UPSC Physics Optional PYQs (CSE & IFoS)PYQs

What is the spin wave function (for s=\frac{1}{2}) if the spin component in the direction of unit vector \eta has a value of \frac{1}{2}\hbar?

Derive an approximate expression for the transmission coefficient for a rectangular potential barrier for which \frac{a}{\hbar}\sqrt{2m(V_0 - E)} \gg 1.

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})

(i) Find the lowest energy of an electron confined in a three-dimensional cubic box of each side 1\text{ \AA}. (ii) Find the temperature at which the average energy of the molecule would be equal to the lowest energy of the electron. [Given : k = 1\cdot 38 \times 10^{-23}\text{ J/K}]

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?

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

The normalized eigenfunction for the ground state of hydrogen atom is \Psi_{100} = \frac{1}{\sqrt{\pi}} \left( \frac{Z}{a_0} \right)^{3/2} e^{-Zr/a_0} Calculate the expectation value of the radius vector r of the electron in the ground state.

The lifetime of a given atom in an excited state is 10^{-8}\text{ s}. It comes to the ground state by emitting a photon of wavelength 5800\text{ \AA}. Find the energy uncertainty and wavelength uncertainty of the photon.

Find the values of the following : (i) \vec{L} \times \vec{L} (ii) L_+ L_- (iii) [L_z, L_+] where \vec{L} is the angular momentum operator. L_+ and L_- are raising and lowering operators respectively.

An electron is confined to an infinite well whose width L (= 100\text{ pm}) is roughly the size of an atom. (i) What are the energies of four least energetic quantum states? (ii) What energy must be imparted to the electron to raise it from a state with n = 12 to a higher energy state with n = 25?

Describe normal and anomalous Zeeman effect. Explain how it lifts the degeneracy in hydrogen atom.

Normalised wave function of hydrogen atom for 1s state is \psi_{100}=\frac{1}{\sqrt{\pi a_0^3}}e^{-r/a_0},\text{ where }a_0=\frac{\hbar^2}{me^2} being the Bohr radius. Calculate the expectation value of potential energy in this state.

A beam of 12\,\mathrm{eV} electron is incident on a potential barrier of height 25\,\mathrm{eV} and width 0.05\,\mathrm{nm}. Calculate the transmission coefficient.

(i) Derive the Schrodinger time-independent wave equation for matter waves. (ii) An electron is confined to move in a one-dimensional potential well of length 5\text{ \AA}. Find the quantized energy values for the three lowest energy states.

Find the minimum magnetic field needed for the Zeeman effect to be observed in a spectral line of 400\,\mathrm{nm} wavelength when a spectrometer whose resolution is 0.010\,\mathrm{nm} is used. Write the answer in the nearest high integer.

To illustrate the idea that the zero point energy gets larger by going from macroscopic to microscopic systems, calculate the zero point energy for a particle in an infinite potential well for the following three 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.

A particle is moving in a one-dimensional box of width 50\,\mathring{\mathrm{A}} and infinite height. Calculate the probability of finding the particle within an interval of 15\,\mathring{\mathrm{A}} at the centres of the box when it is in its state of least energy.

Calculate the Larmor precessional frequency for a magnetic induction field of 0.5\,\mathrm{T}. Hence calculate the splitting in wave numbers of a spectral line due to normal Zeeman effect for the same field.

Use WKB method to estimate the energy levels of a one-dimensional harmonic oscillator.

Find the uncertainty in the momentum of a particle when its position is determined within 0.02 cm. Find also the uncertainty in the velocity of an electron and \alpha-particle respectively when they are located within 15 \times 10^{-8}\,cm.