UPSC Physics Optional PYQs (CSE & IFoS)PYQs

What do you understand by Raman effect ? Explain with help of a diagram, how it can be observed experimentally.

Write full form of acronyms EPR and NMR. Give underlying principle of EPR.

On the basis of electronic spectra of molecules explain, how fluorescence and phosphorescence occur. Distinguish between the two.

Assume a diatomic molecule consisting of two at- oms of masses m_1 and m_2, separated by a distance \vec{r}. Write down the Hamiltonian operator for the molecule. Determine the rotational energy levels of the molecule.

Draw the potential energy of a diatomic molecule as a function of interatomic distance. Mark the vibrational and rotational energy levels. Explain the selection rule for transition between vibrational states.

For transition to the ground state what is the longest wavelength that can be emitted by hydrogen ?

The Hamiltonian of a particle moving along the x-axis is given by \hat{H} = -\alpha \frac{d^2}{dx^2} + 16\alpha \hat{x}^2, where \alpha is a real and positive constant having dimensions of energy. (i) If \psi(x) = A e^{-2x^2}, find the normalization constant A. Check whether \psi is an eigen function of \hat{H}. If yes, find the corresponding eigen value. (ii) Calculate the probability of finding the particle anywhere along the negative x-axis. (iii) Find the eigen value of \hat{H} corresponding to the eigen function \phi(x) = x\psi(x), where \psi(x) is the same as in part (i). (iv) Are the wave functions \psi(x) and \phi(x) orthogonal ?

Explain normal and anomalous Zeeman effect. Obtain expression for Zeeman splitting of an alkali metal spectral line, and illustrate with an example.

Write (do not derive) the formula for the energy levels of a particle in a three dimensional cubical box of side L. How many electrons can occupy the level having energy 66h^2/8mL^2 ?

Show that for the one dimensional wave function \psi(x) = \begin{cases} \frac{1}{\sqrt{2a}} & , \quad |x| < a \\ 0 & , \quad |x| > a \end{cases} where a is a real constant, the rms uncertainty in momentum is infinite.

An electron is moving freely in a one-dimensional infinite potential box with walls at x = 0 and x = a. If the electron is initially in the ground state of the box and if suddenly the wall at x = a is moved x = 4a, calculate the probability of finding the particle in the ground state of the new box.

Write the commutation relations of angular momentum operator \hat{L}_x, \hat{L}_y and \hat{L}_z and calculate the commutators [\hat{L}_+, \hat{L}_z] and [\hat{L}_-, \hat{L}_z]. Show that \hat{L}_+ |l, m\rangle = \sqrt{l(l+1) - m(m+1)} |l, m+1\rangle, where |l, m\rangle is the state with definite values for L^2 and L_z.

An electron is in the spin state \chi = A \begin{pmatrix} 3i \\ 4 \end{pmatrix}. Determine the normalization constant A. Find the expectation value of the spin operator \hat{S}_x and also the uncertainty in the value of S_x in this state.

Show that the probability of transmission across the step barrier represented by the potential V(x) = \begin{cases} 0 & \text{for } x < 0 \\ V_0 & \text{for } x > 0 \end{cases} is T = \frac{4 k_1 k_2}{(k_1 + k_2)^2}, where k_1 and k_2 are wave numbers in regions x < 0 and x > 0, respectively.

State second law of thermodynamics. Prove that no engine operating between two given temperatures is more efficient than a carnot engine operating between same two temperatures.

Explain the Debye model of specific heat of solids. What are its success and failures ?

Show that at T = 0, the Fermi distribution function has a value 1 for energies less than the Fermi energy \epsilon_F and is zero above it. For a system of non-interacting electrons at T = 0, show that the ground state energy of the system of N particles is \frac{3}{5} N \epsilon_F.

Consider N independent particles, each of which can be in either of two energy states +\epsilon and -\epsilon. Derive an expression for the entropy of a microcanonical ensemble for this system, where f = \sum_i \epsilon_i / N\epsilon is fixed, where \epsilon_i is the energy of the i-th particle.

State the law of equipartition of energy. Show how this law can be used to calculate specific heat of gases and hence find the ratio \gamma = C_p/C_v for diatomic and triatomic gases.

One mole of an ideal gas is compressed at constant temperature T from a volume V_1 to a volume V_2. Find the work done and heat absorbed by the gas. The gas now expands adiabatically to a volume 2V. Taking the gas to be diatomic, calculate the final temperature of the gas.