Physics 6311: Statistical Mechanics - Homework 7

Problem 1: Quantum mechanical three-level system (10 points)

A quantum-mechanical system has three energy eigenstates $|0⟩,|-1⟩$ and $|1⟩$ and energies ${𝜖}_0=0$, ${𝜖}_1={𝜖}_{-1}=𝜖$ ($𝜖>0$).

a)

Use the canonical ensemble to calculate the Helmholtz free energy, the entropy and the heat capacity as functions of temperature.

b)

Calculate the occupation probabilities $p_0,p_1$ and $p_{-1}$ of the three levels as functions of temperature. At what temperature is $p_1∕p_0=2$? Discuss the sign of this temperature.

Problem 2: Ideal gas in linear potential well (15 points)

Consider a classical ideal gas of $N$ non-interacting particles at temperature $T$. The particles are subject to the potential energy $U(\overrightarrow{r})=A|\overrightarrow{r}|$ ($A$ is a constant).

a)

Calculate the partition function and the Helmholtz free energy of the gas.

b)

Determine the internal energy and the specific heat. Compare with the equipartition theorem.

c)

Calculate the particle density $n(\overrightarrow{r})$ of the gas (particles per volume) as a function of the position. [Hint: the particle density $n(\overrightarrow{r})$ can be related to a reduced probability density of the single-particle phase space density $\rho (\overrightarrow{r},\overrightarrow{p}$).]

Problem 3: Ideal gas with nonstandard energy-momentum relation (15 points)

Using the canonical ensemble, consider a gas of non-interacting, indistinguishable classical particles in a cubic box of linear size $L$. The classical Hamiltonian is $H={\sum <!--\; FUNCTION\; APPLICATION\; -->}_{i}A|{\overrightarrow{p}}_i{|}^4$ ($A$ is a constant).

a)

Calculate the partition function and the Helmholtz free energy.

b)

Calculate the caloric equation of state (energy-temperature relation).

c)

Calculate the thermodynamic equation of state (relation between $p,V,T$).

d)

Determine the ratio of the specific heats $C_p∕C_v$ and compare to the conventional case with a quadratic energy-momentum relation).