UPSC Physics Optional PYQs (CSE & IFoS)PYQs

Explain spin-orbit coupling. Discuss the splitting of spectral lines of H-atom due to spin-orbit coupling.

The quantum numbers of two electrons in a two-valence electron atom are; n_1=8\quad l_1=4\quad s_1=\frac{1}{2} n_2=7\quad l_2=2\quad s_2=\frac{1}{2}

(i) Assuming L-S coupling, find the possible value of L and hence of J.

(ii) Assuming j-j coupling, find the possible values of J.

The first line in the pure rotational spectrum of HCl appears at 21\mathbin{\cdot}18\,\mathrm{cm}^{-1}. Calculate bond length of the molecule. Given atomic masses of H and Cl are 1.008 and 35.45\,\mathrm{amu}, respectively.

In observing the Raman spectrum of a sample using 3637\,\mathring{\mathrm{A}} as the exciting line, one gets stoke line at 3980\,\mathring{\mathrm{A}}. Deduce the Raman shift in \mathrm{m}^{-1} units. Compute the wavelength in \mathring{\mathrm{A}} for corresponding stokes and anti-stokes lines if the exciting line is 6465\,\mathring{\mathrm{A}}.

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.

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?

Using Pauli spin matrices prove that,

(i) \sigma_x\sigma_y+\sigma_y\sigma_x=0;\ \sigma_y\sigma_z+\sigma_z\sigma_y=0;\ \sigma_x\sigma_z+\sigma_z\sigma_x=0

(ii) \sigma_+\sigma_-=2(1+\sigma_z)

(iii) \sigma_\alpha+\sigma_\beta=i\sigma_\gamma where \alpha\ne\beta\ne\gamma

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.

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.

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 probability of finding a simple harmonic oscillator within the classical limits if the oscillator is in its normal state. Also show that if the oscillator is in its normal state, then the probability of finding the particle outside the classical limits is approximately 16\%.

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.

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.

Write the expression for the Fermi-Dirac distribution. Plot the Fermi-Dirac distribution at T=0 and for T_1>T_2>0. Now from the plot propose two alternative definitions of the Fermi level.

Calculate the probability of an electron occupying an energy level 0.02\,\mathrm{eV} above the Fermi level at T=300\,\mathrm{K}

Explain the characteristics of the following thermodynamic processes for a perfect gas:

(i) Isothermal process

(ii) Adiabatic process

(iii) Isobaric process

(iv) Isochoric process Obtain the expression for the work done by the gas during the above processes.

Eight indistinguishable balls are to be arranged in six distinguishable boxes. Calculate the total number of ways in which the above can be done.

Using the expression for internal energy U = 3N \frac{\hbar\omega}{e^{\hbar\omega/k_B T}-1}, show that Einstein specific heat capacity is given by; C = 3R \left(\frac{\hbar\omega}{k_B T}\right)^2 \frac{e^{\hbar\omega/k_B T}}{\left(e^{\hbar\omega/k_B T}-1\right)^2} Also show that Einstein specific heat capacity given above is proportional to e^{-\hbar\omega/k_B T} at very low temperature.

The melting point of tin is 232^{\circ}\mathrm{C}, its latent heat of fusion is 14\ \mathrm{cal/g} and the specific heat of solid and molten tin are 0.055 and 0.064\ \mathrm{cal/g}\,{}^{\circ}\mathrm{C} respectively. Calculate the change in entropy when 1.0\mathrm{gm} of tin is heated from 100^{\circ}\mathrm{C} to 300^{\circ}\mathrm{C}.