UPSC Physics Optional PYQs (CSE & IFoS)PYQs

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.

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.

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.

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

A particle of rest mass m_0 has a kinetic energy K, show that its de Broglie wavelength is given by \lambda=\frac{hc}{\sqrt{\left[K\left(K+2m_0c^2\right)\right]}}. Hence calculate the wavelength of an electron of kinetic energy 2\mathrm{MeV}. What will be the value of \lambda if K << m_0c^2?

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.

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.

(i) What do you mean by expectation value of a physical quantity ? How does it help to extract information from a wave function ? (ii) A particle limited to move along x-axis has the wave function \psi = ax between x = 0 and x = 1, \psi = 0 elsewhere. Find the probability that the particle can be found between x = 0\cdot 45 and x = 0\cdot 55. Find the expectation value <x> of the particles position x = 0 to x = 1.

Derive energies and wave functions for motion of electron in a hydrogen atom.

Draw a neat sketch showing potential wells and energy levels of the following : (i) Harmonic oscillator (ii) Hydrogen atom (iii) Particle in a box List the contrasting points and mention the variation of $E_n$' with n' of the above plots.

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}.

(i) Draw a neat sketch of a finite rectangular potential barrier well of height U' and width L' that contains a particle whose energy E' is less than U'. Solve the Schrödinger equations of the particles outside the well. (ii) Plot wave functions and probability densities |\psi|^2 of the particle for the first three states. Compare the results with that of a particle in a box of infinite potential (U = \infty).

(i) Discuss about the validity criterion for the WKB approximation. (ii) Qualitatively explain alpha (\alpha) particle emission by WKB approximation.

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

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

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.

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}}.

(i) State and explain position momentum uncertainty principle. Justify that this principle is not just a negative statement rather a useful tool, with one example. (ii) The lifetime of an excited state of an atom is about 10^{-8}\text{ sec.} Calculate the minimum uncertainty in the energy of the excited state.

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

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}.