UPSC Physics Optional PYQs (CSE & IFoS)PYQs

The raising (J_+) and lowering (J_-) operators of total angular momentum are defined by J_+ = J_x + iJ_y and J_- = J_x - iJ_y. Find the values of the following : (i) [J_x, J_+] (ii) [J_y, J_+] (iii) J_- J_+

A particle constrained to move along x-axis in the domain 0 \le x \le L has a wave function \psi(x) = \sin \left( \frac{n \pi x}{L} \right), where n is an integer. Normalize the wave function and evaluate the expectation value of momentum of the particle.

The time independent wave function of a system is \psi(x) = A \exp(ikx), where k is a constant. (i) Is this wave function normalizable in the domain -\infty < x < \infty? (ii) Calculate the probability current density for this function.

For the n^{\text{th}} state of linear harmonic oscillator, evaluate the uncertainty product (\Delta x) \cdot (\Delta p).

A particle of mass m is in a spherically symmetric attractive potential of radius a. Find the minimum depth of the potential needed to have two bound states of zero angular momentum.

Consider a particle of mass m and charge q moving under the influence of a one dimensional harmonic oscillator potential. Assume it is placed in a constant electric field E. The Hamiltonian of this particle is therefore given by H = \frac{p^2}{2m} + \frac{1}{2} m \omega^2 X^2 - qEX. Obtain the energy expression and the wave function of the n\text{th} excited state of the particle.

Find the condition at which de Broglie wavelength equals the Compton wavelength for a particle.

Evaluate the commutator \left[\frac{\partial^2}{\partial x^2}, x\right] and hence show that [p^2, x] = -2 i\hbar p.

By assuming the nucleus as a cubical box of length equal to the nuclear diameter 10^{-12}\text{ cm}, calculate the kinetic energy of the highest level occupied nucleon of iron-56 nucleus.

(i) Derive an expression for the period of Bloch oscillation for a one-dimensional crystal having lattice period a and electric field \varepsilon. (ii) Consider an electron in a perfectly periodic lattice, wherein the energy-wavenumber relationship in the first Brillouin zone is expressed as E = \frac{\hbar^2 k^2}{5\mathrm{m}_e} where \mathrm{m}_e is the mass of an electron in free space. Find the effective mass of the electron and hence write down the time-independent Schrodinger equation for the electron. Also determine the velocity of the electron. Ignore all interactions except between the electron and the lattice.

Consider the potential V(x) = \begin{cases} 0, & 0 < x < a \\ \infty, & \text{elsewhere} \end{cases} (a) Estimate the energies of the ground state as well as those of the first and the second excited states for (i) an electron enclosed in a box of size a = 10^{-10}\text{ m}. (ii) a 1\text{ g} metallic sphere which is moving in a box of size a = 10\text{ cm}. (b) Discuss the importance of the Quantum effects for both of these systems. (c) Estimate the velocities of the electron and the metallic sphere using uncertainty principle.

Consider a stream of particles of mass m each moving in the positive x-direction with kinetic energy E towards the potential barrier V(x) = 0 \quad \text{for } x \le 0 V(x) = \frac{3E}{4} \quad \text{for } x > 0 Find the fraction of particles reflected at x = 0.

Evaluate the most probable distance of the electron of the hydrogen atom in its 2p state. What is the radial probability density at that distance ?

Calculate the zero point energy for a particle in an infinite potential well for the following cases : (i) a 100\text{ g} ball confined on a 5\text{ m} long line. (ii) an oxygen atom confined to a 2 \times 10^{-10}\text{ m} lattice. (iii) an electron confined to a 10^{-10}\text{ m} atom. Why zero point energy is not important for macroscopic objects ? Comment.

Set up the Schrodinger's wave equation for one dimensional potential barrier and obtain the probability of tunnelling.

An electron in a one-dimensional infinite potential well, defined by V(x)=0 for -a\leq x\leq a and V(x)=\infty otherwise, goes from n=4 to n=2 level and emits photon of frequency 3.43\times10^{14}\,\mathrm{Hz}. Calculate the width of the well. (Assume Plank's constant h=6.626\times10^{-34}\,\mathrm{J.S.} and mass of electron m=9.11\times10^{-31}\,\mathrm{kg})

The components of the angular momenta \vec{J}_1 and \vec{J}_2 satisfy commutation rules. Show that the components of the sum \vec{J} = \vec{J}_1 + \vec{J}_2 also satisfy the commutation rules. Whether the difference \vec{J}_1 - \vec{J}_2 satisfies an angular momentum?

A one-dimensional oscillator of mass m = 10^{-20}\text{ kg} oscillates under a force constant k = 10^{-4}\text{ N/m}. (i) Evaluate the zero-point energy. (ii) Calculate the classical amplitude for which the oscillator can have this energy. (iii) What is the energy for the second excited state?

Show that E_n=\langle V\rangle in the stationary states of the hydrogen atom.

What is de Broglie concept of matter wave? Evaluate de Broglie wavelength of Helium that is accelerated through 300\mathrm{V}. (Given mass of proton = mass of neutron = 1.67\times10^{-27}\,kg)