STATISTICAL MECHANICS | IGNOU Master of Science (Physics) Programme (MSCPH)

Download IGNOU MSCPH MPH-011 solved assignment for the January 2026 session — STATISTICAL MECHANICS. This paper contains 4 questions with 4 detailed solved answers including key points and references. Available in English. Free PDF download for Master of Science (Physics) Programme students.

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MPH-011—January 2026 / July 2026

Q: Answer the following sub-questions.

The following response addresses three distinct sub-questions related to statistical mechanics and probability. Sub-question (a) explores the conditions under which the Poisson distribution approximates the Normal distribution, a fundamental concept in limit theorems. Sub-question (b) involves calculating the normalization constant, mean, and variance for a given continuous probability distribution function, which is crucial for characterizing random variables. Finally, sub-question (c) applies

STATISTICAL MECHANICS

AssignmentMaster of Science (Physics) Programme (MSCPH)Management Studies

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Answer Status

Questions & Answers

Q1✓ AnswerLong Answer1000 words

Answer the following sub-questions.

3 sub-questions

Q2✓ AnswerLong Answer1000 words

Answer the following sub-questions.

4 sub-questions

Q3✓ AnswerLong Answer1000 words

Answer the following sub-questions.

6 sub-questions

Q4✓ AnswerLong Answer1000 words

Answer the following sub-questions.

4 sub-questions

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MPH-011 Assignment Questions — January 2026

Q1. Answer the following sub-questions.

(a)) Show under what condition Poisson distribution tends to Normal distribution. (400 words)
(b)) A monthly demand for commodity is a continuous random distribution with probability distribution function given as: p(x) = { 3N(x^2-1) for 1 < x < 2; 0 elsewhere } where N is normalization constant. Obtain the value of N so that the function is normalized and hence, find the mean and the variance. (400 words)
(c)) N particles obey Maxwell-Boltzmann distribution. They are distributed among three states with energies E₁ = 0, E₂ = kBT, and E₃ = 4kBT. If the equilibrium energy of the system is approximately 3000kBT, calculate the total number of particles. (200 words)

Key Points:

Poisson distribution approaches Normal distribution when its mean (λ) is large.
Normalization constant N ensures PDF integrates to 1 over its domain.
Mean (E[X]) is the expected value of the random variable.
Variance (Var[X]) measures the spread of the distribution, calculated as E[X²] - (E[X])².

Answer: The following response addresses three distinct sub-questions related to statistical mechanics and probability. Sub-question (a) explores the conditions under which the Poisson distribution approximates the Normal distribution, a fundamental concept in limit theorems. Sub-question (b) involves calculating the normalization constant, mean, and variance for a given continuous probability distribution function, which is crucial for characterizing random variables. Finally, sub-question (c) applies ...

Q2. Answer the following sub-questions.

(a)) Obtain the phase space area enclosed by a classical harmonic oscillator for energies ranging from 0 to E. (200 words)
(b)) The Hamiltonian for the non-relativistic, non-interacting, monoatomic ideal gas is given by H(q,p) = sum(i=1 to 3)(pi^2 / 2m) + U(qi) where m is the mass of the particle and U(qi) 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. (400 words)
(c)) Obtain expression for average energy U(= <E>) for the canonical ensemble in terms of partition function Z. (200 words)
(d)) For the grand canonical ensemble, obtain an expression for the ensemble average energy and ensemble average particle numbers. (200 words)

Key Points:

Classical SHO phase space area Γ(E) = 2πE/ω, directly proportional to energy E.
Phase space volume Φ(E) for N ideal gas particles is V^N * (π^(3N/2) / Γ(3N/2 + 1)) * (2mE)^(3N/2).
Number of microstates Ω(E, ΔE) = ΔΦ(E) / h^(3N), linking classical phase space to quantum cells.
Canonical ensemble average energy U = - d(ln Z)/dβ, relating energy to canonical partition function Z.

Answer: This comprehensive answer addresses fundamental concepts in statistical mechanics, including phase space analysis for classical systems and the derivation of key thermodynamic quantities within the canonical and grand canonical ensembles using partition functions. Understanding these derivations is crucial for mastering the statistical description of macroscopic systems from microscopic principles, as covered in MPH-011 Statistical Mechanics.

Q3. Answer the following sub-questions.

a) i)) A proton (mass = 1.67 × 10^-27 kg) inside a nucleus (radius = 10^-14 m) may have speed upto 2.0 × 10^8 ms^-1. How many quantum states are available to it? (80 words)
a) ii)) The time evolution of the quantum state, ψ(q,t) is given by the Schrödinger wave equation: iħ (∂ψ/∂t) = Hψ where H is the Hamiltonian operator. Using this equation, obtain time evolution of the wave function when the energy eigenvalue is E. (120 words)
(b)) State and prove quantum Liouville equation. (200 words)
(c)) Using the relation of thermodynamic probability W = Π(i=1 to N) (gi! / Ni!(gi-Ni)!) Obtain an expression of Fermi Dirac distribution function. Show a plot of f(ε) (the occupation index of a state corresponding to energy εi) versus ε for different temperatures and discuss it. What is the physical interpretation of Fermi energy? (200 words)
(d)) Show that the probability that a state with energy δE above the Fermi level EF filled is equal to the probability of a state with energy δE below the Fermi level EF is empty. (200 words)
(e)) What is the Bose-Einstein condensation? Write an expression of number of particles in the excited state (Nex) in terms of Bose-Einstein condensation temperature (Tc), 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 Tc. (200 words)

Key Points:

Quantum Liouville equation: iħ (dρ/dt) = [H, ρ] describes the time evolution of the density operator.
Fermi-Dirac distribution: f(ε) = 1 / (e^((ε - μ)/kT) + 1) is the occupation probability for fermions.
Fermi energy (EF): Maximum fermion energy at T=0K, or energy where f(ε)=1/2 at T>0K.
Fermi-Dirac symmetry: Probability of a state δE above EF being filled equals probability of state δE below EF being empty.

Answer: This response comprehensively addresses the sub-questions based on principles of statistical mechanics and quantum mechanics, as expected in the MPH-011 course. It covers calculations for quantum states, derivations of time evolution and distribution functions, and descriptions of key phenomena like Bose-Einstein condensation and the physical interpretation of Fermi energy, adhering to specified word limits and formatting guidelines.

Q4. Answer the following sub-questions.

(a)) State the Virial theorem and prove Boyle's law using it. (200 words)
(b)) What is meant by Cluster Integrals? Express B₂ and B₃ in terms of Cluster Integrals. (200 words)
(c)) Using entropy as a function of temperature and pressure, obtain the first Ehrenfest's equation. (200 words)
(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 f ≈ 1/√N and hence, justify why thermodynamic calculations are valid for ordinary macroscopic systems. (400 words)

Key Points:

Virial Theorem: ⟨K⟩ = -½ ⟨Σ_i r_i ⋅ F_i⟩, relates kinetic and potential energy averages.
Boyle's Law Proof: PV = NkT derived from Virial theorem for ideal gases, P ∝ 1/V.
Cluster Integrals: Functions representing interactions between clusters of molecules, used in virial expansion.
B₂ & B₃: Virial coefficients expressed as integrals of Mayer functions (f_ij = e^(-βu_ij) - 1).

Answer: This answer comprehensively covers the Virial theorem and its application to Boyle's law, the concept of cluster integrals and their relation to virial coefficients, the derivation of the first Ehrenfest's equation, and the discussion of thermodynamic fluctuations in the canonical ensemble, including the derivation of energy fluctuation and its implication for macroscopic systems. **(a) State the Virial theorem and prove Boyle's law using it.** The Virial theorem states that for a stable, boun...

STATISTICAL MECHANICS | IGNOU Master of Science (Physics) Programme (MSCPH)

Back to MPH-011Back to MPH-011 Course Details

MPH-011—January 2026 / July 2026

STATISTICAL MECHANICS

AssignmentMaster of Science (Physics) Programme (MSCPH)Management Studies

Total Questions

Answers Available

With Sub-parts

Available

Answer Status

Questions & Answers

Q1✓ AnswerLong Answer1000 words

Answer the following sub-questions.

3 sub-questions

Q2✓ AnswerLong Answer1000 words

Answer the following sub-questions.

4 sub-questions

Q3✓ AnswerLong Answer1000 words

Answer the following sub-questions.

6 sub-questions

Q4✓ AnswerLong Answer1000 words

Answer the following sub-questions.

4 sub-questions

Download Study Materials

Get all questions and answers in PDF for offline study.

Related Courses

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Back to MPH-011 All Master of Science (Physics) Programme Courses

MPH-011 Assignment Questions — January 2026

Q1. Answer the following sub-questions.

(a)) Show under what condition Poisson distribution tends to Normal distribution. (400 words)
(b)) A monthly demand for commodity is a continuous random distribution with probability distribution function given as: p(x) = { 3N(x^2-1) for 1 < x < 2; 0 elsewhere } where N is normalization constant. Obtain the value of N so that the function is normalized and hence, find the mean and the variance. (400 words)
(c)) N particles obey Maxwell-Boltzmann distribution. They are distributed among three states with energies E₁ = 0, E₂ = kBT, and E₃ = 4kBT. If the equilibrium energy of the system is approximately 3000kBT, calculate the total number of particles. (200 words)

Key Points:

Poisson distribution approaches Normal distribution when its mean (λ) is large.
Normalization constant N ensures PDF integrates to 1 over its domain.
Mean (E[X]) is the expected value of the random variable.
Variance (Var[X]) measures the spread of the distribution, calculated as E[X²] - (E[X])².

Q2. Answer the following sub-questions.

(a)) Obtain the phase space area enclosed by a classical harmonic oscillator for energies ranging from 0 to E. (200 words)
(b)) The Hamiltonian for the non-relativistic, non-interacting, monoatomic ideal gas is given by H(q,p) = sum(i=1 to 3)(pi^2 / 2m) + U(qi) where m is the mass of the particle and U(qi) 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. (400 words)
(c)) Obtain expression for average energy U(= <E>) for the canonical ensemble in terms of partition function Z. (200 words)
(d)) For the grand canonical ensemble, obtain an expression for the ensemble average energy and ensemble average particle numbers. (200 words)

Key Points:

Classical SHO phase space area Γ(E) = 2πE/ω, directly proportional to energy E.
Phase space volume Φ(E) for N ideal gas particles is V^N * (π^(3N/2) / Γ(3N/2 + 1)) * (2mE)^(3N/2).
Number of microstates Ω(E, ΔE) = ΔΦ(E) / h^(3N), linking classical phase space to quantum cells.
Canonical ensemble average energy U = - d(ln Z)/dβ, relating energy to canonical partition function Z.

Q3. Answer the following sub-questions.

a) i)) A proton (mass = 1.67 × 10^-27 kg) inside a nucleus (radius = 10^-14 m) may have speed upto 2.0 × 10^8 ms^-1. How many quantum states are available to it? (80 words)
a) ii)) The time evolution of the quantum state, ψ(q,t) is given by the Schrödinger wave equation: iħ (∂ψ/∂t) = Hψ where H is the Hamiltonian operator. Using this equation, obtain time evolution of the wave function when the energy eigenvalue is E. (120 words)
(b)) State and prove quantum Liouville equation. (200 words)
(c)) Using the relation of thermodynamic probability W = Π(i=1 to N) (gi! / Ni!(gi-Ni)!) Obtain an expression of Fermi Dirac distribution function. Show a plot of f(ε) (the occupation index of a state corresponding to energy εi) versus ε for different temperatures and discuss it. What is the physical interpretation of Fermi energy? (200 words)
(d)) Show that the probability that a state with energy δE above the Fermi level EF filled is equal to the probability of a state with energy δE below the Fermi level EF is empty. (200 words)
(e)) What is the Bose-Einstein condensation? Write an expression of number of particles in the excited state (Nex) in terms of Bose-Einstein condensation temperature (Tc), 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 Tc. (200 words)

Key Points:

Quantum Liouville equation: iħ (dρ/dt) = [H, ρ] describes the time evolution of the density operator.
Fermi-Dirac distribution: f(ε) = 1 / (e^((ε - μ)/kT) + 1) is the occupation probability for fermions.
Fermi energy (EF): Maximum fermion energy at T=0K, or energy where f(ε)=1/2 at T>0K.
Fermi-Dirac symmetry: Probability of a state δE above EF being filled equals probability of state δE below EF being empty.

Q4. Answer the following sub-questions.

(a)) State the Virial theorem and prove Boyle's law using it. (200 words)
(b)) What is meant by Cluster Integrals? Express B₂ and B₃ in terms of Cluster Integrals. (200 words)
(c)) Using entropy as a function of temperature and pressure, obtain the first Ehrenfest's equation. (200 words)
(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 f ≈ 1/√N and hence, justify why thermodynamic calculations are valid for ordinary macroscopic systems. (400 words)

Key Points:

Virial Theorem: ⟨K⟩ = -½ ⟨Σ_i r_i ⋅ F_i⟩, relates kinetic and potential energy averages.
Boyle's Law Proof: PV = NkT derived from Virial theorem for ideal gases, P ∝ 1/V.
Cluster Integrals: Functions representing interactions between clusters of molecules, used in virial expansion.
B₂ & B₃: Virial coefficients expressed as integrals of Mayer functions (f_ij = e^(-βu_ij) - 1).

MPH-011 Solved Assignment — January 2026 / July 2026

STATISTICAL MECHANICS | IGNOU Master of Science (Physics) Programme (MSCPH)

MPH-011—January 2026 / July 2026

STATISTICAL MECHANICS

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MPH-011 Assignment Questions — January 2026

Q1. Answer the following sub-questions.

Q2. Answer the following sub-questions.

Q3. Answer the following sub-questions.

Q4. Answer the following sub-questions.

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MPH-011 Solved Assignment — January 2026 / July 2026

STATISTICAL MECHANICS | IGNOU Master of Science (Physics) Programme (MSCPH)

MPH-011—January 2026 / July 2026

STATISTICAL MECHANICS

Questions & Answers

Download Study Materials

Related Courses

MPH-011 Assignment Questions — January 2026

Q1. Answer the following sub-questions.

Q2. Answer the following sub-questions.

Q3. Answer the following sub-questions.

Q4. Answer the following sub-questions.

More MPH-011 Solved Materials