• # Paper 1, Section II, B

(a) A group $G$ of transformations acts on a quantum system. Briefly explain why the Born rule implies that these transformations may be represented by operators $U(g): \mathcal{H} \rightarrow \mathcal{H}$ obeying

\begin{aligned} U(g)^{\dagger} U(g) &=1_{\mathcal{H}} \\ U\left(g_{1}\right) U\left(g_{2}\right) &=e^{i \phi\left(g_{1}, g_{2}\right)} U\left(g_{1} \cdot g_{2}\right) \end{aligned}

for all $g_{1}, g_{2} \in G$, where $\phi\left(g_{1}, g_{2}\right) \in \mathbb{R}$.

What additional property does $U(g)$ have when $G$ is a group of symmetries of the Hamiltonian? Show that symmetries correspond to conserved quantities.

(b) The Coulomb Hamiltonian describing the gross structure of the hydrogen atom is invariant under time reversal, $t \mapsto-t$. Suppose we try to represent time reversal by a unitary operator $T$ obeying $U(t) T=T U(-t)$, where $U(t)$ is the time-evolution operator. Show that this would imply that hydrogen has no stable ground state.

An operator $A: \mathcal{H} \rightarrow \mathcal{H}$ is antilinear if

$A(a|\alpha\rangle+b|\beta\rangle)=\bar{a} A|\alpha\rangle+\bar{b} A|\beta\rangle$

for all $|\alpha\rangle,|\beta\rangle \in \mathcal{H}$ and all $a, b \in \mathbb{C}$, and antiunitary if, in addition,

$\left\langle\beta^{\prime} \mid \alpha^{\prime}\right\rangle=\overline{\langle\beta \mid \alpha\rangle},$

where $\left|\alpha^{\prime}\right\rangle=A|\alpha\rangle$ and $\left|\beta^{\prime}\right\rangle=A|\beta\rangle$. Show that if time reversal is instead represented by an antiunitary operator then the above instability of hydrogen is avoided.

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• # Paper 2, Section II, B

(a) Let $\{|n\rangle\}$ be a basis of eigenstates of a non-degenerate Hamiltonian $H$, with corresponding eigenvalues $\left\{E_{n}\right\}$. Write down an expression for the energy levels of the perturbed Hamiltonian $H+\lambda \Delta H$, correct to second order in the dimensionless constant $\lambda \ll 1$.

(b) A particle travels in one dimension under the influence of the potential

$V(X)=\frac{1}{2} m \omega^{2} X^{2}+\lambda \hbar \omega \frac{X^{3}}{L^{3}}$

where $m$ is the mass, $\omega$ a frequency and $L=\sqrt{\hbar / 2 m \omega}$ a length scale. Show that, to first order in $\lambda$, all energy levels coincide with those of the harmonic oscillator. Calculate the energy of the ground state to second order in $\lambda$.

Does perturbation theory in $\lambda$ converge for this potential? Briefly explain your answer.

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• # Paper 3, Section II, B

(a) A quantum system with total angular momentum $j_{1}$ is combined with another of total angular momentum $j_{2}$. What are the possible values of the total angular momentum $j$ of the combined system? For given $j$, what are the possible values of the angular momentum along any axis?

(b) Consider the case $j_{1}=j_{2}$. Explain why all the states with $j=2 j_{1}-1$ are antisymmetric under exchange of the angular momenta of the two subsystems, while all the states with $j=2 j_{1}-2$ are symmetric.

(c) An exotic particle $X$ of spin 0 and negative intrinsic parity decays into a pair of indistinguishable particles $Y$. Assume each $Y$ particle has spin 1 and that the decay process conserves parity. Find the probability that the direction of travel of the $Y$ particles is observed to lie at an angle $\theta \in(\pi / 4,3 \pi / 4)$ from some axis along which their total spin is observed to be $+\hbar$ ?

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• # Paper 4, Section II, 33B

(a) A quantum system has Hamiltonian $H=H_{0}+V(t)$. Let $\{|n\rangle\}_{n \in \mathbb{N}_{0}}$ be an orthonormal basis of $H_{0}$ eigenstates, with corresponding energies $E_{n}=\hbar \omega_{n}$. For $t<0$, $V(t)=0$ and the system is in state $|0\rangle$. Calculate the probability that it is found to be in state $|1\rangle$ at time $t>0$, correct to lowest non-trivial order in $V$.

(b) Now suppose $\{|0\rangle,|1\rangle\}$ form a basis of the Hilbert space, with respect to which

$\left(\begin{array}{cc} \langle 0|H| 0\rangle & \langle 0|H| 1\rangle \\ \langle 1|H| 0\rangle & \langle 1|H| 1\rangle \end{array}\right)=\left(\begin{array}{cc} \hbar \omega_{0} & \hbar v \Theta(t) e^{i \omega t} \\ \hbar v \Theta(t) e^{-i \omega t} & \hbar \omega_{1} \end{array}\right)$

where $\Theta(t)$ is the Heaviside step function and $v$ is a real constant. Calculate the exact probability that the system is in state $|1\rangle$ at time $t$. For which frequency $\omega$ is this probability maximized?

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• # Paper 1, Section II, A

Let $A=(m \omega X+i P) / \sqrt{2 m \hbar \omega}$ be the lowering operator of a one dimensional quantum harmonic oscillator of mass $m$ and frequency $\omega$, and let $|0\rangle$ be the ground state defined by $A|0\rangle=0$.

a) Evaluate the commutator $\left[A, A^{\dagger}\right]$.

b) For $\gamma \in \mathbb{R}$, let $S(\gamma)$ be the unitary operator $S(\gamma)=\exp \left(-\frac{\gamma}{2}\left(A^{\dagger} A^{\dagger}-A A\right)\right)$ and define $A(\gamma)=S^{\dagger}(\gamma) A S(\gamma)$. By differentiating with respect to $\gamma$ or otherwise, show that

$A(\gamma)=A \cosh \gamma-A^{\dagger} \sinh \gamma$

c) The ground state of the harmonic oscillator saturates the uncertainty relation $\Delta X \Delta P \geqslant \hbar / 2$. Compute $\Delta X \Delta P$ when the oscillator is in the state $|\gamma\rangle=S(\gamma)|0\rangle$.

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• # Paper 2, Section II, A

(a) Consider the Hamiltonian $H(t)=H_{0}+\delta H(t)$, where $H_{0}$ is time-independent and non-degenerate. The system is prepared to be in some state $|\psi\rangle=\sum_{r} a_{r}|r\rangle$ at time $t=0$, where $\{|r\rangle\}$ is an orthonormal basis of eigenstates of $H_{0}$. Derive an expression for the state at time $t$, correct to first order in $\delta H(t)$, giving your answer in the interaction picture.

(b) An atom is modelled as a two-state system, where the excited state $|e\rangle$ has energy $\hbar \Omega$ above that of the ground state $|g\rangle$. The atom interacts with an electromagnetic field, modelled as a harmonic oscillator of frequency $\omega$. The Hamiltonian is $H(t)=H_{0}+\delta H(t)$, where

$H_{0}=\frac{\hbar \Omega}{2}(|e\rangle\langle e|-| g\rangle\langle g|) \otimes 1_{\text {field }}+1_{\text {atom }} \otimes \hbar \omega\left(A^{\dagger} A+\frac{1}{2}\right)$

is the Hamiltonian in the absence of interactions and

$\delta H(t)= \begin{cases}0, & t \leqslant 0 \\ \frac{1}{2} \hbar(\Omega-\omega)\left(|e\rangle\langle g|\otimes A+\beta| g\rangle\langle e| \otimes A^{\dagger}\right), & t>0\end{cases}$

describes the coupling between the atom and the field.

(i) Interpret each of the two terms in $\delta H(t)$. What value must the constant $\beta$ take for time evolution to be unitary?

(ii) At $t=0$ the atom is in state $(|e\rangle+|g\rangle) / \sqrt{2}$ while the field is described by the (normalized) state $e^{-1 / 2} e^{-A^{\dagger}}|0\rangle$ of the oscillator. Calculate the probability that at time $t$ the atom will be in its excited state and the field will be described by the $n^{\text {th }}$excited state of the oscillator. Give your answer to first non-trivial order in perturbation theory. Show that this probability vanishes when $t=\pi /(\Omega-\omega)$.

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• # Paper 3, Section II, 33A

Explain what is meant by the terms boson and fermion.

Three distinguishable spin-1 particles are governed by the Hamiltonian

$H=\frac{2 \lambda}{\hbar^{2}}\left(\mathbf{S}_{1} \cdot \mathbf{S}_{2}+\mathbf{S}_{2} \cdot \mathbf{S}_{3}+\mathbf{S}_{3} \cdot \mathbf{S}_{1}\right)$

where $\mathbf{S}_{i}$ is the spin operator of particle $i$ and $\lambda$ is a positive constant. How many spin states are possible altogether? By considering the total spin operator, determine the eigenvalues and corresponding degeneracies of the Hamiltonian.

Now consider the case that all three particles are indistinguishable and all have the same spatial wavefunction. What are the degeneracies of the Hamiltonian in this case?

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• # Paper 4, Section II,

Briefly explain why the density operator $\rho$ obeys $\rho \geqslant 0$ and $\operatorname{Tr}(\rho)=1$. What is meant by a pure state and a mixed state?

A two-state system evolves under the Hamiltonian $H=\hbar \boldsymbol{\omega} \cdot \boldsymbol{\sigma}$, where $\boldsymbol{\omega}$ is a constant vector and $\sigma$ are the Pauli matrices. At time $t$ the system is described by a density operator

$\rho(t)=\frac{1}{2}\left(1_{\mathcal{H}}+\mathbf{a}(t) \cdot \boldsymbol{\sigma}\right)$

where $1_{\mathcal{H}}$ is the identity operator. Initially, the vector $\mathbf{a}(0)=\mathbf{a}$ obeys $|\mathbf{a}|<1$ and $\mathbf{a} \cdot \boldsymbol{\omega}=0$. Find $\rho(t)$ in terms of a and $\boldsymbol{\omega}$. At what time, if any, is the system definitely in the state $\left|\uparrow_{x}\right\rangle$ that obeys $\sigma_{x}\left|\uparrow_{x}\right\rangle=+\left|\uparrow_{x}\right\rangle ?$

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• # Paper 1, Section II, B

A $d=3$ isotropic harmonic oscillator of mass $\mu$ and frequency $\omega$ has lowering operators

$\mathbf{A}=\frac{1}{\sqrt{2 \mu \hbar \omega}}(\mu \omega \mathbf{X}+\mathrm{i} \mathbf{P})$

where $\mathbf{X}$ and $\mathbf{P}$ are the position and momentum operators. Assuming the standard commutation relations for $\mathbf{X}$ and $\mathbf{P}$, evaluate the commutators $\left[A_{i}^{\dagger}, A_{j}^{\dagger}\right],\left[A_{i}, A_{j}\right]$ and $\left[A_{i}, A_{j}^{\dagger}\right]$, for $i, j=1,2,3$, among the components of the raising and lowering operators.

How is the ground state $|\mathbf{0}\rangle$ of the oscillator defined? How are normalised higher excited states obtained from $|\mathbf{0}\rangle$ ? [You should determine the appropriate normalisation constant for each energy eigenstate.]

By expressing the orbital angular momentum operator $\mathbf{L}$ in terms of the raising and lowering operators, show that each first excited state of the isotropic oscillator has total orbital angular momentum quantum number $\ell=1$, and find a linear combination $|\psi\rangle$ of these first excited states obeying $L_{z}|\psi\rangle=+\hbar|\psi\rangle$ and $\||\psi\rangle \|=1$.

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• # Paper 2, Section II, B

(a) Let $|i\rangle$ and $|j\rangle$ be two eigenstates of a time-independent Hamiltonian $H_{0}$, separated in energy by $\hbar \omega_{i j}$. At time $t=0$ the system is perturbed by a small, time independent operator $V$. The perturbation is turned off at time $t=T$. Show that if the system is initially in state $|i\rangle$, the probability of a transition to state $|j\rangle$ is approximately

$P_{i j}=4|\langle i|V| j\rangle|^{2} \frac{\sin ^{2}\left(\omega_{i j} T / 2\right)}{\left(\hbar \omega_{i j}\right)^{2}}$

(b) An uncharged particle with spin one-half and magnetic moment $\mu$ travels at speed $v$ through a region of uniform magnetic field $\mathbf{B}=(0,0, B)$. Over a length $L$ of its path, an additional perpendicular magnetic field $b$ is applied. The spin-dependent part of the Hamiltonian is

$H(t)= \begin{cases}-\mu\left(B \sigma_{z}+b \sigma_{x}\right) & \text { while } 0

where $\sigma_{z}$ and $\sigma_{x}$ are Pauli matrices. The particle initially has its spin aligned along the direction of $\mathbf{B}=(0,0, B)$. Find the probability that it makes a transition to the state with opposite spin

(i) by assuming $b \ll B$ and using your result from part (a),

(ii) by finding the exact evolution of the state.

[Hint: for any 3-vector $\mathbf{a}, e^{i \mathbf{a} \cdot \boldsymbol{\sigma}}=(\cos a) I+(i \sin a) \hat{\mathbf{a}} \cdot \boldsymbol{\sigma}$, where $I$ is the $2 \times 2$ unit matrix, $\boldsymbol{\sigma}=\left(\sigma_{x}, \sigma_{y}, \sigma_{z}\right), \quad a=|\mathbf{a}|$ and $\left.\hat{\mathbf{a}}=\mathbf{a} /|\mathbf{a}| .\right]$

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• # Paper 3, Section II, B

Consider the Hamiltonian $H=H_{0}+V$, where $V$ is a small perturbation. If $H_{0}|n\rangle=E_{n}|n\rangle$, write down an expression for the eigenvalues of $H$, correct to second order in the perturbation, assuming the energy levels of $H_{0}$ are non-degenerate.

In a certain three-state system, $H_{0}$ and $V$ take the form

$H_{0}=\left(\begin{array}{ccc} E_{1} & 0 & 0 \\ 0 & E_{2} & 0 \\ 0 & 0 & E_{3} \end{array}\right) \quad \text { and } \quad V=V_{0}\left(\begin{array}{ccc} 0 & \epsilon & \epsilon^{2} \\ \epsilon & 0 & 0 \\ \epsilon^{2} & 0 & 0 \end{array}\right)$

with $V_{0}$ and $\epsilon$ real, positive constants and $\epsilon \ll 1$.

(a) Consider first the case $E_{1}=E_{2} \neq E_{3}$ and $\left|\epsilon V_{0} /\left(E_{3}-E_{2}\right)\right| \ll 1$. Use the results of degenerate perturbation theory to obtain the energy eigenvalues correct to order $\epsilon$.

(b) Now consider the different case $E_{3}=E_{2} \neq E_{1}$ and $\left|\epsilon V_{0} /\left(E_{2}-E_{1}\right)\right| \ll 1$. Use the results of non-degenerate perturbation theory to obtain the energy eigenvalues correct to order $\epsilon^{2}$. Why is it not necessary to use degenerate perturbation theory in this case?

(c) Obtain the exact energy eigenvalues in case (b), and compare these to your perturbative results by expanding to second order in $\epsilon$.

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• # Paper 4, Section II, B

Define the spin raising and spin lowering operators $S_{+}$and $S_{-}$. Show that

$S_{\pm}|s, \sigma\rangle=\hbar \sqrt{s(s+1)-\sigma(\sigma \pm 1)}|s, \sigma \pm 1\rangle,$

where $S_{z}|s, \sigma\rangle=\hbar \sigma|s, \sigma\rangle$ and $S^{2}|s, \sigma\rangle=s(s+1) \hbar^{2}|s, \sigma\rangle$.

Two spin- $\frac{1}{2}$ particles, with spin operators $\mathbf{S}^{(1)}$ and $\mathbf{S}^{(2)}$, have a Hamiltonian

$H=\alpha \mathbf{S}^{(1)} \cdot \mathbf{S}^{(2)}+\mathbf{B} \cdot\left(\mathbf{S}^{(1)}-\mathbf{S}^{(2)}\right)$

where $\alpha$ and $\mathbf{B}=(0,0, B)$ are constants. Express $H$ in terms of the two particles' spin raising and spin lowering operators $S_{\pm}^{(1)}, S_{\pm}^{(2)}$ and the corresponding $z$-components $S_{z}^{(1)}$, $S_{z}^{(2)}$. Hence find the eigenvalues of $H$. Show that there is a unique groundstate in the limit $B \rightarrow 0$ and that the first excited state is triply degenerate in this limit. Explain this degeneracy by considering the action of the combined spin operator $\mathbf{S}^{(1)}+\mathbf{S}^{(2)}$ on the energy eigenstates.

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• # Paper 1, Section II, D

A one-dimensional harmonic oscillator has Hamiltonian

$H=\hbar \omega\left(A^{\dagger} A+\frac{1}{2}\right)$

where $\left[A, A^{\dagger}\right]=1$. Show that $A|n\rangle=\sqrt{n}|n-1\rangle$, where $H|n\rangle=\left(n+\frac{1}{2}\right) \hbar \omega|n\rangle$ and $\langle n \mid n\rangle=1$.

This oscillator is perturbed by adding a new term $\lambda X^{4}$ to the Hamiltonian. Given that

$A=\frac{m \omega X-i P}{\sqrt{2 m \hbar \omega}}$

show that the ground state of the perturbed system is

$\left|0_{\lambda}\right\rangle=|0\rangle-\frac{\hbar \lambda}{4 m^{2} \omega^{3}}\left(3 \sqrt{2}|2\rangle+\sqrt{\frac{3}{2}}|4\rangle\right)$

to first order in $\lambda$. [You may use the fact that, in non-degenerate perturbation theory, a perturbation $\Delta$ causes the first-order shift

$\left|m^{(1)}\right\rangle=\sum_{n \neq m} \frac{\langle n|\Delta| m\rangle}{E_{m}-E_{n}}|n\rangle$

in the $m^{\text {th }}$energy level.]

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• # Paper 2, Section II, D

Explain what is meant by the intrinsic parity of a particle.

In each of the decay processes below, parity is conserved.

A deuteron $\left(d^{+}\right)$has intrinsic parity $\eta_{d}=+1$ and spin $s=1$. A negatively charged pion $\left(\pi^{-}\right)$has spin $s=0$. The ground state of a hydrogenic 'atom' formed from a deuteron and a pion decays to two identical neutrons $(n)$, each of spin $s=\frac{1}{2}$ and parity $\eta_{n}=+1$. Deduce the intrinsic parity of the pion.

The $\Delta^{-}$particle has spin $s=\frac{3}{2}$ and decays as

$\Delta^{-} \rightarrow \pi^{-}+n .$

What are the allowed values of the orbital angular momentum? In the centre of mass frame, the vector $\mathbf{r}_{\pi}-\mathbf{r}_{n}$ joining the pion to the neutron makes an angle $\theta$ to the $\hat{\mathbf{z}}$-axis. The final state is an eigenstate of $J_{z}$ and the spatial probability distribution is proportional to $\cos ^{2} \theta$. Deduce the intrinsic parity of the $\Delta^{-}$.

[Hint: You may use the fact that the first three Legendre polynomials are given by

$\left.P_{0}(x)=1, \quad P_{1}(x)=x, \quad P_{2}(x)=\frac{1}{2}\left(3 x^{2}-1\right) . \quad\right]$

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• # Paper 3, Section II, D

A quantum system is prepared in the ground state $|0\rangle$ at time $t=0$. It is subjected to a time-varying Hamiltonian $H=H_{0}+\Delta(t)$. Show that, to first order in $\Delta(t)$, the system evolves as

$|\psi(t)\rangle=\sum_{k} c_{k}(t) \mathrm{e}^{-i E_{k} t / \hbar}|k\rangle$

where $H_{0}|k\rangle=E_{k}|k\rangle$ and

$c_{k}(t)=\frac{1}{i \hbar} \int_{0}^{t}\left\langle k\left|\Delta\left(t^{\prime}\right)\right| 0\right\rangle \mathrm{e}^{i\left(E_{k}-E_{0}\right) t^{\prime} / \hbar} \mathrm{d} t^{\prime}$

A large number of hydrogen atoms, each in the ground state, are subjected to an electric field

$\mathbf{E}(t)=\left\{\begin{array}{lll} 0 & \text { for } & t<0 \\ \hat{\mathbf{z}} \mathcal{E}_{0} \exp (-t / \tau) & \text { for } & t>0 \end{array}\right.$

where $\mathcal{E}_{0}$ is a constant. Show that the fraction of atoms found in the state $|n, \ell, m\rangle=$ $|2,1,0\rangle$ is, after a long time and to lowest non-trivial order in $\mathcal{E}_{0}$,

$\frac{2^{15}}{3^{10}} \frac{a_{0}^{2} e^{2} \mathcal{E}_{0}^{2}}{\hbar^{2}\left(\omega^{2}+1 / \tau^{2}\right)}$

where $\hbar \omega$ is the energy difference between the $|2,1,0\rangle$ and $|1,0,0\rangle$ states, and $e$ is the electron charge and $a_{0}$ the Bohr radius. What fraction of atoms lie in the $|2,0,0\rangle$ state?

[Hint: You may assume the hydrogenic wavefunctions

$\langle\mathbf{r} \mid 1,0,0\rangle=\frac{2}{\sqrt{4 \pi}} \frac{1}{a_{0}^{3 / 2}} \exp \left(-\frac{r}{a_{0}}\right) \quad \text { and } \quad\langle\mathbf{r} \mid 2,1,0\rangle=\frac{1}{\sqrt{4 \pi}} \frac{1}{\left(2 a_{0}\right)^{3 / 2}} \frac{r}{a_{0}} \cos \theta \exp \left(-\frac{r}{2 a_{0}}\right)$

and the integral

$\int_{0}^{\infty} r^{m} \mathrm{e}^{-\alpha r} \mathrm{~d} r=\frac{m !}{\alpha^{m+1}}$

for $m$ a positive integer.]

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• # Paper 4, Section II, D

The spin operators obey the commutation relations $\left[S_{i}, S_{j}\right]=i \hbar \epsilon_{i j k} S_{k}$. Let $|s, \sigma\rangle$ be an eigenstate of the spin operators $S_{z}$ and $\mathbf{S}^{2}$, with $S_{z}|s, \sigma\rangle=\sigma \hbar|s, \sigma\rangle$ and $\mathbf{S}^{2}|s, \sigma\rangle=s(s+1) \hbar^{2}|s, \sigma\rangle$. Show that

$S_{\pm}|s, \sigma\rangle=\sqrt{s(s+1)-\sigma(\sigma \pm 1)} \hbar|s, \sigma \pm 1\rangle,$

where $S_{\pm}=S_{x} \pm i S_{y}$. When $s=1$, use this to derive the explicit matrix representation

$S_{x}=\frac{\hbar}{\sqrt{2}}\left(\begin{array}{lll} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{array}\right)$

in a basis in which $S_{z}$ is diagonal.

A beam of atoms, each with spin 1 , is polarised to have spin $+\hbar$ along the direction $\mathbf{n}=(\sin \theta, 0, \cos \theta)$. This beam enters a Stern-Gerlach filter that splits the atoms according to their spin along the $\hat{\mathbf{z}}$-axis. Show that $N_{+} / N_{-}=\cot ^{4}(\theta / 2)$, where $N_{+}$(respectively, $N_{-}$) is the number of atoms emerging from the filter with spins parallel (respectively, anti-parallel) to $\hat{\mathbf{z}}$.

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• # Paper 1, Section II, C

The position and momentum operators of the harmonic oscillator can be written as

$\hat{x}=\left(\frac{\hbar}{2 m \omega}\right)^{1 / 2}\left(a+a^{\dagger}\right), \quad \hat{p}=\left(\frac{\hbar m \omega}{2}\right)^{1 / 2} i\left(a^{\dagger}-a\right)$

where $m$ is the mass, $\omega$ is the frequency and the Hamiltonian is

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

Assuming that

$[\hat{x}, \hat{p}]=i \hbar$

derive the commutation relations for $a$ and $a^{\dagger}$. Construct the Hamiltonian in terms of $a$ and $a^{\dagger}$. Assuming that there is a unique ground state, explain how all other energy eigenstates can be constructed from it. Determine the energy of each of these eigenstates.

Consider the modified Hamiltonian

$H^{\prime}=H+\lambda \hbar \omega\left(a^{2}+a^{\dagger 2}\right)$

where $\lambda$ is a dimensionless parameter. Use perturbation theory to calculate the modified energy levels to second order in $\lambda$, quoting any standard formulae that you require. Show that the modified Hamiltonian can be written as

$H^{\prime}=\frac{1}{2 m}(1-2 \lambda) \hat{p}^{2}+\frac{1}{2} m \omega^{2}(1+2 \lambda) \hat{x}^{2} .$

Assuming $|\lambda|<\frac{1}{2}$, calculate the modified energies exactly. Show that the results are compatible with those obtained from perturbation theory.

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• # Paper 2, Section II, C

Let $\boldsymbol{\sigma}=\left(\sigma_{1}, \sigma_{2}, \sigma_{3}\right)$ be a set of Hermitian operators obeying

$\left[\sigma_{i}, \sigma_{j}\right]=2 i \epsilon_{i j k} \sigma_{k} \quad \text { and } \quad(\mathbf{n} \cdot \boldsymbol{\sigma})^{2}=1$

where $\mathbf{n}$ is any unit vector. Show that $(*)$ implies that

$(\mathbf{a} \cdot \boldsymbol{\sigma})(\mathbf{b} \cdot \boldsymbol{\sigma})=\mathbf{a} \cdot \mathbf{b}+i(\mathbf{a} \times \mathbf{b}) \cdot \boldsymbol{\sigma}$

for any vectors a and $\mathbf{b}$. Explain, with reference to the properties $(*)$, how $\boldsymbol{\sigma}$ can be related to the intrinsic angular momentum $\mathbf{S}$ for a particle of spin $\frac{1}{2}$.

Show that the operators $P_{\pm}=\frac{1}{2}(1 \pm \mathbf{n} \cdot \boldsymbol{\sigma})$ are Hermitian and obey

$P_{\pm}^{2}=P_{\pm}, \quad P_{+} P_{-}=P_{-} P_{+}=0$

Show how $P_{\pm}$can be used to write any state $|\chi\rangle$ as a linear combination of eigenstates of $\mathbf{n} \cdot \boldsymbol{\sigma}$. Use this to deduce that if the system is in a normalised state $|\chi\rangle$ when $\mathbf{n} \cdot \boldsymbol{\sigma}$ is measured, then the results $\pm 1$ will be obtained with probabilities

$\| P_{\pm}|\chi\rangle \|^{2}=\frac{1}{2}(1 \pm\langle\chi|\mathbf{n} \cdot \boldsymbol{\sigma}| \chi\rangle)$

If $|\chi\rangle$ is a state corresponding to the system having spin up along a direction defined by a unit vector $\mathbf{m}$, show that a measurement will find the system to have spin up along $\mathbf{n}$ with probability $\frac{1}{2}(1+\mathbf{n} \cdot \mathbf{m})$.

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• # Paper 3, Section II, C

The angular momentum operators $\mathbf{J}=\left(J_{1}, J_{2}, J_{3}\right)$ obey the commutation relations

\begin{aligned} &{\left[J_{3}, J_{\pm}\right]=\pm J_{\pm}} \\ &{\left[J_{+}, J_{-}\right]=2 J_{3}} \end{aligned}

where $J_{\pm}=J_{1} \pm i J_{2}$.

A quantum mechanical system involves the operators $a, a^{\dagger}, b$ and $b^{\dagger}$ such that

$\begin{gathered} {\left[a, a^{\dagger}\right]=\left[b, b^{\dagger}\right]=1} \\ {[a, b]=\left[a^{\dagger}, b\right]=\left[a, b^{\dagger}\right]=\left[a^{\dagger}, b^{\dagger}\right]=0 .} \end{gathered}$

Define $K_{+}=a^{\dagger} b, K_{-}=a b^{\dagger}$ and $K_{3}=\frac{1}{2}\left(a^{\dagger} a-b^{\dagger} b\right)$. Show that $K_{\pm}$and $K_{3}$ obey the same commutation relations as $J_{\pm}$and $J_{3}$.

Suppose that the system is in the state $|0\rangle$ such that $a|0\rangle=b|0\rangle=0$. Show that $\left(a^{\dagger}\right)^{2}|0\rangle$ is an eigenstate of $K_{3}$. Let $K^{2}=\frac{1}{2}\left(K_{+} K_{-}+K_{-} K_{+}\right)+K_{3}^{2}$. Show that $\left(a^{\dagger}\right)^{2}|0\rangle$ is an eigenstate of $K^{2}$ and find the eigenvalue. How many other states do you expect to find with same value of $K^{2}$ ? Find them.

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• # Paper 4, Section II, C

The Hamiltonian for a quantum system in the Schrödinger picture is

$H_{0}+\lambda V(t),$

where $H_{0}$ is independent of time and the parameter $\lambda$ is small. Define the interaction picture corresponding to this Hamiltonian and derive a time evolution equation for interaction picture states.

Let $|n\rangle$ and $|m\rangle$ be eigenstates of $H_{0}$ with distinct eigenvalues $E_{n}$ and $E_{m}$ respectively. Show that if the system was in the state $|n\rangle$ in the remote past, then the probability of measuring it to be in a different state $|m\rangle$ at a time $t$ is

$\frac{\lambda^{2}}{\hbar^{2}}\left|\int_{-\infty}^{t} d t^{\prime}\left\langle m\left|V\left(t^{\prime}\right)\right| n\right\rangle e^{i\left(E_{m}-E_{n}\right) t^{\prime} / \hbar}\right|^{2}+O\left(\lambda^{3}\right)$

Let the system be a simple harmonic oscillator with $H_{0}=\hbar \omega\left(a^{\dagger} a+\frac{1}{2}\right)$, where $\left[a, a^{\dagger}\right]=1$. Let $|0\rangle$ be the ground state which obeys $a|0\rangle=0$. Suppose

$V(t)=e^{-p|t|}\left(a+a^{\dagger}\right),$

with $p>0$. In the remote past the system was in the ground state. Find the probability, to lowest non-trivial order in $\lambda$, for the system to be in the first excited state in the far future.

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• # Paper 1, Section II, A

A particle in one dimension has position and momentum operators $\hat{x}$ and $\hat{p}$ whose eigenstates obey

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

For a state $|\psi\rangle$, define the position-space and momentum-space wavefunctions $\psi(x)$ and $\tilde{\psi}(p)$ and show how each of these can be expressed in terms of the other.

Write down the translation operator $U(\alpha)$ and check that your expression is consistent with the property $U(\alpha)|x\rangle=|x+\alpha\rangle$. For a state $|\psi\rangle$, relate the position-space and momentum-space wavefunctions for $U(\alpha)|\psi\rangle$ to $\psi(x)$ and $\tilde{\psi}(p)$ respectively.

Now consider a harmonic oscillator with mass $m$, frequency $\omega$, and annihilation and creation operators

$a=\left(\frac{m \omega}{2 \hbar}\right)^{1 / 2}\left(\hat{x}+\frac{i}{m \omega} \hat{p}\right), \quad a^{\dagger}=\left(\frac{m \omega}{2 \hbar}\right)^{1 / 2}\left(\hat{x}-\frac{i}{m \omega} \hat{p}\right)$

Let $\psi_{n}(x)$ and $\tilde{\psi}_{n}(p)$ be the wavefunctions corresponding to the normalised energy eigenstates $|n\rangle$, where $n=0,1,2, \ldots$.

(i) Express $\psi_{0}(x-\alpha)$ explicitly in terms of the wavefunctions $\psi_{n}(x)$.

(ii) Given that $\tilde{\psi}_{n}(p)=f_{n}(u) \tilde{\psi}_{0}(p)$, where the $f_{n}$ are polynomials and $u=(2 / \hbar m \omega)^{1 / 2} p$, show that

$e^{-i \gamma u}=e^{-\gamma^{2} / 2} \sum_{n=0}^{\infty} \frac{\gamma^{n}}{\sqrt{n !}} f_{n}(u) \text { for any real } \gamma$

[You may quote standard results for a harmonic oscillator. You may also use, without proof, $e^{A+B}=e^{A} e^{B} e^{-\frac{1}{2}[A, B]}$ for operators $A$ and $B$ which each commute with $\left.[A, B] .\right]$

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• # Paper 2, Section II, A

(a) Let $|j m\rangle$ be standard, normalised angular momentum eigenstates with labels specifying eigenvalues for $\mathbf{J}^{2}$ and $J_{3}$. Taking units in which $\hbar=1$,

$J_{\pm}|j m\rangle=\{(j \mp m)(j \pm m+1)\}^{1 / 2}|j m \pm 1\rangle .$

Check the coefficients above by computing norms of states, quoting any angular momentum commutation relations that you require.

(b) Two particles, each of spin $s>0$, have combined spin states $|J M\rangle$. Find expressions for all such states with $M=2 s-1$ in terms of product states.

(c) Suppose that the particles in part (b) move about their centre of mass with a spatial wavefunction that is a spherically symmetric function of relative position. If the particles are identical, what spin states $|J 2 s-1\rangle$ are allowed? Justify your answer.

(d) Now consider two particles of spin 1 that are not identical and are both at rest. If the 3-component of the spin of each particle is zero, what is the probability that their total, combined spin is zero?

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• # Paper 3, Section II, 31 A

A three-dimensional oscillator has Hamiltonian

$H=\frac{1}{2 m}\left(\hat{p}_{1}^{2}+\hat{p}_{2}^{2}+\hat{p}_{3}^{2}\right)+\frac{1}{2} m \omega^{2}\left(\alpha^{2} \hat{x}_{1}^{2}+\beta^{2} \hat{x}_{2}^{2}+\gamma^{2} \hat{x}_{3}^{2}\right),$

where the constants $m, \omega, \alpha, \beta, \gamma$ are real and positive. Assuming a unique ground state, construct the general normalised eigenstate of $H$ and give a formula for its energy eigenvalue. [You may quote without proof results for a one-dimensional harmonic oscillator of mass $m$ and frequency $\omega$ that follow from writing $\hat{x}=(\hbar / 2 m \omega)^{1 / 2}\left(a+a^{\dagger}\right)$ and $\left.\hat{p}=(\hbar m \omega / 2)^{1 / 2} i\left(a^{\dagger}-a\right) .\right]$

List all states in the four lowest energy levels of $H$ in the cases:

(i) $\alpha<\beta<\gamma<2 \alpha$;

(ii) $\alpha=\beta$ and $\gamma=\alpha+\epsilon$, where $0<\epsilon \ll \alpha$.

Now consider $H$ with $\alpha=\beta=\gamma=1$ subject to a perturbation

$\lambda m \omega^{2}\left(\hat{x}_{1} \hat{x}_{2}+\hat{x}_{2} \hat{x}_{3}+\hat{x}_{3} \hat{x}_{1}\right),$

where $\lambda$ is small. Compute the changes in energies for the ground state and the states at the first excited level of the original Hamiltonian, working to the leading order at which nonzero corrections occur. [You may quote without proof results from perturbation theory.]

Explain briefly why some energy levels of the perturbed Hamiltonian will be exactly degenerate. [Hint: Compare with (ii) above.]

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• # Paper 4, Section II, A

(a) Consider a quantum system with Hamiltonian $H=H_{0}+V$, where $H_{0}$ is independent of time. Define the interaction picture corresponding to this Hamiltonian and derive an expression for the time derivative of an operator in the interaction picture, assuming it is independent of time in the Schrödinger picture.

(b) The Pauli matrices $\boldsymbol{\sigma}=\left(\sigma_{1}, \sigma_{2}, \sigma_{3}\right)$ satisfy

$\sigma_{i} \sigma_{j}=\delta_{i j}+i \epsilon_{i j k} \sigma_{k}$

Explain briefly how these properties allow $\sigma$ to be used to describe a quantum system with spin $\frac{1}{2}$.

(c) A particle with spin $\frac{1}{2}$ has position and momentum operators $\hat{\mathbf{x}}=\left(\hat{x}_{1}, \hat{x}_{2}, \hat{x}_{3}\right)$ and $\hat{\mathbf{p}}=\left(\hat{p}_{1}, \hat{p}_{2}, \hat{p}_{3}\right)$. The unitary operator corresponding to a rotation through an angle $\theta$ about an axis $\mathbf{n}$ is $U=\exp (-i \theta \mathbf{n} \cdot \mathbf{J} / \hbar)$ where $\mathbf{J}$ is the total angular momentum. Check this statement by considering the effect of an infinitesimal rotation on $\hat{\mathbf{x}}, \hat{\mathbf{p}}$ and $\boldsymbol{\sigma}$.

(d) Suppose that the particle in part (c) has Hamiltonian $H=H_{0}+V$ with

$H_{0}=\frac{1}{2 m} \hat{\mathbf{p}}^{2}+\alpha \mathbf{L} \cdot \boldsymbol{\sigma} \quad \text { and } \quad V=B \sigma_{3}$

where $\mathbf{L}$ is the orbital angular momentum and $\alpha, B$ are constants. Show that all components of $\mathbf{J}$ are independent of time in the interaction picture. Is this true in the Heisenberg picture?

[You may quote commutation relations of $\mathbf{L}$ with $\hat{\mathbf{x}}$ and $\hat{\mathbf{p}}$.]

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• # Paper 1, Section II, A

If $A$ and $B$ are operators which each commute with their commutator $[A, B]$, show that

$F(\lambda)=e^{\lambda A} e^{\lambda B} e^{-\lambda(A+B)} \quad \text { satisfies } \quad F^{\prime}(\lambda)=\lambda[A, B] F(\lambda)$

By solving this differential equation for $F(\lambda)$, deduce that

$e^{A} e^{B}=e^{\frac{1}{2}[A, B]} e^{A+B}$

The annihilation and creation operators for a harmonic oscillator of mass $m$ and frequency $\omega$ are defined by

$a=\sqrt{\frac{m \omega}{2 \hbar}}\left(\hat{x}+\frac{i}{m \omega} \hat{p}\right), \quad a^{\dagger}=\sqrt{\frac{m \omega}{2 \hbar}}\left(\hat{x}-\frac{i}{m \omega} \hat{p}\right)$

Write down an expression for the general normalised eigenstate $|n\rangle(n=0,1,2, \ldots)$ of the oscillator Hamiltonian $H$ in terms of the ground state $|0\rangle$. What is the energy eigenvalue $E_{n}$ of the state $|n\rangle ?$

Suppose the oscillator is now subject to a small perturbation so that it is described by the modified Hamiltonian $H+\varepsilon V(\hat{x})$ with $V(\hat{x})=\cos (\mu \hat{x})$. Show that

$V(\hat{x})=\frac{1}{2} e^{-\gamma^{2} / 2}\left(e^{i \gamma a^{\dagger}} e^{i \gamma a}+e^{-i \gamma a^{\dagger}} e^{-i \gamma a}\right)$

where $\gamma$ is a constant, to be determined. Hence show that to $O\left(\varepsilon^{2}\right)$ the shift in the ground state energy as a result of the perturbation is

$\varepsilon e^{-\mu^{2} \hbar / 4 m \omega}-\varepsilon^{2} e^{-\mu^{2} \hbar / 2 m \omega} \frac{1}{\hbar \omega} \sum_{p=1}^{\infty} \frac{1}{(2 p) ! 2 p}\left(\frac{\mu^{2} \hbar}{2 m \omega}\right)^{2 p} .$

[Standard results of perturbation theory may be quoted without proof.]

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• # Paper 2, Section II, A

Express the spin operator $\mathbf{S}$ for a particle of spin $\frac{1}{2}$ in terms of the Pauli matrices $\boldsymbol{\sigma}=\left(\sigma_{1}, \sigma_{2}, \sigma_{3}\right)$ where

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

Show that $(\mathbf{n} \cdot \boldsymbol{\sigma})^{2}=\mathbb{I}$ for any unit vector $\mathbf{n}$ and deduce that

$e^{-i \theta \mathbf{n} \cdot \mathbf{S} / \hbar}=\mathbb{I} \cos (\theta / 2)-i(\mathbf{n} \cdot \boldsymbol{\sigma}) \sin (\theta / 2) .$

The space of states $V$ for a particle of spin $\frac{1}{2}$ has basis states $|\uparrow\rangle,|\downarrow\rangle$ which are eigenstates of $S_{3}$ with eigenvalues $\frac{1}{2} \hbar$ and $-\frac{1}{2} \hbar$ respectively. If the Hamiltonian for the particle is $H=\frac{1}{2} \alpha \hbar \sigma_{1}$, find

$e^{-i t H / \hbar}|\uparrow\rangle \quad \text { and } \quad e^{-i t H / \hbar}|\downarrow\rangle$

as linear combinations of the basis states.

The space of states for a system of two spin $\frac{1}{2}$ particles is $V \otimes V$. Write down explicit expressions for the joint eigenstates of $\mathbf{J}^{2}$ and $J_{3}$, where $\mathbf{J}$ is the sum of the spin operators for the particles.

Suppose that the two-particle system has Hamiltonian $H=\frac{1}{2} \lambda \hbar\left(\sigma_{1} \otimes \mathbb{I}-\mathbb{I} \otimes \sigma_{1}\right)$ and that at time $t=0$ the system is in the state with $J_{3}$ eigenvalue $\hbar$. Calculate the probability that at time $t>0$ the system will be measured to be in the state with $\mathbf{J}^{2}$ eigenvalue zero.

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• # Paper 3, Section II, A

Let $|j, m\rangle$ denote the normalised joint eigenstates of $\mathbf{J}^{2}$ and $J_{3}$, where $\mathbf{J}$ is the angular momentum operator for a quantum system. State clearly the possible values of the quantum numbers $j$ and $m$ and write down the corresponding eigenvalues in units with $\hbar=1$.

Consider two quantum systems with angular momentum states $\left|\frac{1}{2}, r\right\rangle$ and $|j, m\rangle$. The eigenstates corresponding to their combined angular momentum can be written as

$|J, M\rangle=\sum_{r, m} C_{r m}^{J M}\left|\frac{1}{2}, r\right\rangle|j, m\rangle,$

where $C_{r m}^{J M}$ are Clebsch-Gordan coefficients for addition of angular momenta $\frac{1}{2}$ and $j$. What are the possible values of $J$ and what is a necessary condition relating $r, m$ and $M$ in order that $C_{r m}^{J M} \neq 0$ ?

Calculate the values of $C_{r m}^{J M}$ for $j=2$ and for all $M \geqslant \frac{3}{2}$. Use the sign convention that $C_{r m}^{J J}>0$ when $m$ takes its maximum value.

A particle $X$ with spin $\frac{3}{2}$ and intrinsic parity $\eta_{X}$ is at rest. It decays into two particles $A$ and $B$ with spin $\frac{1}{2}$ and spin 0 , respectively. Both $A$ and $B$ have intrinsic parity $-1$. The relative orbital angular momentum quantum number for the two particle system is $\ell$. What are the possible values of $\ell$ for the cases $\eta_{X}=+1$ and $\eta_{X}=-1$ ?

Suppose particle $X$ is prepared in the state $\left|\frac{3}{2}, \frac{3}{2}\right\rangle$ before it decays. Calculate the probability $P$ for particle $A$ to be found in the state $\left|\frac{1}{2}, \frac{1}{2}\right\rangle$, given that $\eta_{X}=+1$.

What is the probability $P$ if instead $\eta_{X}=-1$ ?

[Units with $\hbar=1$ should be used throughout. You may also use without proof

$\left.J_{-}|j, m\rangle=\sqrt{(j+m)(j-m+1)}|j, m-1\rangle .\right]$

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• # Paper 4, Section II, A

The Hamiltonian for a quantum system in the Schrödinger picture is $H_{0}+\lambda V(t)$, where $H_{0}$ is independent of time and the parameter $\lambda$ is small. Define the interaction picture corresponding to this Hamiltonian and derive a time evolution equation for interaction picture states.

Suppose that $|\chi\rangle$ and $|\phi\rangle$ are eigenstates of $H_{0}$ with distinct eigenvalues $E$ and $E^{\prime}$, respectively. Show that if the system is in state $|\chi\rangle$ at time zero then the probability of measuring it to be in state $|\phi\rangle$ at time $t$ is

$\frac{\lambda^{2}}{\hbar^{2}}\left|\int_{0}^{t} d t^{\prime}\left\langle\phi\left|V\left(t^{\prime}\right)\right| \chi\right\rangle e^{i\left(E^{\prime}-E\right) t^{\prime} / \hbar}\right|^{2}+O\left(\lambda^{3}\right)$

Let $H_{0}$ be the Hamiltonian for an isotropic three-dimensional harmonic oscillator of mass $m$ and frequency $\omega$, with $\chi(r)$ being the ground state wavefunction (where $r=|\mathbf{x}|$ ) and $\phi_{i}(\mathbf{x})=(2 m \omega / \hbar)^{1 / 2} x_{i} \chi(r)$ being wavefunctions for the states at the first excited energy level $(i=1,2,3)$. The oscillator is in its ground state at $t=0$ when a perturbation

$\lambda V(t)=\lambda \hat{x}_{3} e^{-\mu t}$

is applied, with $\mu>0$, and $H_{0}$ is then measured after a very large time has elapsed. Show that to first order in perturbation theory the oscillator will be found in one particular state at the first excited energy level with probability

$\frac{\lambda^{2}}{2 \hbar m \omega\left(\mu^{2}+\omega^{2}\right)},$

but that the probability that it will be found in either of the other excited states is zero (to this order).

$\left[\right.$ You may use the fact that $\left.4 \pi \int_{0}^{\infty} r^{4}|\chi(r)|^{2} d r=\frac{3 \hbar}{2 m \omega} .\right]$

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• # Paper 1, Section II, A

Let $\hat{x}, \hat{p}$ and $H(\hat{x}, \hat{p})=\hat{p}^{2} / 2 m+V(\hat{x})$ be the position operator, momentum operator and Hamiltonian for a particle moving in one dimension. Let $|\psi\rangle$ be the state vector for the particle. The position and momentum eigenstates have inner products

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

Show that

$\langle x|\hat{p}| \psi\rangle=-i \hbar \frac{\partial}{\partial x} \psi(x) \quad \text { and } \quad\langle p|\hat{x}| \psi\rangle=i \hbar \frac{\partial}{\partial p} \tilde{\psi}(p)$

where $\psi(x)$ and $\tilde{\psi}(p)$ are the wavefunctions in the position representation and momentum representation, respectively. Show how $\psi(x)$ and $\tilde{\psi}(p)$ may be expressed in terms of each other.

For general $V(\hat{x})$, express $\langle p|V(\hat{x})| \psi\rangle$ in terms of $\tilde{\psi}(p)$, and hence write down the time-independent Schrödinger equation in the momentum representation satisfied by $\tilde{\psi}(p)$.

Consider now the case $V(x)=-\left(\hbar^{2} \lambda / m\right) \delta(x), \lambda>0$. Show that there is a bound state with energy $E=-\varepsilon, \varepsilon>0$, with wavefunction $\tilde{\psi}(p)$