UPSC Physics Optional PYQs (CSE & IFoS)PYQs

What are the field quanta of weak and electromagnetic interactions ? Show that weak interaction is a short range force while electromagnetic interaction is a long range one.

At what energies, do we anticipate the unification of strong and electroweak interactions ? What will happen to coupling constants of these interactions in this situation ? How does unification indicate the decay of proton and existence of leptoquark ?

Write down the Yukawa potential and derive an expression for the mass of \pi-meson in terms of range of nuclear force. What part of the Nucleon – Nucleon interaction is explained by this potential ?

2^2S_{y_2} level in H atom is 1058 \text{ MHz} above the 2^2P_{y_2} level. (i) What is this known as ? (ii) Express the above frequency in \text{cm}^{-1}. (iii) Calculate the energy difference between the above two levels in eV.

(i) Establish that hc = 1240 \text{ eV. nm} = 1240 \text{ MeV. fm} (ii) The energy levels of a hydrogen atom are given by E_n = (-1/n^2) \text{ Ryd,} where 1 \text{ Ryd} = hcR. Show that R = 1 \cdot 097 \times 10^7 \text{ m}^{-1}. What exactly is R ?

(i) Find the ground state total orbital (L) and total spin (S) quantum numbers for nitrogen. (ii) Find the L and S values of the two "first excited states" of helium atom.

What was the aim of Stern – Gerlach experiment ? Why were silver atoms chosen instead of electrons in the experiment ? What was the outcome of the experiment ?

Explain fluorescence and phosphorescence in electronically excited molecules.

Discuss the vibrational spectra of a diatomic molecule treating it as a harmonic oscillator as well as an anharmonic oscillator and compare them.

Explain Born – Oppenheimer approximation. Discuss the intensity distribution in the vibrational electronic spectra of a diatomic molecule on the basis of Franck – Condon principle.

Show that the Pauli Spin Matrices obey the following relations : (i) \text{Tr} (\sigma_x) = \text{Tr} (\sigma_y) (ii) \det (\sigma_y) = \det (\sigma_z) (iii) The eigenvalues of \sigma_z and \sigma_x are the same. (iv) Write down the y-component of the spin angular momentum matrix corresponding to an antineutrino.

Show that the time-dependent part of all the solutions of the Schrödinger equation in one-dimension has the structure \phi(t) = \exp (- i E t / h), provided the potential is not an explicit function of time.

(i) Consider a particle in a three-dimensional box. Derive an expression for g(E), the density of states. (ii) Show that \frac{g(p)}{g(E)} = \frac{dE}{dp} , where g(p) is the density of states in the momentum space. Deduce that g(p) is proportional to p^2 for a free non-relativistic particle.

Using dimensional analysis, explain why the angular momentum of a particle cannot be \hbar^2.

(i) The quantum mechanical probability distribution function of an electron in the ground state of the hydrogen atom is P(r) = N r^2 \exp (-2br). Using the result \int_0^\infty P(r) \, dr = 1, deduce that N is proportional to b^3. (ii) Prove that the value of 40 \, k_B T at T = 300 \text{ K} is nearly 1 \text{ eV}. Hence determine the Fermi temperature of a metal whose Fermi energy is 9 \cdot 4 \text{ eV}. (iii) Show that the Fermi velocity is related to the Fermi energy of electrons through the relation \frac{v_F}{c} = 1 \cdot 98 \left( \frac{E_F}{1 \text{ MeV}} \right)^{1/2} .

(i) Consider a positron in a box. If the energy released is 60 \text{ eV} when it jumps from the third excited state to the ground state, show that the width of the potential is nearly 0 \cdot 3 \text{ nm}. (ii) Prove that the most probable distance of an electron from the proton (in the hydrogen atom) is the Bohr radius of the hydrogen atom. Consider only the ground state.

(i) Explain spin-orbit coupling of an atomic electron. (ii) Show that the 2p state in the H atom splits up into two substates due to spin-orbit coupling. (iii) Calculate the energy of separation in eV, resulting from the spin-orbit coupling when the magnetic field experienced by the electron is 0 \cdot 4 \text{ T}.

Derive an expression for the specific heat of a solid on the basis of Debye's model. Show that it converges to Dulong and Petit's law at high temperatures.

Energy distribution for n_i particles in classical statistical mechanics is given by n_i = g_i e^{-\alpha -\beta \varepsilon_i} where \alpha and \beta are constants. g_i is the single particle states in the i\text{ th} level. Using equipartition theorem, show that correct thermodynamic interpretation is \beta = \frac{1}{kT} (Use \int_0^\infty e^{-x} x^{1/2} \, dx = \frac{\sqrt{\pi}}{2} and \int_0^\infty e^{-\beta x} x^{3/2} \, dx = \frac{3\sqrt{\pi}}{4\beta^{5/2}})

Define entropy. How is it related to disorder? Hence, derive the Boltzmann relation S = k \log \Omega, where \Omega is the probability and k is the Boltzmann constant. Show that for any type of process, involving a closed system \Delta S \ge \frac{\Delta Q}{T} where the equality sign applies for internally reversible processes and the inequality for internally irreversible processes.