UPSC Physics Optional PYQs (CSE & IFoS)PYQs

State the quantum numbers I_3, Y and S for the uds quarks and antiquarks. Which combination of these leads to the formation of (1) proton and (2) neutron ?

A star converts all its hydrogen to helium, achieving 100% helium composition. It then converts the helium to carbon via the reaction {}^{4}_{2}\text{He} + \text{He} \rightarrow {}^{12}_{6}\text{C} + 7.27\text{ MeV} The mass of the star is 5 \times 10^{32}\text{ kg} and it generates energy at the rate of 5 \times 10^{30}\text{ W}. How long will it take to convert all helium to carbon at this rate ?

What are mirror nuclei ? How does the charge independence of nuclear force emerge from this concept?

Write down the Bethe-Weizsacker semi-empirical mass formula for a nucleus. Explain the significance of each term occurring in it. Discuss the stability of a nucleus against \beta-decay. What is the effect of pairing term on stability?

(i) Calculate the Q-value of the reaction {}^9_4\text{Be} (^4_2\text{He}, n) ^{12}_6\text{C} Given \text{Mass } (^9\text{Be}) = 9.012183\text{ u} \text{Mass } (^4\text{He}) = 4.002603\text{ u} \text{Mass } (^{12}\text{C}) = 12.000\text{ u} Write down the nucleonic configuration of {}^{7}\text{Li}, {}^{12}\text{C}, {}^{17}\text{O} and {}^{27}\text{Al} in the ground state of the nuclear shell model and hence calculate the corresponding ground state angular momenta and parities. How do the observed ground state angular momenta and parities agree with those predicted on the basis of the shell model ?

Write an expression for the energy of a diatomic molecule executing both rotation and vibration in a given electronic state. Explain the observed spectra giving a schematic diagram. What are O, P, Q, R and S branches ?

(i) State and explain Franck-Condon principle. (5) (ii) In three separate pairs of electronic states (one ground and another upper state) of diatomic molecules, the average internuclear distances are -

1. equal;
2. slightly greater;
3. appreciably greater. In accordance with Franck-Condon principle, obtain most favorable transitions from the ground state vibrational levels in each of the above situations. Illustrate the transitions by drawing suitable energy-level diagrams. (15)

Describe Stern-Gerlach experiment. Explain how it demonstrates the discrete nature of the magnetic moment of an atom.

(i) What is Lamb shift ? Has it been observed in any atom other than hydrogen? (10) (ii) How does the 21 cm hydrogen line originate? Comment on its astronomical applications. (10)

(i) State and explain Heisenberg's uncertainty principle. Experimental data reveal that no electron in an atom has energy greater than 4 MeV. Assuming that the radius of a nucleus is 10^{-14}\text{ m}, show using Heisenberg's uncertainty principle that an electron cannot exist inside the nucleus. (10) (ii) Calculate the de Broglie wavelength of thermal neutrons at 300 K. (10)

(i) Define angular momentum, Express it in operator form and show that \vec{L} \times \vec{L} = i\hbar \vec{L} Explain the physical significance of this relation. (10) (ii) Let Y_{lm} be an eigenstate of L^2 and L_z with eigenvalues l(l + 1)\hbar^2 and m\hbar, respectively. Show that \phi = (L_x + i L_y) Y_{lm} is likewise an eigenstate of L^2 and L_z, and determine the eigenvalues. (10)

A one-dimensional potential barrier is represented by the function V(x) = 0 \quad \text{for } x < 0 = V_0 \quad \text{for } x > 0 Where V_0 is positive. Find the transmission coefficient for particles of mass m incident from the left on the barrier.

Determine the discrete energy levels and the corresponding eigenfunctions for a particle in an infinitely deep potential well inside a cube of dimension L, i.e., assume \begin{gathered} V(x,y,z) = 0 \quad \text{for } 0 < x < L;\\ 0 < y < L;\\ 0 < z < L = \infty \quad \text{elsewhere} \end{gathered}

Solve the Schrodinger equation for a linear harmonic oscillator. Obtain the eigenvalues and the corresponding eigenfunctions.

In the free electron theory of metals, a conductor is regarded as consisting of free electrons in a three-dimensional box. Using the results of (a), obtain an expression for the density of states,

Establish the relation C_p - C_v = \left[ P + \left(\frac{\partial U}{\partial V}\right)_T \right] \left(\frac{\partial V}{\partial T}\right)_P Use it to find out an expression for C_p of one mole of a gas whose internal energy is given by U = cT - \frac{a}{V} and which satisfies the equation of state \left[ P + \frac{a}{V^2} \right](V - b) = RT. Here a, b and c are constants.

Derive a relation between the total number of Fermions in terms of Fermi momentum and hence obtain the expression for the total energy E of the system at absolute zero. Combine this expression with the equation of state PV = \frac{2}{3} E to show that the pressure of an ideal Fermi gas at T = 0 is proportional to 5/3 power of its number density.

Prove the law of increase of entropy. Show that for a system at fixed temperature and pressure to be in equilibrium, its Gibbs free energy should be minimum.

A system at temperature T_1 is brought in contact with a reservoir at temperature T_2 > T_1. When the system and the reservoir reach thermal equilibrium, calculate the change in entropy of the universe assuming the heat capacity of C_p of the system to be constant. Discuss whether the considered change is positive or not.

State Amperes law of magnetostatics. Using this law, find the magnetic field at a point due to an infinitely long filamentary current.