Solve your IGNOU Doubts

Question:

What is meant by Cluster Integrals? Express B₂ and B₃ in terms of Cluster Integrals.

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Question:

The susceptibility of $\text{O}_2$ is $0.39 \times 10^{-3}$ and its density is $1.43\ \text{kg m}^{-3}$ . Calculate its total polarisability. Assume that the mass number for $\text{O}_2$ is 32.

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Question:

In an n-type semiconductor the Fermi level lies $0.4\ \text{eV}$ below the conduction band at $T = 300\ \text{K}$ . Calculate the position of the Fermi level when the temperature is raised to $330\ \text{K}$ .

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Question:

State the Virial theorem and prove Boyle’s law using it.

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Question:

Calculate the resistivity of Silicon at $60^\circ \text{C}$ given that its resistivity at $30^\circ \text{C}$ is $2.5\ \Omega\text{m}$ .

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Question:

What is the Bose-Einstein condensation? Write an expression of number of particles in the excited state (N_ex) in terms of Bose-Einstein condensation temperature (T_c), total number of bosons (N) in an assembly and temperature T. Show a plot of distribution of bosons as a function of temperature below and above the T_c.

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Question:

Calculate the energy dispersion for s-band in the bcc lattice for the tight binding approximation.
Note: For the central atom located at (0,0,0) in the bcc unit cell, the nearest neighbours are located at $\frac{a}{2}(\pm1, \pm1, \pm1)$ , where a is the lattice constant.

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Question:

Show that the probability that a state with energy $\delta\epsilon$ above the Fermi level $\epsilon_F$ filled is equal to the probability of a state with energy $\delta\epsilon$ below the Fermi level $\epsilon_F$ is empty.

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Question:

Using the relation of thermodynamic probability

$$W = \prod_{i=1}^N \frac{g_i}{N_i!(g_i - N_i)!}$
Obtain an expression of Fermi Dirac distribution function. Show a plot of $f(\epsilon)$ (the occupation index of a state corresponding to energy $\epsilon_i$ ) versus $\epsilon$ for different temperatures and discuss it. What is the physical interpretation of Fermi energy?

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Question:

For the energy dispersion relation:
$\varepsilon(\vec{k}) = 2t_x \cos(k_x a_x) - 2t_y \cos(k_y a_y) - 2t_z \cos(k_z a_z)$
calculate the inverse mass tensor.

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Question:

State and prove quantum Liouville equation.

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Question:

The time evolution of the quantum state, $\psi(q,t)$ is given by the Schrödinger wave equation:
$$i\hbar \frac{\partial \psi}{\partial t} = \hat{H}\psi$

where H is the Hamiltonian operator. Using this equation, obtain time evolution of the wave function when the energy eigenvalue is E.

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Question:

In Sommerfeld free electron theory, show that at a temperature T
(i) $\mu = \varepsilon_F \left[ 1 - \frac{1}{3} \left( \frac{\pi k_B T}{2 \varepsilon_F} \right)^2 \right]$ and (ii) $c_v = \frac{\pi^2}{2} \left( \frac{k_B T}{\varepsilon_F} \right) n k_B$

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Question:

The time evolution of the quantum state, $\psi(q,t)$ is given by the Schrödinger wave equation:
$$i\hbar \frac{\partial \psi}{\partial t} = \hat{H}\psi$

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Question:

A proton $(\text{mass} = 1.67 \times 10^{-27} \text{ kg})$ inside a nucleus $(\text{radius} = 10^{-14} \text{ m})$ may have speed upto $2.0 \times 10^8 \text{ ms}^{-1}$ . How many quantum states are available to it?

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Question:

A divalent metal crystallizes in an fcc structure with a lattice constant of 4.5 A. Calculate the number density of conduction electrons and the Fermi velocity.

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Question:

For the grand canonical ensemble, obtain an expression for the ensemble average energy and ensemble average particle numbers.

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Question:

Calculate the temperature at which the lattice contribution to the specific heat and the electronic contribution to the specific heat become equal in a metal which has a Debye temperature of $390 \text{ K}$ and a Fermi energy $11.7 \text{ eV}$ .

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Question:

For a linear chain of identical atoms of mass $10^{-26} \text{ kg}$ calculate the maximum value of the angular frequency of the longitudinal wave and the group velocity at $k=0$ , given that the inter-atomic distance is 2.0 A and the spring constant is $20 \text{ Nm}^{-1}$ .