce that the potential $V(r)$ has a bound state of zero angular momentum and compute its energy. Give the form of the (un-normalised) bound state wavefunction in the region $r>r_{0}$.

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• # Paper 1, Section II, A

A particle of mass $m$ moves in one dimension in a periodic potential $V(x)$ satisfying $V(x+a)=V(x)$. Define the Floquet matrix $F$. Show that $\operatorname{det} F=1$ and explain why Tr $F$ is real. Show that allowed bands occur for energies such that $(\operatorname{Tr} F)^{2}<4$. Consider the potential

$V(x)=-\frac{\hbar^{2} \lambda}{m} \sum_{n=-\infty}^{+\infty} \delta(x-n a)$

For states of negative energy, construct the Floquet matrix with respect to the basis of states $e^{\pm \mu x}$. Derive an inequality for the values of $\mu$ in an allowed energy band.

For states of positive energy, construct the Floquet matrix with respect to the basis of states $e^{\pm i k x}$. Derive an inequality for the values of $k$ in an allowed energy band.

Show that the state with zero energy lies in a forbidden region for $\lambda a>2$.

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• # Paper 2, Section II, A

Consider a one-dimensional chain of $2 N \gg 1$ atoms, each of mass $m$. Impose periodic boundary conditions. The forces between neighbouring atoms are modelled as springs, with alternating spring constants $\lambda$ and $\alpha \lambda$. In equilibrium, the separation between the atoms is $a$.

Denote the position of the $n^{\text {th }}$atom as $x_{n}(t)$. Let $u_{n}(t)=x_{n}(t)-n a$ be the displacement from equilibrium. Write down the equations of motion of the system.

Show that the longitudinal modes of vibration are labelled by a wavenumber $k$ that is restricted to lie in a Brillouin zone. Find the frequency spectrum. What is the frequency gap at the edge of the Brillouin zone? Show that the gap vanishes when $\alpha=1$. Determine approximations for the frequencies near the centre of the Brillouin zone. Plot the frequency spectrum. What is the speed of sound in this system?

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• # Paper 3, Section II, A

A beam of particles of mass $m$ and momentum $p=\hbar k$ is incident along the $z$-axis. The beam scatters off a spherically symmetric potential $V(r)$. Write down the asymptotic form of the wavefunction in terms of the scattering amplitude $f(\theta)$.

The incoming plane wave and the scattering amplitude can be expanded in partial waves as,

$\begin{gathered} e^{i k r \cos \theta} \sim \frac{1}{2 i k r} \sum_{l=0}^{\infty}(2 l+1)\left(e^{i k r}-(-1)^{l} e^{-i k r}\right) P_{l}(\cos \theta) \\ f(\theta)=\sum_{l=0}^{\infty} \frac{2 l+1}{k} f_{l} P_{l}(\cos \theta) \end{gathered}$

where $P_{l}$ are Legendre polynomials. Define the $S$-matrix. Assuming that the S-matrix is unitary, explain why we can write

$f_{l}=e^{i \delta_{l}} \sin \delta_{l}$

for some real phase shifts $\delta_{l}$. Obtain an expression for the total cross-section $\sigma_{T}$ in terms of the phase shifts $\delta_{l}$.

[Hint: You may use the orthogonality of Legendre polynomials:

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

Consider the repulsive, spherical potential

$V(r)=\left\{\begin{array}{cc} +V_{0} & ra \end{array}\right.$

where $V_{0}=\hbar^{2} \gamma^{2} / 2 m$. By considering the s-wave solution to the Schrödinger equation, show that

$\frac{\tan \left(k a+\delta_{0}\right)}{k a}=\frac{\tanh \left(\sqrt{\gamma^{2}-k^{2}} a\right)}{\sqrt{\gamma^{2}-k^{2}} a}$

For low momenta, $k a \ll 1$, compute the s-wave contribution to the total cross-section. Comment on the physical interpretation of your result in the limit $\gamma a \rightarrow \infty$.

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• # Paper 4, Section II, A

Define a Bravais lattice $\Lambda$ in three dimensions. Define the reciprocal lattice $\Lambda^{\star}$. Define the Brillouin zone.

An FCC lattice has a basis of primitive vectors given by

$\mathbf{a}_{1}=\frac{a}{2}\left(\mathbf{e}_{2}+\mathbf{e}_{3}\right), \quad \mathbf{a}_{2}=\frac{a}{2}\left(\mathbf{e}_{1}+\mathbf{e}_{3}\right), \quad \mathbf{a}_{3}=\frac{a}{2}\left(\mathbf{e}_{1}+\mathbf{e}_{2}\right),$

where $\mathbf{e}_{i}$ is an orthonormal basis of $\mathbb{R}^{3}$. Find a basis of reciprocal lattice vectors. What is the volume of the Brillouin zone?

The asymptotic wavefunction for a particle, of wavevector $\mathbf{k}$, scattering off a potential $V(\mathbf{r})$ is

$\psi(\mathbf{r}) \sim e^{i \mathbf{k} \cdot \mathbf{r}}+f_{\mathrm{V}}\left(\mathbf{k} ; \mathbf{k}^{\prime}\right) \frac{e^{i k r}}{r}$

where $\mathbf{k}^{\prime}=k \hat{\mathbf{r}}$ and $f_{\mathrm{V}}\left(\mathbf{k} ; \mathbf{k}^{\prime}\right)$ is the scattering amplitude. Give a formula for the Born approximation to the scattering amplitude.

Scattering of a particle off a single atom is modelled by a potential $V(\mathbf{r})=V_{0} \delta(r-d)$ with $\delta$-function support on a spherical shell, $r=|\mathbf{r}|=d$ centred at the origin. Calculate the Born approximation to the scattering amplitude, denoting the resulting expression as $\tilde{f}_{V}\left(\mathbf{k} ; \mathbf{k}^{\prime}\right)$.

Scattering of a particle off a crystal consisting of atoms located at the vertices of a lattice $\Lambda$ is modelled by a potential

$V_{\Lambda}=\sum_{\mathbf{R} \in \Lambda} V(\mathbf{r}-\mathbf{R})$

where $V(\mathbf{r})=V_{0} \delta(r-d)$ as above. Calculate the Born approximation to the scattering amplitude giving your answer in terms of your approximate expression $\tilde{f}_{\mathrm{V}}$ for scattering off a single atom. Show that the resulting amplitude vanishes unless the momentum transfer $\mathbf{q}=\mathbf{k}-\mathbf{k}^{\prime}$ lies in the reciprocal lattice $\Lambda^{\star}$.

For the particular FCC lattice considered above, show that, when $k=|\mathbf{k}|>2 \pi / a$, scattering occurs for two values of the scattering angle, $\theta_{1}$ and $\theta_{2}$, related by

$\frac{\sin \left(\frac{\theta_{1}}{2}\right)}{\sin \left(\frac{\theta_{2}}{2}\right)}=\frac{2}{\sqrt{3}}$

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• # Paper 1, Section II, C

A one-dimensional lattice has $N$ sites with lattice spacing $a$. In the tight-binding approximation, the Hamiltonian describing a single electron is given by

$H=E_{0} \sum_{n}|n\rangle\langle n|-J \sum_{n}(|n\rangle\langle n+1|+| n+1\rangle\langle n|)$

where $|n\rangle$ is the normalised state of the electron localised on the $n^{\text {th }}$lattice site. Using periodic boundary conditions $|N+1\rangle \equiv|1\rangle$, solve for the spectrum of this Hamiltonian to derive the dispersion relation

$E(k)=E_{0}-2 J \cos (k a)$

Define the Brillouin zone. Determine the number of states in the Brillouin zone.

Calculate the velocity $v$ and effective mass $m^{\star}$ of the particle. For which values of $k$ is the effective mass negative?

In the semi-classical approximation, derive an expression for the time-dependence of the position of the electron in a constant electric field.

Describe how the concepts of metals and insulators arise in the model above.

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• # Paper 2, Section II, C

Give an account of the variational method for establishing an upper bound on the ground-state energy of a Hamiltonian $H$ with a discrete spectrum $H|n\rangle=E_{n}|n\rangle$, where $E_{n} \leqslant E_{n+1}, n=0,1,2 \ldots$

A particle of mass $m$ moves in the three-dimensional potential

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

where $A, \mu>0$ are constants and $r$ is the distance to the origin. Using the normalised variational wavefunction

$\psi(r)=\sqrt{\frac{\alpha^{3}}{\pi}} e^{-\alpha r}$

show that the expected energy is given by

$E(\alpha)=\frac{\hbar^{2} \alpha^{2}}{2 m}-\frac{4 A \alpha^{3}}{(\mu+2 \alpha)^{2}}$

Explain why there is necessarily a bound state when $\mu. What can you say about the existence of a bound state when $\mu \geqslant A m / \hbar^{2}$ ?

[Hint: The Laplacian in spherical polar coordinates is

$\left.\nabla^{2}=\frac{1}{r^{2}} \frac{\partial}{\partial r}\left(r^{2} \frac{\partial}{\partial r}\right)+\frac{1}{r^{2} \sin \theta} \frac{\partial}{\partial \theta}\left(\sin \theta \frac{\partial}{\partial \theta}\right)+\frac{1}{r^{2} \sin ^{2} \theta} \frac{\partial^{2}}{\partial \phi^{2}}\right]$

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• # Paper 3, Section II, C

A particle of mass $m$ and charge $q$ moving in a uniform magnetic field $\mathbf{B}=\boldsymbol{\nabla} \times \mathbf{A}=$ $(0,0, B)$ is described by the Hamiltonian

$H=\frac{1}{2 m}(\mathbf{p}-q \mathbf{A})^{2}$

where $\mathbf{p}$ is the canonical momentum, which obeys $\left[x_{i}, p_{j}\right]=i \hbar \delta_{i j}$. The mechanical momentum is defined as $\boldsymbol{\pi}=\mathbf{p}-q \mathbf{A}$. Show that

$\left[\pi_{x}, \pi_{y}\right]=i q \hbar B$

Define

$a=\frac{1}{\sqrt{2 q \hbar B}}\left(\pi_{x}+i \pi_{y}\right) \quad \text { and } \quad a^{\dagger}=\frac{1}{\sqrt{2 q \hbar B}}\left(\pi_{x}-i \pi_{y}\right) \text {. }$

Derive the commutation relation obeyed by $a$ and $a^{\dagger}$. Write the Hamiltonian in terms of $a$ and $a^{\dagger}$ and hence solve for the spectrum.

In symmetric gauge, states in the lowest Landau level with $k_{z}=0$ have wavefunctions

$\psi(x, y)=(x+i y)^{M} e^{-q B r^{2} / 4 \hbar}$

where $r^{2}=x^{2}+y^{2}$ and $M$ is a positive integer. By considering the profiles of these wavefunctions, estimate how many lowest Landau level states can fit in a disc of radius $R$.

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• # Paper 4, Section II, C

(a) In one dimension, a particle of mass $m$ is scattered by a potential $V(x)$ where $V(x) \rightarrow 0$ as $|x| \rightarrow \infty$. For wavenumber $k>0$, the incoming $(\mathcal{I})$ and outgoing $(\mathcal{O})$ asymptotic plane wave states with positive $(+)$ and negative $(-)$ parity are given by

$\begin{array}{rr} \mathcal{I}_{+}(x)=e^{-i k|x|}, & \mathcal{I}_{-}(x)=\operatorname{sign}(x) e^{-i k|x|} \\ \mathcal{O}_{+}(x)=e^{+i k|x|}, & \mathcal{O}_{-}(x)=-\operatorname{sign}(x) e^{+i k|x|} \end{array}$

(i) Explain how this basis may be used to define the $S$-matrix,

$\mathcal{S}^{P}=\left(\begin{array}{cc} S_{++} & S_{+-} \\ S_{-+} & S_{--} \end{array}\right)$

(ii) For what choice of potential would you expect $S_{+-}=S_{-+}=0$ ? Why?

(b) The potential $V(x)$ is given by

$V(x)=V_{0}[\delta(x-a)+\delta(x+a)]$

with $V_{0}$ a constant.

(i) Show that

$S_{--}(k)=e^{-2 i k a}\left[\frac{\left(2 k-i U_{0}\right) e^{i k a}+i U_{0} e^{-i k a}}{\left(2 k+i U_{0}\right) e^{-i k a}-i U_{0} e^{i k a}}\right]$

where $U_{0}=2 m V_{0} / \hbar^{2}$. Verify that $\left|S_{--}\right|^{2}=1$. Explain the physical meaning of this result.

(ii) For $V_{0}<0$, by considering the poles or zeros of $S_{--}(k)$, show that there exists one bound state of negative parity if $a U_{0}<-1$.

(iii) For $V_{0}>0$ and $a U_{0} \gg 1$, show that $S_{--}(k)$ has a pole at

$k a=\pi+\alpha-i \gamma$

where $\alpha$ and $\gamma$ are real and

$\alpha=-\frac{\pi}{a U_{0}}+O\left(\frac{1}{\left(a U_{0}\right)^{2}}\right) \quad \text { and } \quad \gamma=\left(\frac{\pi}{a U_{0}}\right)^{2}+O\left(\frac{1}{\left(a U_{0}\right)^{3}}\right)$

Explain the physical significance of this result.

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• # Paper 1, Section II, A

A particle in one dimension of mass $m$ and energy $E=\hbar^{2} k^{2} / 2 m(k>0)$ is incident from $x=-\infty$ on a potential $V(x)$ with $V(x) \rightarrow 0$ as $x \rightarrow-\infty$ and $V(x)=\infty$ for $x>0$. The relevant solution of the time-independent Schrödinger equation has the asymptotic form

$\psi(x) \sim \exp (i k x)+r(k) \exp (-i k x), \quad x \rightarrow-\infty$

Explain briefly why a pole in the reflection amplitude $r(k)$ at $k=i \kappa$ with $\kappa>0$ corresponds to the existence of a stable bound state in this potential. Indicate why a pole in $r(k)$ just below the real $k$-axis, at $k=k_{0}-i \rho$ with $k_{0} \gg \rho>0$, corresponds to a quasi-stable bound state. Find an approximate expression for the lifetime $\tau$ of such a quasi-stable state.

Now suppose that

$V(x)= \begin{cases}\left(\hbar^{2} U / 2 m\right) \delta(x+a) & \text { for } x<0 \\ \infty & \text { for } x>0\end{cases}$

where $U>0$ and $a>0$ are constants. Compute the reflection amplitude $r(k)$ in this case and deduce that there are quasi-stable bound states if $U$ is large. Give expressions for the wavefunctions and energies of these states and compute their lifetimes, working to leading non-vanishing order in $1 / U$ for each expression.

[ You may assume $\psi=0$ for $x \geqslant 0$ and $\lim _{\epsilon \rightarrow 0+}\left\{\psi^{\prime}(-a+\epsilon)-\psi^{\prime}(-a-\epsilon)\right\}=U \psi(-a)$.]

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• # Paper 2, Section II, A

A particle of mass $m$ moves in three dimensions subject to a potential $V(\mathbf{r})$ localised near the origin. The wavefunction for a scattering process with incident particle of wavevector $\mathbf{k}$ is denoted $\psi(\mathbf{k}, \mathbf{r})$. With reference to the asymptotic form of $\psi$, define the scattering amplitude $f\left(\mathbf{k}, \mathbf{k}^{\prime}\right)$, where $\mathbf{k}^{\prime}$ is the wavevector of the outgoing particle with $\left|\mathbf{k}^{\prime}\right|=|\mathbf{k}|=k$.

By recasting the Schrödinger equation for $\psi(\mathbf{k}, \mathbf{r})$ as an integral equation, show that

$f\left(\mathbf{k}, \mathbf{k}^{\prime}\right)=-\frac{m}{2 \pi \hbar^{2}} \int d^{3} \mathbf{r}^{\prime} \exp \left(-i \mathbf{k}^{\prime} \cdot \mathbf{r}^{\prime}\right) V\left(\mathbf{r}^{\prime}\right) \psi\left(\mathbf{k}, \mathbf{r}^{\prime}\right)$

[You may assume that

$\mathcal{G}(k ; \mathbf{r})=-\frac{1}{4 \pi|\mathbf{r}|} \exp (i k|\mathbf{r}|)$

is the Green's function for $\nabla^{2}+k^{2}$ which obeys the appropriate boundary conditions for a scattering solution.]

Now suppose $V(\mathbf{r})=\lambda U(\mathbf{r})$, where $\lambda \ll 1$ is a dimensionless constant. Determine the first two non-zero terms in the expansion of $f\left(\mathbf{k}, \mathbf{k}^{\prime}\right)$ in powers of $\lambda$, giving each term explicitly as an integral over one or more position variables $\mathbf{r}, \mathbf{r}^{\prime}, \ldots$

Evaluate the contribution to $f\left(\mathbf{k}, \mathbf{k}^{\prime}\right)$ of order $\lambda$ in the case $U(\mathbf{r})=\delta(|\mathbf{r}|-a)$, expressing the answer as a function of $a, k$ and the scattering angle $\theta$ (defined so that $\left.\mathbf{k} \cdot \mathbf{k}^{\prime}=k^{2} \cos \theta\right)$.

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• # Paper 3, Section II, A

(a) A spinless charged particle moves in an electromagnetic field defined by vector and scalar potentials $\mathbf{A}(\mathbf{x}, t)$ and $\phi(\mathbf{x}, t)$. The wavefunction $\psi(\mathbf{x}, t)$ for the particle satisfies the time-dependent Schrödinger equation with Hamiltonian

$\hat{H}_{0}=\frac{1}{2 m}(-i \hbar \boldsymbol{\nabla}+e \mathbf{A}) \cdot(-i \hbar \boldsymbol{\nabla}+e \mathbf{A})-e \phi .$

Consider a gauge transformation

$\mathbf{A} \rightarrow \tilde{\mathbf{A}}=\mathbf{A}+\nabla f, \quad \phi \rightarrow \tilde{\phi}=\phi-\frac{\partial f}{\partial t}, \quad \psi \rightarrow \tilde{\psi}=\exp \left(-\frac{i e f}{\hbar}\right) \psi$

for some function $f(\mathbf{x}, t)$. Define covariant derivatives with respect to space and time, and show that $\tilde{\psi}$ satisfies the Schrödinger equation with potentials $\tilde{\mathbf{A}}$ and $\tilde{\phi}$.

(b) Suppose that in part (a) the magnetic field has the form $\mathbf{B}=\boldsymbol{\nabla} \times \mathbf{A}=(0,0, B)$, where $B$ is a constant, and that $\phi=0$. Find a suitable $\mathbf{A}$ with $A_{y}=A_{z}=0$ and determine the energy levels of the Hamiltonian $\hat{H}_{0}$ when the $z$-component of the momentum of the particle is zero. Suppose in addition that the particle is constrained to lie in a rectangular region of area $\mathcal{A}$ in the $(x, y)$-plane. By imposing periodic boundary conditions in the $x$-direction, estimate the degeneracy of each energy level. [You may use without proof results for a quantum harmonic oscillator, provided they are clearly stated.]

(c) An electron is a charged particle of spin $\frac{1}{2}$ with a two-component wavefunction $\psi(\mathbf{x}, t)$ governed by the Hamiltonian

$\hat{H}=\hat{H}_{0} \mathbb{I}_{2}+\frac{e \hbar}{2 m} \mathbf{B} \cdot \boldsymbol{\sigma}$

where $\mathbb{I}_{2}$ is the $2 \times 2$ unit matrix and $\sigma=\left(\sigma_{1}, \sigma_{2}, \sigma_{3}\right)$ denotes the Pauli matrices. Find the energy levels for an electron in the constant magnetic field defined in part (b), assuming as before that the $z$-component of the momentum of the particle is zero.

Consider $N$ such electrons confined to the rectangular region defined in part (b). Ignoring interactions between the electrons, show that the ground state energy of this system vanishes for $N$ less than some integer $N_{\max }$ which you should determine. Find the ground state energy for $N=(2 p+1) N_{\max }$, where $p$ is a positive integer.

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• # Paper 4, Section II, A

Let $\Lambda \subset \mathbb{R}^{2}$ be a Bravais lattice. Define the dual lattice $\Lambda^{*}$ and show that

$V(\mathbf{x})=\sum_{\mathbf{q} \in \Lambda^{*}} V_{\mathbf{q}} \exp (i \mathbf{q} \cdot \mathbf{x})$

obeys $V(\mathbf{x}+l)=V(\mathbf{x})$ for all $l \in \Lambda$, where $V_{\mathbf{q}}$ are constants. Suppose $V(\mathbf{x})$ is the potential for a particle of mass $m$ moving in a two-dimensional crystal that contains a very large number of lattice sites of $\Lambda$ and occupies an area $\mathcal{A}$. Adopting periodic boundary conditions, plane-wave states $|\mathbf{k}\rangle$ can be chosen such that

$\langle\mathbf{x} \mid \mathbf{k}\rangle=\frac{1}{\mathcal{A}^{1 / 2}} \exp (i \mathbf{k} \cdot \mathbf{x}) \quad \text { and } \quad\left\langle\mathbf{k} \mid \mathbf{k}^{\prime}\right\rangle=\delta_{\mathbf{k} \mathbf{k}^{\prime}}$

The allowed wavevectors $\mathbf{k}$ are closely spaced and include all vectors in $\Lambda^{*}$. Find an expression for the matrix element $\left\langle\mathbf{k}|V(\mathbf{x})| \mathbf{k}^{\prime}\right\rangle$ in terms of the coefficients $V_{\mathbf{q}}$. [You need not discuss additional details of the boundary conditions.]

Now suppose that $V(\mathbf{x})=\lambda U(\mathbf{x})$, where $\lambda \ll 1$ is a dimensionless constant. Find the energy $E(\mathbf{k})$ for a particle with wavevector $\mathbf{k}$ to order $\lambda^{2}$ in non-degenerate perturbation theory. Show that this expansion in $\lambda$ breaks down on the Bragg lines in k-space defined by the condition

$\mathbf{k} \cdot \mathbf{q}=\frac{1}{2}|\mathbf{q}|^{2} \quad \text { for } \quad \mathbf{q} \in \Lambda^{*}$

and explain briefly, without additional calculations, the significance of this for energy levels in the crystal.

Consider the particular case in which $\Lambda$ has primitive vectors

$\mathbf{a}_{1}=2 \pi\left(\mathbf{i}+\frac{1}{\sqrt{3}} \mathbf{j}\right), \quad \mathbf{a}_{2}=2 \pi \frac{2}{\sqrt{3}} \mathbf{j}$

where $\mathbf{i}$ and $\mathbf{j}$ are orthogonal unit vectors. Determine the polygonal region in $\mathbf{k}$-space corresponding to the lowest allowed energy band.

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• # Paper 1, Section II, A

Define the Rayleigh-Ritz quotient $R[\psi]$ for a normalisable state $|\psi\rangle$ of a quantum system with Hamiltonian $H$. Given that the spectrum of $H$ is discrete and that there is a unique ground state of energy $E_{0}$, show that $R[\psi] \geqslant E_{0}$ and that equality holds if and only if $|\psi\rangle$ is the ground state.

A simple harmonic oscillator (SHO) is a particle of mass $m$ moving in one dimension subject to the potential

$V(x)=\frac{1}{2} m \omega^{2} x^{2}$

Estimate the ground state energy $E_{0}$ of the SHO by using the ground state wavefunction for a particle in an infinite potential well of width $a$, centred on the origin (the potential is $U(x)=0$ for $|x| and $U(x)=\infty$ for $|x|>a / 2)$. Take $a$ as the variational parameter.

Perform a similar estimate for the energy $E_{1}$ of the first excited state of the SHO by using the first excited state of the infinite potential well as a trial wavefunction.

Is the estimate for $E_{1}$ necessarily an upper bound? Justify your answer.

$\left[\right.$ You may use : $\int_{-\pi / 2}^{\pi / 2} y^{2} \cos ^{2} y d y=\frac{\pi}{4}\left(\frac{\pi^{2}}{6}-1\right) \quad$ and $\left.\quad \int_{-\pi}^{\pi} y^{2} \sin ^{2} y d y=\pi\left(\frac{\pi^{2}}{3}-\frac{1}{2}\right) \cdot\right]$

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• # Paper 2, Section II, A

A beam of particles of mass $m$ and energy $\hbar^{2} k^{2} / 2 m$ is incident on a target at the origin described by a spherically symmetric potential $V(r)$. Assuming the potential decays rapidly as $r \rightarrow \infty$, write down the asymptotic form of the wavefunction, defining the scattering amplitude $f(\theta)$.

Consider a free particle with energy $\hbar^{2} k^{2} / 2 m$. State, without proof, the general axisymmetric solution of the Schrödinger equation for $r>0$ in terms of spherical Bessel and Neumann functions $j_{\ell}$ and $n_{\ell}$, and Legendre polynomials $P_{\ell}(\ell=0,1,2, \ldots)$. Hence define the partial wave phase shifts $\delta_{\ell}$ for scattering from a potential $V(r)$ and derive the partial wave expansion for $f(\theta)$ in terms of phase shifts.

Now suppose

$V(r)=\left\{\begin{array}{cc} \hbar^{2} \gamma^{2} / 2 m & ra \end{array}\right.$

with $\gamma>k$. Show that the S-wave phase shift $\delta_{0}$ obeys

$\frac{\tanh (\kappa a)}{\kappa a}=\frac{\tan \left(k a+\delta_{0}\right)}{k a}$

where $\kappa^{2}=\gamma^{2}-k^{2}$. Deduce that for an S-wave solution

$f \rightarrow \frac{\tanh \gamma a-\gamma a}{\gamma} \quad \text { as } \quad k \rightarrow 0$

[You may assume : $\quad \exp (i k r \cos \theta)=\sum_{\ell=0}^{\infty}(2 \ell+1) i^{\ell} j_{\ell}(k r) P_{\ell}(\cos \theta)$

and $j_{\ell}(\rho) \sim \frac{1}{\rho} \sin (\rho-\ell \pi / 2), \quad n_{\ell}(\rho) \sim-\frac{1}{\rho} \cos (\rho-\ell \pi / 2) \quad$ as $\left.\quad \rho \rightarrow \infty .\right]$

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• # Paper 3, Section II, A

A particle of mass $m$ and energy $E=-\hbar^{2} \kappa^{2} / 2 m<0$ moves in one dimension subject to a periodic potential

$V(x)=-\frac{\hbar^{2} \lambda}{m} \sum_{\ell=-\infty}^{\infty} \delta(x-\ell a) \quad \text { with } \quad \lambda>0 .$

Determine the corresponding Floquet matrix $\mathcal{M}$. [You may assume without proof that for the Schrödinger equation with potential $\alpha \delta(x)$ the wavefunction $\psi(x)$ is continuous at $x=0$ and satisfies $\left.\psi^{\prime}(0+)-\psi^{\prime}(0-)=\left(2 m \alpha / \hbar^{2}\right) \psi(0) .\right]$

Explain briefly, with reference to Bloch's theorem, how restrictions on the energy of a Bloch state can be derived from $\mathcal{M}$. Deduce that for the potential $V(x)$ above, $\kappa$ is confined to a range whose boundary values are determined by

$\tanh \left(\frac{\kappa a}{2}\right)=\frac{\kappa}{\lambda} \quad \text { and } \quad \operatorname{coth}\left(\frac{\kappa a}{2}\right)=\frac{\kappa}{\lambda} .$

Sketch the left-hand and right-hand sides of each of these equations as functions of $y=\kappa a / 2$. Hence show that there is exactly one allowed band of negative energies with either (i) $E_{-} \leqslant E<0$ or (ii) $E_{-} \leqslant E \leqslant E_{+}<0$ and determine the values of $\lambda a$ for which each of these cases arise. [You should not attempt to evaluate the constants $E_{\pm} .$]

Comment briefly on the limit $a \rightarrow \infty$ with $\lambda$ fixed.

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• # Paper 4, Section II,

Let $\Lambda$ be a Bravais lattice with basis vectors $\mathbf{a}_{1}, \mathbf{a}_{2}, \mathbf{a}_{3}$. Define the reciprocal lattice $\Lambda^{*}$ and write down basis vectors $\mathbf{b}_{1}, \mathbf{b}_{2}, \mathbf{b}_{3}$ for $\Lambda^{*}$ in terms of the basis for $\Lambda$.

A finite crystal consists of identical atoms at sites of $\Lambda$ given by

$\ell=n_{1} \mathbf{a}_{1}+n_{2} \mathbf{a}_{2}+n_{3} \mathbf{a}_{3} \quad \text { with } \quad 0 \leqslant n_{i}

A particle of mass $m$ scatters off the crystal; its wavevector is $\mathbf{k}$ before scattering and $\mathbf{k}^{\prime}$ after scattering, with $|\mathbf{k}|=\left|\mathbf{k}^{\prime}\right|$. Show that the scattering amplitude in the Born approximation has the form

$-\frac{m}{2 \pi \hbar^{2}} \Delta(\mathbf{q}) \tilde{U}(\mathbf{q}), \quad \mathbf{q}=\mathbf{k}^{\prime}-\mathbf{k}$

where $U(\mathbf{x})$ is the potential due to a single atom at the origin and $\Delta(\mathbf{q})$ depends on the crystal structure. [You may assume that in the Born approximation the amplitude for scattering off a potential $V(\mathbf{x})$ is $-\left(m / 2 \pi \hbar^{2}\right) \tilde{V}(\mathbf{q})$ where tilde denotes the Fourier transform.]

Derive an expression for $|\Delta(\mathbf{q})|$ that is valid when $e^{-i \mathbf{q} \cdot \mathbf{a}_{i}} \neq 1$. Show also that when $\mathbf{q}$ is a reciprocal lattice vector $|\Delta(\mathbf{q})|$ is equal to the total number of atoms in the crystal. Comment briefly on the significance of these results.

Now suppose that $\Lambda$ is a face-centred-cubic lattice:

$\mathbf{a}_{1}=\frac{a}{2}(\hat{\mathbf{y}}+\hat{\mathbf{z}}), \quad \mathbf{a}_{2}=\frac{a}{2}(\hat{\mathbf{z}}+\hat{\mathbf{x}}), \quad \mathbf{a}_{3}=\frac{a}{2}(\hat{\mathbf{x}}+\hat{\mathbf{y}})$

where $a$ is a constant. Show that for a particle incident with $|\mathbf{k}|>2 \pi / a$, enhanced scattering is possible for at least two values of the scattering angle, $\theta_{1}$ and $\theta_{2}$, related by

$\frac{\sin \left(\theta_{1} / 2\right)}{\sin \left(\theta_{2} / 2\right)}=\frac{\sqrt{3}}{2}$

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• # Paper 1, Section II, A

A particle of mass $m$ scatters on a localised potential well $V(x)$ in one dimension. With reference to the asymptotic behaviour of the wavefunction as $x \rightarrow \pm \infty$, define the reflection and transmission amplitudes, $r$ and $t$, for a right-moving incident particle of wave number $k$. Define also the corresponding amplitudes, $r^{\prime}$ and $t^{\prime}$, for a left-moving incident particle of wave number $k$. Derive expressions for $r^{\prime}$ and $t^{\prime}$ in terms of $r$ and $t$.

(a) Define the $S$-matrix, giving its elements in terms of $r$ and $t$. Using the relation

$|r|^{2}+|t|^{2}=1$

(which you need not derive), show that the S-matrix is unitary. How does the S-matrix simplify if the potential well satisfies $V(-x)=V(x)$ ?

(b) Consider the potential well

$V(x)=-\frac{3 \hbar^{2}}{m} \frac{1}{\cosh ^{2}(x)}$

The corresponding Schrödinger equation has an exact solution

$\psi_{k}(x)=\exp (i k x)\left[3 \tanh ^{2}(x)-3 i k \tanh (x)-\left(1+k^{2}\right)\right]$

with energy $E=\hbar^{2} k^{2} / 2 m$, for every real value of $k$. [You do not need to verify this.] Find the S-matrix for scattering on this potential. What special feature does the scattering have in this case?

(c) Explain the connection between singularities of the S-matrix and bound states of the potential well. By analytic continuation of the solution $\psi_{k}(x)$ to appropriate complex values of $k$, find the wavefunctions and energies of the bound states of the well. [You do not need to normalise the wavefunctions.]

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• # Paper 2, Section II, A

(a) A classical particle of mass $m$ scatters on a central potential $V(r)$ with energy $E$, impact parameter $b$, and scattering angle $\theta$. Define the corresponding differential cross-section.

For particle trajectories in the Coulomb potential,

$V_{C}(r)=\frac{e^{2}}{4 \pi \epsilon_{0} r}$

the impact parameter is given by

$b=\frac{e^{2}}{8 \pi \epsilon_{0} E} \cot \left(\frac{\theta}{2}\right)$