IGNOU MPH 11 SOLVED ASSIGNMENT

Ques 1.

Show under what condition Poisson distribution tends to Normal distribution.

Ques 2.

 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.

Ques 3.

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.

Ques 4.

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

Ques 5.

 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.

Ques 6.

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

Ques 7.

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

Ques 8.

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

Ques 9.

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

Ques 10.

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.

Ques 11.

State and prove quantum Liouville equation.

Ques 12.

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?

Ques 13.

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.

Ques 14.

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.

Ques 15.

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

Ques 16.

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

Ques 17.

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

Ques 18.

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.

Ques 19.

Show under what condition Poisson distribution tends to Normal distribution.

Ques 20.

 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.

Ques 21.

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.

Ques 22.

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

Ques 23.

 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.

Ques 24.

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

Ques 25.

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

Ques 26.

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

Ques 27.

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

Ques 28.

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.

Ques 29.

State and prove quantum Liouville equation.

Ques 30.

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?

Ques 31.

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.

Ques 32.

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.

Ques 33.

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

Ques 34.

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

Ques 35.

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

Ques 36.

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.

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