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KOÇ UNIVERSITY Phys.506: Quantum Statistical Mechanics Instr: Alkan Kabakçıoğlu
HOMEWORK PROBLEMS
Q1. Carnot Engine: Derive the efficiency $\eta_{CE}$ of the ideal gas Carnot engine in terms of the ideal gas temperature $\theta$. Use your result to show the equivalence of $\theta$ and the thermodynamic temperature defined by $1 - \eta_{CE} = T_c/T_h$.
For the derivation, start with the isotherms for an ideal gas given by the equation of state $PV = Nk_B\theta$, where $N$ is the number of gas molecules and $k_B$ is the Boltzmann constant. For the adiabatic curves, use the fact that the internal energy of the ideal gas is a function of $\theta$ only, but don't assume that $E \propto \theta$.
Q2. Equation of state for a non-ideal gas: For a typical gas, experiments show that
$$\left. \frac{\partial P}{\partial V} \right|_T = 2a\frac{N^2}{V^3} - \frac{Nk_BT}{(V-Nb)^2} \quad \text{and} \quad \left. \frac{\partial T}{\partial P} \right|_V = \frac{(V-Nb)}{Nk_B},$$ where $a$ and $b$ are material-dependent, positive constants.
(a) Find an expression for $P(T,V)$. Can you guess the physical significance of the constants $a$ and $b$?
(b) Use the chain rule to combine the experimental observations above and derive the thermal expansivity of the gas,
$$\alpha = \frac{1}{V} \left. \frac{\partial V}{\partial T} \right|_P.$$ Make sure that your result reduces to the value you find for the ideal gas in the limit $a, b \to 0$.
(c) Find an expression for $C_P - C_V$ in terms of $\partial E/\partial V|_T, P, V,$ and $\alpha$.
Q3. (Kardar 1.8) Hard core gas: a gas obeys the equation of state $P(V - Nb) = Nk_BT$, and has a heat capacity $C_V$ independent of temperature. ($N$ is kept fixed in the following.)
(a) Find the Maxwell relation involving $\partial S/\partial V|_{T,N}$.
(b) By calculating $dE(T,V)$, show that $E$ is a function of $T$ (and $N$) only.
(c) Show that $\gamma \equiv C_P/C_V = 1 + Nk_B/C_V$ (independent of $T$ and $V$).
(d) By writing an expression for $E(P,V)$, or otherwise, show that an adiabatic change satisfies the equation $P(V-Nb)^\gamma = \text{constant}$.
Q4. Internal energy and adiabatic processes in paramagnets:
(a) A Curie paramagnet at temperature $T$ and uniform magnetic field $H$ has the equation of state
$$M = aH/T$$ where $M$ is the magnetization and $a$ is a constant. Starting from $dE = TdS + HdM$, show that the internal energy $E(M,T)$ is a function of temperature only.
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(b) Find the most general form of the equation of state for a Curie paramagnet.
(c) Assuming the heat capacity $C_M$ is a constant, show that for a reversible and adiabatic transformation,
$$\frac{e^{M^2/2aC_M}}{T} = \text{const} .$$ Q5. (Kardar 1.9) Superconducting transition: many metals become superconductors at low temperatures $T$, and magnetic fields $B$. The heat capacities of the two phases at zero magnetic field are approximately given by
$$\begin{cases} C_s(T) = V \alpha T^3 & \text{in the superconducting phase} \ C_n(T) = V [\beta T^3 + \gamma T] & \text{in the normal phase} \end{cases} \quad (1)$$ where $V$ is the volume, and ${\alpha, \beta, \gamma}$ are constants. (There is no appreciable change in volume at this transition, and mechanical work can be ignored throughout this problem.)
(a) Calculate the entropies $S_s(T)$ and $S_n(T)$ of the two phases at zero field, using the third law of thermodynamics.
(b) Experiments indicate that there is no latent heat ($L=0$) for the transition between the normal and superconducting phases at zero field. Use this information to obtain the transition temperature $T_c$, as a function of $\alpha, \beta,$ and $\gamma$.
(c) At zero temperature, the electrons in the superconductor form bound Cooper pairs. As a result, the internal energy of the superconductor is reduced by an amount $V\Delta$, that is, $E_n(T=0) = E_0$, and $E_s(T=0) = E_0 - V\Delta$ for the metal and superconductor, respectively. Calculate the internal energies of both phases at finite temperatures.
(d) By comparing the Gibbs free energies (or chemical potential) in the two phases, obtain an expression for the energy gap $\Delta$ in terms of $\alpha, \beta,$ and $\gamma$.
(e) In the presence of a magnetic field $B$, inclusion of magnetic work results in $dE = TdS + BdM + \mu dN$, where $M$ is the magnetization. The superconducting phase is a perfect diamagnet, expelling the magnetic field from its interior, such that $M_s = -VB/(4\pi)$ in the appropriate units. The normal metal can be regarded as approximately non-magnetic, with
(c) Calculate the heat capacity $C(T)$ and sketch it.
(d) What is the probability that a specific molecule is standing up?
(e) What is the largest possible value of the internal energy at any positive temperature?
Q9. (Kardar 4.5) Non-harmonic gas: Consider a gas of $N$ non-interacting atoms in a $d$-dimensional box of "volume" $V$, with a kinetic energy
$$\mathcal{H} = \sum_{i=1}^{N} A |\vec{p}_i|^s,$$ where $\vec{p}_i$ is the momentum of the $i$th particle.
(a) Calculate the classical partition function $Z(N,T)$ at a temperature $T$. (You don't have to keep track of numerical constants in the integration.)
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(b) Calculate the pressure and the internal energy of this gas. (Note how the usual equipartition theorem is modified for non-quadratic degrees of freedom.)
(c) Now consider a diatomic gas of $N$ molecules, each with energy
$$\mathcal{H}_i = A \left( |\vec{p}_i^{(1)}|^8 + |\vec{p}_i^{(2)}|^8 \right) + K |\vec{q}_i^{(1)} - \vec{q}_i^{(2)}|^t ,$$ where the superscripts refer to the two particles in the molecule. (Note that this unrealistic potential allows the two atoms to occupy the same point.) Calculate the expectation value $\langle |\vec{q}_i^{(1)} - \vec{q}_i^{(2)}|^t \rangle$, at temperature T.
(d) Calculate the heat capacity ratio $\gamma = C_P/C_V$, for the above diatomic gas.
- You may find the answer to this question at the back of the book. However, attempt to solve it by yourselves and consult the book only if you get stuck.
Q10. (Kardar 4.10) Molecular oxygen: Molecular oxygen has a net magnetic spin $\vec{S}$ of unity, that is, $S^z$ is quantized to -1, 0, or +1. The Hamiltonian for an ideal gas of $N$ such molecules in a magnetic field $\vec{B} \parallel \hat{z}$ is
$$\mathcal{H} = \sum_{i=1}^{N} \left[ \frac{\vec{p}_i^2}{2m} - \mu B S_i^z \right]$$ where ${\vec{p}_i}$ are the center of mass momenta of the molecules. The corresponding coordinates ${\vec{q}_i}$ are confined to a volume V. (Ignore all other degrees of freedom.)
(a) Treating ${\vec{p}_i, \vec{q}_i}$ classically, but the spin degrees of freedom as quantized, calculate the partition function, $Z(T, N, V, B)$.
(a) What are the probabilities for $S_i^z$ of a specific molecule to take on values of -1, 0, +1 at a temperature T?
(b) Find the average magnetic dipole moment, $\langle M \rangle / V$, where $M = \mu \sum_{i=1}^{N} S_i^z$.
(c) Calculate the zero field susceptibility $\chi = \left. \partial \langle M \rangle / \partial B \right|_{B=0}$.
Q11. (Kardar 4.11) One-dimensional polymer: Consider a polymer formed by connecting $N$ disc-shaped molecules into a one-dimensional chain. Each molecule can align along either its long axis (of length $2a$) or short axis (length $a$). The energy of the monomer aligned along its shorter axis is higher by $\epsilon$, that is, the total energy is $\mathcal{H} = \epsilon U$, where $U$ is the number of monomers standing up.
(a) Calculate the partition function, $Z(T, N)$, of the polymer.
(b) Find the relative probabilities for a monomer to be aligned along its short or long axis.
(c) Calculate the average length, $\langle L(T, N) \rangle$, of the polymer.
(d) Obtain the variance, $\langle L(T, N)^2 \rangle_c$.
(e) What does the central limit theorem say about the probability distribution for the length $L(T, N)$?
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Q12. (Kardar 4.9) Langmuir isotherms: An ideal gas of particles is in contact with the surface of a catalyst.
- Show that the chemical potential of the gas particles is related to their temperature and pressure via $\mu = k_B T [ \ln(P/T^{5/2}) + A_0 ]$, where $A_0$ is a constant.
- If there are $\mathcal{N}$ distinct adsorption sites on the surface, and each adsorbed particle gains an energy $\epsilon$ upon adsorption, calculate the grand partition function for the two-dimensional gas with a chemical potential $\mu$.
- In equilibrium, the gas and surface particles are at the same temperature and chemical potential. Show that the fraction of occupied surface sites is then given by $f(T, P) = P / (P + P_0(T))$. Find $P_0(T)$.
- In the grand canonical ensemble, the particle number $N$ is a random variable. Calculate its characteristic function $\langle \exp(-ikN) \rangle$ in terms of $\mathcal{Q}(\beta \mu)$ and hence show that $$\langle N^m \rangle_c = -(k_B T)^{m-1} \left. \frac{\partial^m \mathcal{G}}{\partial \mu^m} \right|_T ,$$ where $\mathcal{G}$ is the grand potential.
(d) The surface tension of water without surfactants is $\sigma_0$, approximately independent of temperature. Calculate the surface tension $\sigma(n, T)$ in the presence of surfactants.
(e) By considering the isothermal compressibility of the surfactants, show that below a certain temperature $T_c$, the expression for $\sigma$ is manifestly incorrect. What do you think happens at low temperatures?
Q16. Two particles in a harmonic potential: Consider a one-dimensional harmonic potential containing two identical, noninteracting particles of mass $m$ (ignore spin). The single-particle Hamiltonian
$$\mathcal{H} = -\frac{\hbar^2}{2m} \frac{d^2}{dq^2} + \frac{m \omega^2}{2} q^2$$ has normalized eigenfunctions $\psi_n(q)$ with eigenvalues (energy) $\epsilon_n = (n + \frac{1}{2}) \hbar \omega$, where $n = 0, 1, 2, ...$
(Precise form of the eigenfunctions is given in terms of Hermite polynomials $H_n(q)$:
$\psi_n(q) = \frac{1}{\sqrt{2^n n!}} \left( m \omega / \pi \hbar \right)^{1/4} \exp \left[ -m \omega q^2 / 2 \hbar \right] H_n(\sqrt{m \omega / \hbar} q)$, with $H_0(x) = 1, H_1(x) = 2x, H_2(x) = x^
- 1$, etc., although this information is irrelevant for this problem.)
(a) Find the two-particle wave function $\psi_{gs}(q_1, q_2)$ for the ground state if the particles are (i) fermions (ii) bosons. Express your answer in terms of $\psi_n(q)$.
(b) Find the two-particle wave function $\psi_{exc}(q_1, q_2)$ for the first-excited state if the particles are (i) fermions (ii) bosons. Express your answer in terms of $\psi_n(q)$.
(c) Explicitly calculate the two-particle canonical partition function $Z_2(T)$ for fermions and bosons, separately. Express $Z_2(T)$ in terms of the one-particle partition function, $Z_1$ for each case.
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Q17. Spin-1 particle in a magnetic field: A spin-1 particle in a magnetic field $\vec{B}$ is described by the Hamiltonian
$$\mathcal{H} = -\mu_B \vec{J} \cdot \vec{B}$$ where
$$J_x = \frac{1}{\sqrt{2}} \begin{pmatrix} 0 & 1 & 0 \ 1 & 0 & 1 \ 0 & 1 & 0 \end{pmatrix}, \quad J_y = \frac{1}{\sqrt{2}} \begin{pmatrix} 0 & -i & 0 \ i & 0 & -i \ 0 & i & 0 \end{pmatrix}, \quad J_z = \begin{pmatrix} 1 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & -1 \end{pmatrix} .$$ are spin operators expressed in the basis $|-\rangle, |0\rangle, |+\rangle$. Note that, $J_\alpha^3 = J_\alpha$.
(a) In the canonical ensemble, evaluate the density matrix $\rho$ for $\vec{B}$ along $\hat{z}$. (Check that $\rho$ satisfies the properties of a general density matrix.)
(b) Calculate the average energy.
(c) In the limit $T \to 0$, is the canonical ensemble pure or mixed? Would your answer change if $\vec{B}$ were along $\hat{x}$? Why?
Q18. (Kardar, Problem 6.8) Quantum mechanical entropy: a quantum mechanical system (defined by a Hamiltonian $\mathcal{H}$), at temperature $T$, is described by a density matrix $\rho(t)$, which has an associated entropy $S(t) = -\text{tr} \rho(t) \ln \rho(t)$.
(a) Write down the time evolution equation for the density matrix, and calculate $dS/dt$.
(b) Using the method of Lagrange multipliers, find the density operator $\rho_{max}$ that maximizes the functional $S[\rho]$, subject to the constraint of fixed average energy $\langle \mathcal{H} \rangle = \text{tr} \rho \mathcal{H} = E$.
(c) Show that the solution to part (b) is stationary, that is, $\partial \rho_{max} / \partial t = 0$.
Q19. (Kardar, Problem 7.10) Solar interior: according to astrophysical data, the plasma at the center of the Sun has temperature $T = 1.6 \times 10^7$ K, hydrogen density $\rho_H = 6 \times 10^4 \text{ kg m}^{-3}$, and helium density $\rho_{He} = 1 \times 10^5 \text{ kg m}^{-3}$.
(a) Obtain the thermal wavelength for electrons, protons, and $\alpha$-particles (nuclei of He).
(b) Assuming the gas is ideal, determine whether the electron, proton, or $\alpha$-particle gases are degenerate in the quantum mechanical sense.
(c) Estimate the total gas pressure due to these particles near the center of the Sun.
(d) Estimate the total radiation pressure close to the center of the Sun. Is gravitational collapse prevented by matter pressure or radiation pressure?
Q20. Bose-Einstein condensation in a harmonic trap: Consider a gas of $N$ nonrelativistic, noninteracting bosons with mass $m$ and zero spin in a two-dimensional harmonic potential $V(x,y) = m \omega^2(x^2 + y^2)/2$. The energy spectrum (ignoring the ground-state energy) is given by $\epsilon_{n_x, n_y} = (n_x + n_y) \hbar \omega$, where $n_{x,y} = 0, 1, 2, ...$
(a) Express the average occupation number of a state with energy $\epsilon$ as a function of temperature and chemical potential.
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