List of top Questions asked in GATE PH

A spin $\dfrac{1}{2}$ particle is in a spin up state along the $x$-axis (with unit vector $\hat{x}$) and is denoted as $\left|\dfrac{1}{2}, \dfrac{1}{2}\right\rangle_x$. What is the probability of finding the particle to be in a spin up state along the direction $\hat{x'}$, which lies in the $xy$-plane and makes an angle $\theta$ with respect to the positive $x$-axis, if such a measurement is made?
Consider two non-identical spin-$\dfrac{1}{2}$ particles labelled 1 and 2 in the spin product state $\left[\left|\dfrac{1}{2}, -\dfrac{1}{2}\right\rangle \left|\dfrac{1}{2}, \dfrac{1}{2}\right\rangle\right]$. The Hamiltonian of the system is $H = \dfrac{4\lambda}{\hbar^2} \, \vec{S_1} \cdot \vec{S_2}$, where $\vec{S_1}$ and $\vec{S_2}$ are the spin operators of particles 1 and 2, respectively, and $\lambda$ is a constant with appropriate dimensions. What is the expectation value of $H$ in the above state?
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?