List of top Physics Questions asked in GATE PH

A rocket of length 18.0 m is moving at speed \(0.9c\) (where \(c\) is the speed of light) parallel to its own length, relative to the earth. The length of the rocket measured in meters by an observer on earth (rounded off to two decimal places) is

The function \(f(z)\) of the complex variable \(z\) given below,\[ f(z) = \dfrac{z^{2} - 5z + 4}{z^{3} + 4z - z^{2} - 4} \]has singular points at \(z =\)

Consider the Pauli matrices \(\sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\), \(\sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}\), \(\sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\).
The value of \(\text{Tr}\big(\sigma_z [\sigma_x, \sigma_y]\big)\) is

The Hamiltonian for a quantum particle of mass \(m\) is given below, where \(\omega < \Omega\). The Schrodinger equation for this system can be solved exactly using the orthogonal transformation \(x = \dfrac{x_1 - x_2}{\sqrt2}\) and \(y = \dfrac{x_1+x_2}{\sqrt2}\).
\[ H = -\dfrac{\hbar^2}{2m}\left[\dfrac{\partial^2}{\partial x^2} + \dfrac{\partial^2}{\partial y^2}\right] + \dfrac12 m\Omega^2(x^2+y^2) + m\omega^2 xy \]The ground state energy of this system is

Which one of the following statements is true?

Consider two particles with angular momenta \(j_1 = 2\hbar\) and \(j_2 = \hbar/2\). If the expression
\[ |j=\tfrac52, m=\tfrac32\rangle = c_1|j_1=2,m_1=1\rangle|j_2=\tfrac12,m_2=\tfrac12\rangle + c_2|j_1=2,m_1=2\rangle|j_2=\tfrac12,m_2=-\tfrac12\rangle \]gives an eigenstate of the total angular momentum of the two particles, using standard notation, which of the following is true?
(Hint: \(\hat J_{\pm}|j,m\rangle = \sqrt{j(j+1)-m(m\pm1)}\,|j,m\pm1\rangle\))

Consider an electron in the energy eigenstate \(\psi_{211}(\vec{r})\) of the hydrogen atom. Given that the radial probability distribution of the electron in such a state takes its maximum value at \(r = n_0 a\), where \(a\) is the Bohr radius, and \(n_0\) is an integer. The value of \(n_0\) (in integer) is . The radial part of the wavefunction \(\psi_{211}(\vec{r})\) is given by \(R_{21}(r) = \dfrac{1}{\sqrt{24a^{5}}}\,re^{-r/2a}\).

Raman spectrum of a molecule was recorded using a source of wavelength 5000 Angstrom. The first Stokes line is observed at 5100 Angstrom. The first anti-Stokes line will appear at a wavelength \(L\) (in Angstrom). The value of \(L\) (rounded off to nearest integer) is

The specific heat \(C_p(T)\) of one mole of a material as a function of temperature \(T\) is given as \(C_p(T) = AT + BT^3\), where \(A = 0.695\) mJ.mol\(^{-1}\).K\(^{-2}\) and \(B = 0.045\) mJ.mol\(^{-1}\).K\(^{-4}\). When \(T\) is changed from 1 K to 10 K at constant pressure, the change in entropy \(\Delta S\) in mJ.mol\(^{-1}\).K\(^{-1}\) (rounded off to one decimal place) is

Consider a metal-superconductor junction connected to a dc voltage \(V\). At \(T < T_c\), where \(T_c\) is the superconductor's transition temperature, the current \(I\) versus \(V\) behavior of this junction is shown schematically in the figure below.

If the superconducting energy gap is \(D\) meV, the value of \(D\) (rounded off to one decimal place) is

A dielectric sphere carries a uniform polarization \(P = 26\ \mu\text{C}.\text{cm}^{-2}\). The magnitude of the electric field at the center of the sphere is \(E \times 10^{9}\ \text{N}.\text{C}^{-1}\). The value of \(E\) (rounded off to one decimal place) is . \((\epsilon_0 = 8.85 \times 10^{-12}\ \text{C}^{2}.\text{N}^{-1}.\text{m}^{-2})\)

For the energy dispersion of an electron in a one-dimensional solid \(E(k) = E_0 - 2\gamma \cos(ka)\), the ratio of the effective mass of the electron in the solid to the free electron mass (\(m_e\)) at \(k = 0\) is \(R_0\). Taking \(\gamma = 0.5\) eV and \(a = 0.5\) nm, the value of \(R_0\) (rounded off to two decimal places) is
(\(\hbar = 1.054 \times 10^{-34}\) J.s, \(m_e = 9.1 \times 10^{-31}\) kg, electron charge \(= 1.6 \times 10^{-19}\) C)

For which of the following functions does the Laplacian vanish?

Given \(Q\) is the electromagnetic charge and \(S\) is the strangeness quantum number, identify the particle(s) for which \((Q - S) = 0\) is satisfied.

Schematic variation of the specific heat \(C_p\) of an ideal gas of diatomic molecules with temperature \(T\) is shown in the figure below. For rotational energy \(E_R\) and vibrational energy \(E_v\) of the molecule, which of the following options is/are correct? Here \(k_B\) is the Boltzmann constant.

A projectile of mass \(m\) is launched from the ground with the initial speed \(v_0\) at an angle \(30^{\circ}\) from the horizontal. Take the ground to be horizontal. Ignoring the drag, the magnitude of Hamilton's action \(\int L\,dt\) for the particle from the beginning till it hits the ground is \(f \times \left(\dfrac{mv_0^{3}}{g}\right)\). The value of \(f\) (rounded off to two decimal places) is .

If the perturbation \(V = \lambda x^3\) is added to the Hamiltonian of a one dimensional harmonic oscillator, the matrix element \(\langle m|V|0\rangle\) is/are non-zero for which of the following states? Here, the eigenstates of the harmonic oscillator are denoted by \(|n\rangle\).

Which one of the following is an allowed process?

The formula for energy \(E\) of a photon gas at temperature \(T\) in a two-dimensional box at equilibrium with \(g_{2d}(\nu)\) denoting the density of states of photons is given below, where symbols \(\nu\), \(h\) and \(k_B\) have their standard meaning. The specific heat (\(C_V\)) of this photon gas obeys
\[ E = \int_0^{\infty} d\nu \, g_{2d}(\nu) \, \frac{h\nu}{\exp\left(\dfrac{h\nu}{k_B T}\right) - 1} \]

A gas of non-interacting \(^4\text{He}\) atoms (of mass \(m\)) is in a three-dimensional trap whose energy levels can be approximated by those of a harmonic oscillator potential \(V(x,y,z) = \dfrac{1}{2} m \omega^2 (x^2+y^2+z^2)\). The chemical potential of the gas at \(T=0\) K is

Sketch of a two-dimensional vector field \(\vec{V}\) is shown below. Here, length and arrow head of the arrows denote magnitude and direction of the vector field, respectively.

Which of the following statements is correct for \( \nabla \times \vec{V} \)?

For the electric field of an electromagnetic wave given below, which of the following statements is correct?
\[ \vec{E} = \hat{x}\, E_0 \cos(\omega t) + \hat{y}\, 2E_0 \cos\left(\omega t + \frac{\pi}{2}\right) \]

Given \( |v_1\rangle = \dfrac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ i \end{pmatrix} \) and \( |v_2\rangle = \dfrac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ -i \end{pmatrix} \), the tensor product \( |v_1\rangle \otimes |v_2\rangle \) is

An infinitely large non-conducting thin sheet in the \(xy\) plane (\(z=0\)) carries a uniform surface charge density \(\sigma = 17.70\times10^{-12}\) C.m\(^{-2}\). The electric field in the region \(z < 0\) is \(\vec{E}_2 = \hat{x}+2\hat{y}+3\hat{z}\). Then, the electric field \(\vec{E}_1\) in the region \(z > 0\) will be (\(\epsilon_0 = 8.85\times10^{-12}\) C\(^2\).N\(^{-1}\).m\(^{-2}\))

Consider the Otto cycle for an ideal gas engine consisting of two quasistatic adiabatic and two quasistatic isochoric processes. The correct temperature-entropy (T-S) phase diagram for the cycle is: