PYQ Bank

(i) What angles do the \vec{L} \text{(vector)} make with the z-axis when l = 2 for an electron ? (ii) Determine the values of the total angular momentum for a 3d electron.

What are Pauli spin matrices ? Show that : (\vec{\sigma}\cdot\vec{A})(\vec{\sigma}\cdot\vec{B}) = \vec{A}\cdot\vec{B} + i\vec{\sigma}\cdot(\vec{A}\times\vec{B}) where \vec{\sigma} are the Pauli spin matrices and \vec{A} and \vec{B} are vector operators which commute with \vec{\sigma}, but do not necessarily commute with each other.

Solve the eigen value equation L^2 Y(\theta, \phi) = \lambda \hbar^2 Y(\theta, \phi) and obtain the eigen values and eigen functions of L^2.

The normalized wave function for the electron in the ground state of the hydrogen atom is given by \psi(r)=\frac{1}{(\pi a_0^3)^{1/2}}e^{-r/a_0} where a_0 is the radius of the first Bohr orbit. Calculate \langle r\rangle and \left\langle\frac{1}{r}\right\rangle.

For a quantum mechanical system prove that all energy eigen-values E_n are real and if E_n \neq E_k, then the corresponding eigen functions are orthogonal.

Show that {}^2S_{\frac{1}{2}}, {}^2P_{\frac{1}{2}} and {}^2P_{\frac{3}{2}} levels of sodium spectrum are split in the ratio of 3:1:2 due to anomalous Zeeman effect.

Calculate (\Delta x)^2, where \Delta x = x - \langle x \rangle.

What led van der Waals to modify the ideal gas equation? Using the concepts of critical temperature T_c, pressure P_c and volume V_c, show that the critical constant for a real gas is 8/3.

Calculate the values of van der Waals constants a and b for oxygen with T_c=154.2\,\mathrm{K}, P_c=49.7 atmosphere and R=80\,\mathrm{cm^3\,atmosphere/K}.

Show that the elemental quantity of heat \delta Q is not a total differential.

Find out the expressions for van der Waals constants a and b.

Write down the expressions for Bose-Einstein and Fermi-Dirac distribution functions, and show how Fermi-Dirac distribution leads to the explanation of Pauli's exclusion principle.

Write down the expression for the Bose-Einstein distribution function and explain the meaning of the symbols used.

Consider a gas of photons in equilibrium and contained in a volume V at a temperature T. Using the Bose-Einstein statistics, calculate the total energy of the photon gas.

The electric potential of a grounded conducting sphere of radius a in a uniform electric field is given as \phi(r,\theta)=-E_0r\left[1-\left(\frac{a}{r}\right)^3\cos\theta\right] Find the surface charge distribution.

In deriving the Rayleigh-Jeans law, we count the number of modes dn corresponding to a wave number k for a photon gas in a cubical box. Consider a cubical container of volume V containing such gas in equilibrium. Calculate the differential number of allowable normal modes of frequency \omega.

Find whether the discharge of a condenser through the inductive circuit is oscillatory when C = 0.1\,\mu\mathrm{F}, L:=10\,\mathrm{mH} and R = 200\,\Omega. If it is oscillatory, calculate its frequency.

An electrical circuit consists of a resistance R, inductance L and capacitance C in series. If a charge is put on the capacitor at some instant, determine the condition that V_C, the voltage across the capacitor, is subsequently oscillatory. Derive an expression for the quality factor Q of the circuit by considering the decay of the oscillation, using the result that the amplitude falls by a factor of e in \left(\frac{Q}{\pi}\right) period.

A long solenoid has 220 turns /cm; its diameter is 3.2 cm. Inside the solenoid at its centre, we place a 130-turn closepacked coil of diameter 2.1 cm along its axis. The current in the solenoid is increased from zero to 1.5 amperes at a steady rate over a period of 0.16 second. What is the magnitude of the induced e.m.f. that appears in the central coil when the current in the solenoid is being changed?

Consider the Earth as a black body. Radiations from the Sun arrive at the surface of the Earth with an average intensity of S\text{ watts/m}^2. If the reflection coefficient of the Earth's surface is \alpha, determine the temperature of the Earth under equilibrium conditions.