UPSC Physics Optional PYQs (CSE & IFoS)PYQs

Explain how the magnetic moments of atoms, the space quantization of angular momentum and the spin of electron are measured using Stern–Gerlach experiment.

A particle is described by the wave function \Psi(x)=\left(\frac{\pi}{2}\right)^{-1/4}e^{-ax^2/2}. Calculate \Delta x and \Delta p for the particle, and verify the uncertainty relation \Delta x\Delta p=\frac{\hbar}{2}.

A blue lamp emits light of mean wavelength of 4500\ \mathring{\mathrm{A}}. The rating of the lamp is 150 W and its 8% of the energy appears as light. How many photons are emitted per second by the lamp?

Consider a Hermitian operator A with property A^3=1. Show that A=1.

Prove the commutation relation for the angular momentum: [L^2,L_z]=0 Also show that (\vec{L}\times\vec{L})=i\hbar\vec{L}.

If the z-component of an electron spin is +\frac{\hbar}{2}, what is the probability that its component along a direction z' (forming an angle \theta with z-axis) is \frac{\hbar}{2} or -\frac{\hbar}{2}? What is the average value of spin along z'?

Calculate the Hall coefficient of sodium based on the free electron model. Sodium has BCC structure and the side of the cube is 4.28\,\mathring{\mathrm{A}}.

Prove that Bohr hydrogen atom approaches classical conditions, when n becomes very large and small quantum jumps are involved.

Using the uncertainty principle \Delta x\Delta p \geq \hbar/2, estimate the ground state energy of a harmonic oscillator.

Find the probability current density for the wave function \Psi(x,t)=\left[Ae^{ipx/\hbar}+Be^{-ipx/\hbar}\right]e^{-ip^2t/2m\hbar} Interpret the result physically.

Write the wave functions for a particle on both sides of a step potential, for E>V_0: V(x)=\begin{cases} V_0, & x>0\\ 0, & x<0 \end{cases}

Interpret the results physically.

What is Zeeman effect? Explain Zeeman effect on the basis of classical electron theory.

(i) The energy level of a quantum harmonic oscillator with frequency \nu is given by E_n=\left(n+\frac{1}{2}\right)h\nu,\quad \text{where } n=0,1,2,\ldots Calculate its partition function.

(ii) Calculate the partition function of a two level system.

State the first law of thermodynamics for a diffusively interacting system. The temperature of 10\ \mathrm{g} of air is raised by 2^\circ\mathrm{C} at constant volume. Calculate the increase in its internal energy. Given: C_v=0.172\ \mathrm{cal\ g^{-1}\ {}^\circ C^{-1}}.

Discuss the principle of adiabatic demagnetization process to achieve low temperatures. Determine the fall in temperature produced by adiabatic demagnetization of a paramagnetic material at initial temperature of 3\ \mathrm{K} when the magnetic field is switched off from 10{,}000 oersted to zero. Given: heat capacity at constant magnetic field =0.2\ \mathrm{J\ g^{-1}\ K^{-1}} and Curie constant per gram mole per \mathrm{cm^3} =0.042\ \mathrm{erg\ K^{-1}\ g^{-1}\ Oe^{-2}}.

Obtain the Clausius-Clapeyron equation. Using this equation, show that for the phase boundary of the liquid and vapour phases, p–T relation can be written as p=p_0e^{-L/kT}. Here it has been assumed that the latent heat L is independent of temperature, that vapour is treated as an ideal gas and that V_{\mathrm{vapour}}=V\gg V_{\mathrm{liquid}} and that p\to p_0 as T\to\infty.

Starting from Maxwell-Boltzmann distribution for a free particle in 3-dimension, obtain the expression for root mean square (rms) speed of a particle. Calculate the rms speed of nitrogen \mathrm{N}_2 molecule at room temperature 27^\circ\mathrm{C}.

Using the two-fluid model of a conductor (normal and superconducting) and the Maxwell's equations, derive the two London equations of superconductivity.

Write Maxwell's equations in free space in both differential and integral forms. Obtain wave equations and show that electromagnetic waves can travel in free space with a speed of light. Can one get the wave equations from the integral form of the Maxwell's equations?

Describe the oscillations of electric and magnetic fields in an ideal LC circuit. The applied voltage phasor in a circuit is (4+3i) volt and resulting current phasor is (3+4i) ampere. Draw the phasor diagram. Determine the impedance of the circuit and indicate whether it is inductive or capacitive in nature. Also find the power dissipation in the circuit.