List of top Questions asked in GATE PH

Apart from the acoustic modes, 9 optical modes are identified from the measurements of phonon dispersions of a solid with the chemical formula \( A_nB_m \), where \( A \) and \( B \) denote the atomic species, and \( n \) and \( m \) are integers. Which of the following combinations of \( n \) and \( m \) is/are possible?

A material behaves as a superconductor below a critical temperature \( T_c \) and as a normal conductor above \( T_c \). A magnetic field \(\overrightarrow{B} = B\hat{z}\) is applied when \( T > T_c \). The material is then cooled below \( T_c \) in the presence of \( \vec{B} \). Which of the following figures represents the correct configuration of magnetic field lines?

Consider a vector field \( \vec{F} = (2xz + 3y^2)\hat{y} + 4yz^2\hat{z} \). The closed path (\( \Gamma \): \( A \rightarrow B \rightarrow C \rightarrow D \rightarrow A \)) in the \( z = 0 \) plane is shown in the figure.

\( \oint_\Gamma \vec{F} \cdot d\vec{l} \) denotes the line integral of \( \vec{F} \) along the closed path \( \Gamma \). Which of the following options is/are true?

The non-relativistic Hamiltonian for a single electron atom is
\[ H_0 = \frac{p^2}{2m} - V(r) \]
where \( V(r) \) is the Coulomb potential and \( m \) is the mass of the electron.Considering the spin-orbit interaction term:
\[ H' = \frac{1}{2m^2c^2} \frac{1}{r} \frac{dV}{dr} \overrightarrow{L} \cdot \overrightarrow{S} \]
added to \( H_0 \), which of the following statements is/are true?

Decays of mesons and baryons can be categorized as weak, strong and electromagnetic decays depending upon the interactions involved in the processes. Which of the following option is/are true?

The equation of motion for the forced simple harmonic oscillator is \( \ddot{x}(t) + \omega^2 x(t) = F \cos(\omega t) \) where \( x(t = 0) = 0 \) and \( \dot{x}(t = 0) = 0 \). Which one of the following options is correct?

An infinite one-dimensional lattice extends along the x-axis. At each lattice site, there exists an ion with spin 1/2. The spin can point either in the +z or -z direction only. Let \( S_P \), \( S_F \), and \( S_A \) denote the entropies of paramagnetic, ferromagnetic, and antiferromagnetic configurations, respectively. Which of the following relations is/are true?

Consider two matrices: \( P = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix} \) and \( Q = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \). Which of the following statements is/are true?

In a parallel plate capacitor, the plate at \( x = 0 \) is grounded and the plate at \( x = d \) is maintained at a potential \( V_0 \). The space between the two plates is filled with a linear dielectric of permittivity \( \epsilon = \epsilon_0 \left(1 + \frac{x}{d}\right) \), where \( \epsilon_0 \) is the permittivity of free space. Neglecting the edge effects, the electric field (\(\vec{E}\)) inside the capacitor is .

The spin-orbit interaction in a hydrogen-like atom is given by the Hamiltonian \( H' = -k \vec{L} \cdot \vec{S} \), where \( k \) is a real constant. The splitting between levels \( ^2P_{3/2} \) and \( ^2P_{1/2} \) due to this interaction is _______.

The electric field in a region depends only on \( x \) and \( y \) coordinates as \( \vec{E} = k \frac{x \hat{x} + y \hat{y}}{x^2 + y^2} \) where \( k \) is a constant. The flux of \( \vec{E} \) through the surface of a sphere of radius \( R \) with its center at the origin is \( n\pi k R \), where the value of \( n \) is ___ (in integer).

The Hamiltonian of a system of \( N \) particles in volume \( V \) at temperature \( T \) is \( H = \sum_{i=1}^{2N} a_i q_i^2 + \sum_{i=1}^{2N} b_i p_i^2 \) where \( a_i \) and \( b_i \) are positive constants. The ensemble average of the Hamiltonian is \( aNk_B T \), where \( k_B \) is the Boltzmann constant. The value of \( a \) is _____ (in integer).

The vapor pressure (P) of solid ammonia is given by \( \ln(P) = 23.03 - \frac{3754}{T} \), while that of liquid ammonia is given by \( \ln(P) = 19.49 - \frac{3063}{T} \), where \( T \) is the temperature in K. The temperature of the triple point of ammonia is ________ K (rounded off to two decimal places).

Let P, Q and R be three different nuclei. Which one of the following nuclear processes is possible?

The complex function \( e^{-\left(\frac{2}{z-1}\right)} \) has:

Consider a volume integral
\(I=\int_V\nabla^2(\frac{1}{r})dv\)
over a volume \( V \), where \( r = \sqrt{x^2 + y^2 + z^2} \). Which of the following statements is/are correct?

The minimum number of basic logic gates required to realize the Boolean expression \( B \cdot (A + B) + A \cdot (B + A) \) is ____ (in integer).

The temperature dependence of the electrical conductivity (σ) of three intrinsic semiconductors A, B and C is shown in figure

Let \(E_A, E_B\) and \(E_C\) be the bandgaps of A, B and C, respectively. Which one of the following relations is correct?

The wavefunction for a particle is given by the form \( e^{-i(ax+\beta)} \), where \( a \) and \( \beta \) are real constants. In which one of the following potentials \( V(x) \) does the particle move?

Following trial wave functions \( \phi_1 = e^{-Z' (r_1 + r_2)} \) and \( \phi_2 = e^{-Z' (r_1 + r_2) (1 + g |\vec{r}_1 - \bar{\vec{r}}_2|)} \) are used to obtain variational estimates \( E_1 \) and \( E_2 \) for the ground state energy \( E_0 \) of the helium atom. Here, \( Z' \) and \( g \) are variational parameters, \( \bar{\vec{r}}_1 \) and \( \bar{\vec{r}}_2 \) are the position vectors of the electrons. Let \( E_0 \) be the exact ground state energy of the helium atom.\( E_1 \) and \( E_2 \) are the variational estimates of the ground state energy of the helium atom corresponding to \( \phi_1 \) and \( \phi_2 \), respectively, Which one of the following options is true?

An inertial observer sees two spacecrafts S and T flying away from each other along x-axis with individual speed 0.5c, where c is the speed of light. The speed of T with respect to S is

Consider the following circuit:

Suppose the input signal P is

and the input signal Q is 

Which one of the following output signals is correct?

A particle is subjected to a potential described as follows:
\[ V(x) = \begin{cases} \infty, & \text{if } x \leq 0 \\ V_0, & \text{if } a \leq x \leq b \\ 0, & \text{elsewhere} \end{cases} \]
Here, \(a > 0\) and \(b > a\). If the energy of the particle \(E < V_0\), which one of the following schematics is a valid quantum mechanical wavefunction (\(\Psi\)) for the system?

An infinitely long cylinder of radius \( R \) carries a frozen-in magnetization \(\vec{M} = ke^{-s}\hat{z}\), where \( k \) is a constant and \( s \) is the distance from the axis of the cylinder. The magnetic permeability of free space is \(\mu_0\). There is no free current present anywhere. The magnetic flux density (\(\vec{B}\)) inside the cylinder is

Atomic numbers of V, Cr, Fe and Zn are 23, 24, 26 and 30, respectively. Which one of the following materials does NOT show an electron spin resonance (ESR) spectra?