IGNOU MPH 11 SOLVED ASSIGNMENT

Included:

MPH 011 (January 2024 - July 2024) - ENGLISH

Tutor Marked Assignment

STATISTICAL MECHANICS

Course Code: MPH-011

Assignment Code: MPH-011/TMA/2024-25

Max. Marks: 100

Note: Attempt all questions. The marks for each question are indicated against it.

PART A

1. a) Calculate the mean, and the variance of the following probability density functions:

i) Uniform 

ii) Rayleigh distribution 

b) Show that for any distribution where N is large, the distribution tends to a Gaussian or a normal distribution.

2. a) Consider a single classical 1-D non-relativistic harmonic oscillator. Obtain x(t) and p(t) as a function of time t. Assume that initially at t = 0, the oscillator is at the maximum displacement  .Plot its phase space trajectory for fixed total energy E

b) What is Gibb’s paradox? Derive Sackur-Tetrode equation of entropy for a classical, ideal gas.

c) Obtain an expression of probability of finding the system in the microstate k corresponding to energy E_{k i}n canonical ensemble. Also, write the expression of canonical partition function.

d) Calculate relative fluctuations in the energy of the system about the mean value in canonical ensemble.

e) Consider a gas in lattice-gas model, in which gas contained in volume V is assumed to be divided into fixed small cubic cells of volume V0 such that each of the (cell) can contain at most one atom. The system behaves as a set of distinguishable and independent cells in contact with a reservoir at temperature T and chemical potential .Obtain the expressions of canonical partition function and average number of particles

3. Derive quantum Liouville equation. Write this equation for the Steady state.

PART B

4. a) Using the result   show that the expression of density matrix of a

free particle in box of volume V in the canonical ensemble in the coordinate representation is given by

b) Obtain Fermi-Dirac distribution function. Draw it for  What is its physical significance? 

c) Explain Pauli’s paramagnetism. Show that at T = 0 Pauli’s paramagnetic susceptibility is given by

where  is the intrinsic magnetic moment of the particle of mass m.

d) How many photons are present in 3 1cm of radiation at 727^oC? What is their average energy?

5. a) Derive the expression for the second virial coefficient B₂ in terms of the intermolecular potential  using the approximation  Use the Lennard-Jones potential to outline its calculation.

b) Using the Mayer f-function, show that the second virial coefficient B₂ for  gas

   is the

diameter of the sphere. Here  

c) State Ising model and obtain the partition function (ln Z) for 1D in the absence of an external magnetic field up to second order perturbation. Explain the physical significance of each term in the Hamiltonian and discuss the role of the coupling constant J.

MPH 011 (January 2026 - July 2026) - ENGLISH

Tutor Marked Assignment
STATISTICAL MECHANICS

Course Code: MPH-011
Assignment Code: MPH-011/TMA/2026

Max. Marks: 100

Note: Attempt all questions. The marks for each question are indicated against it.

PART A

1. a) Show under what condition Poisson distribution tends to Normal distribution.

b) A monthly demand for commodity is a continuous random distribution with probability distribution function given as:

where N is normalization constant. Obtain the value of N so that the function is normalized and hence, find the mean and the variance.

c) N particles obey Maxwell-Boltzmann distribution. They are distributed among three states with energies and . If the equilibrium energy of the system is approximately 3000k_BT, calculate the total number of particles.

2. a) Obtain the phase space area enclosed by a classical harmonic oscillator for energies ranging from 0 to E.

b) The Hamiltonian for the non-relativistic, non-interacting, monoatomic ideal gas is given by

where m is the mass of the particle and U(q_i) is the potential energy. Obtain an expression of the phase space volume of the energy shell and hence obtain an expression of number of microstates.

c) Obtain expression for average energy  for the canonical ensemble in terms of partition function Z.

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

PART B

3. a) i) A proton  inside a nucleus  may have speed upto . How many quantum states are available to it?

ii) The time evolution of the quantum state,  is given by the Schrödinger wave equation:

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

b) State and prove quantum Liouville equation.

c) Using the relation of thermodynamic probability

Obtain an expression of Fermi Dirac distribution function. Show a plot of  (the occupation index of a state corresponding to energy ) versus  for different temperatures and discuss it. What is the physical interpretation of Fermi energy?

d) Show that the probability that a state with energy  above the Fermi level  filled is equal to the probability of a state with energy  below the Fermi level  is empty.

e) 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.

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

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

c) Using entropy as a function of temperature and pressure, obtain the first Ehrenfest’s equation.

d) Discuss the concept of thermodynamic fluctuations in the canonical ensemble. Define energy fluctuations and derive an expression for the mean square fluctuation of energy. Using a classical ideal gas, show that the relative fluctuation in energy varies as  and hence, justify why thermodynamic calculations are valid for ordinary macroscopic systems.

Q: How will I receive the PDF?
A: Immediately after payment, the download link will appear and be sent to your email.

Q: Is this hand-written or typed?
A: This is a professional typed computer PDF. You can use it as a reference for your handwritten submission.