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MPH-011 Solved Assignment July 2025 | STATISTICAL MECHANICS | IGNOU MSCPH

Q: Derive quantum Liouville equation. Write this equation for the Steady state.

The quantum Liouville equation describes the time evolution of the density operator ρ, which statistically represents a quantum system. For a pure state, ρ is defined as ρ = |ψ⟩⟨ψ|. The equation is derived by considering the time-dependent Schrödinger equation, iħ ∂|ψ⟩/∂t = H|ψ⟩, and its adjoint for ⟨ψ|. To find ∂ρ/∂t, we differentiate ρ with respect to time: ∂ρ/∂t = (∂|ψ⟩/∂t)⟨ψ| + |ψ⟩(∂⟨ψ|/∂t) Substituting the Schrödinger equation and ∂⟨ψ|/∂t = -(i/ħ)⟨ψ|H, we obtain: ∂ρ/∂t = (-i/ħ)H|ψ⟩⟨ψ| +

Q1. PART A

(a)) Calculate the mean, and the variance of the following probability density functions: i) Uniform p(x) = 1/(2a), -a < x < a, and p(x) = 0 elsewhere. ii) Rayleigh distribution p(x) = x/(a^2) * e^(-x^2/(2a^2)) (400 words)
(b)) Show that for any distribution where N is large, the distribution tends to a Gaussian or a normal distribution. (400 words)

Key Points:

Mean (E[x]) for uniform distribution p(x)=1/(2a) over [-a,a] is 0.
Variance (Var[x]) for uniform distribution p(x)=1/(2a) over [-a,a] is a²/3.
Mean (E[x]) for Rayleigh distribution p(x)=x/(a²)e^(-x²/(2a²)) for x≥0 is a√(π/2).
Variance (Var[x]) for Rayleigh distribution is a²(4-π)/2.

Answer:

Q3. Derive quantum Liouville equation. Write this equation for the Steady state.

Key Points:

Quantum Liouville equation describes time evolution of the density operator ρ.
Derived from time-dependent Schrödinger equation and definition of ρ.
Formula: iħ ∂ρ/∂t = [H, ρ], where H is the Hamiltonian.
The term [H, ρ] represents the commutator Hρ - ρH.

Answer: The quantum Liouville equation describes the time evolution of the density operator ρ, which statistically represents a quantum system. For a pure state, ρ is defined as ρ = |ψ⟩⟨ψ|. The equation is derived by considering the time-dependent Schrödinger equation, iħ ∂|ψ⟩/∂t = H|ψ⟩, and its adjoint for ⟨ψ|. To find ∂ρ/∂t, we differentiate ρ with respect to time: ∂ρ/∂t = (∂|ψ⟩/∂t)⟨ψ| + |ψ⟩(∂⟨ψ|/∂t) Substituting the Schrödinger equation and ∂⟨ψ|/∂t = -(i/ħ)⟨ψ|H, we obtain: ∂ρ/∂t = (-i/ħ)H|ψ⟩⟨ψ| + ...

Q4. PART B

(a)) Using the result p = e^(-вĤ) / Tr(e^(-вĤ)), show that the expression of density matrix (p) of a free particle in box of volume V in the canonical ensemble in the coordinate representation is given by Tr(e^(-βH)) = V (m / (2πħ^2))^(3/2) (200 words)
(b)) Obtain Fermi-Dirac distribution function. Draw it for T = 0K and T > 0K. What is its physical significance? (200 words)
(c)) Explain Pauli's paramagnetism. Show that at T = 0 Pauli's paramagnetic susceptibility is given by χ0 = (3/2) nµ*² / εF where µ* is the intrinsic magnetic moment of the particle of mass m. (200 words)
(d)) How many photons are present in 1cm³ of radiation at 727°C? What is their average energy? (Take: ∫(from 0 to ∞) x²dx / (e^x - 1) = 2.405) (200 words)

Key Points:

Canonical partition function Tr(e^(-βĤ)) for a free particle in volume V is V(m k T / (2πħ²))^(3/2).
Fermi-Dirac distribution f(ε) = 1 / (e^((ε - μ)/kT) + 1) describes fermion occupancy probability.
At T=0K, Fermi-Dirac distribution is a step function; at T>0K, it's a smeared S-curve around μ.
Pauli paramagnetism arises from spin alignment of free electrons near Fermi surface in a magnetic field.

Answer: The following response addresses the four sub-questions from the MPH-011 Statistical Mechanics course, covering topics such as the density matrix in the canonical ensemble, Fermi-Dirac distribution, Pauli's paramagnetism, and properties of blackbody radiation. Each sub-question is answered comprehensively within its specified word limit, using established principles of statistical mechanics.

MPH-011 Solved Assignment — July 2025

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

MPH-011—July 2025

STATISTICAL MECHANICS

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MPH-011 Assignment Questions — July 2025

Q1. PART A

Q3. Derive quantum Liouville equation. Write this equation for the Steady state.

Q4. PART B

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