List of practice Questions

Consider a particle in a two dimensional infinite square well potential of side $L$, with $0 \leq x \leq L$ and $0 \leq y \leq L$. The wavefunction of the particle is zero only along the line $y = \dfrac{L}{2}$, apart from the boundaries of the well. If the energy of the particle in this state is $E$, what is the energy of the ground state?

The wavefunction of a particle in one dimension is given by \[ \psi(x) = \begin{cases} M, & \text{for } -a < x < a, \\ 0, & \text{otherwise}. \end{cases} \] Here $M$ and $a$ are positive constants. If $\varphi(p)$ is the corresponding momentum space wavefunction, which one of the following plots best represents $|\varphi(p)|^2$?

In a hadronic interaction, $\pi^0$'s are produced with different momenta, and they immediately decay into two photons with an opening angle $\theta$ between them. Assuming that all these decays occur in one plane, which one of the following figures depicts the behaviour of $\theta$ as a function of the $\pi^0$ momentum $p$?

In the first Brillouin zone of a rectangular lattice (lattice constants $a = 6$ \AA\ and $b = 4$ \AA), three incoming phonons with the same wave vector $\langle 1.2$ \AA$^{-1}, 0.6$ \AA$^{-1}\rangle$ interact to give one phonon. Which one of the following is the CORRECT wave vector of the resulting phonon?

A neutron beam with a wave vector $\vec{k}$ and an energy 20.4 meV diffracts from a crystal with an outgoing wave vector $\vec{k'}$. One of the diffraction peaks is observed for the reciprocal lattice vector $\vec{G}$ of magnitude $3.14$ \AA$^{-1}$. What is the diffraction angle in degrees (rounded off to the nearest integer) that $\vec{k}$ makes with the plane? (Use mass of neutron = $1.67 \times 10^{-27}$ Kg)

A rod $PQ$ of proper length $L$ lies along the $X$-axis and moves towards the positive $X$-direction with speed $v = \frac{3c}{5}$ with respect to the ground (see figure), where $c$ is the speed of light in vacuum. An observer on the ground measures the positions of $P$ and $Q$ at different times $t_P$ and $t_Q$ respectively in the ground frame, and finds the difference between them to be $\frac{9L}{10}$. What is the value of $t_Q - t_P$?

Young’s double slit experiment is performed using a beam of $C_{60}$ (fullerene) molecules, each molecule being made up of 60 carbon atoms. When the slit separation is 50 nm, fringes are formed on a screen kept at a distance of 1 m from the slits. Now, the experiment is repeated with $C_{70}$ molecules with a slit separation of 92.5 nm. The kinetic energies of both the beams are the same. The position of the 4$^{\text{th}}$ bright fringe for $C_{60}$ will correspond to the $n^{\text{th}}$ bright fringe for $C_{70}$. What is the value of $n$ (rounded off to the nearest integer)?

A particle of mass $m$ is free to move on a frictionless horizontal two-dimensional $(r, \theta)$ plane, and is acted upon by a force $\vec{F} = -\dfrac{k}{r^3}\hat{r}$ with $k$ being a positive constant. If $p_r$ and $p_\theta$ are the generalized momenta corresponding to $r$ and $\theta$ respectively, then what is the value of $\dfrac{dp_r}{dt}$?

A symmetric top has principal moments of inertia $I_1 = I_2 = \dfrac{2\alpha}{3}$, $I_3 = 2\alpha$ about a set of principal axes 1, 2, 3 respectively, passing through its center of mass, where $\alpha$ is a positive constant. There is no force acting on the body and the angular speed of the body about the 3-axis is $\omega_3 = \dfrac{1}{8}$ rad/s. With what angular frequency in rad/s does the angular velocity vector $\vec{\omega}$ precess about the 3-axis?

Which one of the following options is the most appropriate match between the items given in Column 1 and Column 2?

A simple harmonic oscillator with an angular frequency $\omega$ is in thermal equilibrium with a reservoir at absolute temperature $T$, with $\omega = \dfrac{2\pi k_B T}{h}$. Which one of the following is the partition function of the system?

An input voltage in the form of a square wave of frequency 1 kHz is given to a circuit, which results in the output shown schematically below. Which one of the following options is the CORRECT representation of the circuit?

An electric field as a function of radial coordinate $r$ has the form $\vec{E} = \alpha \dfrac{e^{-r^2}}{r} \hat{r}$, where $\alpha$ is a constant. Assume that dimensions are appropriately taken care of. The electric flux through a sphere of radius $\sqrt{2}$, centered at the origin, is $\Phi$. What is the value of $\dfrac{\Phi}{2\pi\alpha}$ (rounded off to two decimal places)?

Consider an isolated magnetized sphere of radius $R$ with a uniform magnetization $\vec{M}$ along the positive $z$ direction, with the north and south poles of the sphere lying on the $z$ axis. It is given that the magnetic field inside the sphere is $\vec{B} = \dfrac{2\mu_0}{3}\vec{M}$, where $\mu_0$ is the permeability of vacuum. Which of the following statements is(are) CORRECT?

The $\Xi^{0*}$ particle is a member of the Baryon decuplet with isospin state \[ | I, I_3 \rangle = \left| \tfrac{1}{2}, \tfrac{1}{2} \right\rangle \] and strangeness quantum number $-2$. In the quark model, which one of the following is the flavour part of the $\Xi^{0*}$ wavefunction?

A $4 \times 4$ matrix $M$ has the property $M^{\dagger} = -M$ and $M^4 = 1$, where $1$ is the $4 \times 4$ identity matrix. Which one of the following is the CORRECT set of eigenvalues of the matrix $M$?

An atom with non-zero magnetic moment has an angular momentum of magnitude $\sqrt{12}\,\hbar$. When a beam of such atoms is passed through a Stern-Gerlach apparatus, how many beams does it split into?

Which one of the following options is CORRECT for the given logic circuit?

A $^{60}$Co nucleus emits a $\beta$-particle and is converted to $^{60}$Ni$^{*}$ with $J^P = 4^+$, which in turn decays to the $^{60}$Ni ground state with $J^P = 0^+$ by emitting two photons in succession, as shown in the figure. Which one of the following statements is CORRECT?

Graphene is a two-dimensional material, in which carbon atoms are arranged in a honeycomb lattice with lattice constant $a$. As shown in the figure, $\vec{a_1}$ and $\vec{a_2}$ are two lattice vectors. Which one of the following is the area of the first Brillouin zone for this lattice?

The figure schematically shows the M (magnetization) - H (magnetic field) plots for certain types of materials. Here M and H are plotted in the same scale and units. Which one of the following is the most appropriate combination?

For nonrelativistic electrons in a solid, different energy dispersion relations with effective masses $m_a^*, m_b^*, m_c^*$ are schematically shown in the plots. Which one of the following options is CORRECT?

For a non-magnetic metal, which one of the following graphs best represents the behaviour of $\frac{C}{T}$ vs. $T^2$, where $C$ is the heat capacity and $T$ is the temperature?

Consider a two-dimensional Cartesian coordinate system in which a rank 2 contravariant tensor is represented by the matrix \[ \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \] The coordinate system is rotated anticlockwise by an acute angle $\theta$ with the origin fixed. Which one of the following matrices represents the tensor in the new coordinate system?

The Hall experiment is carried out with a non-magnetic semiconductor. The current $I$ is along the X-axis and the magnetic field $B$ is along the Z-axis. Which one of the following is the CORRECT representation of the variation of the magnitude of the Hall resistivity $\rho_{xy}$ as a function of the magnetic field?