Pb 7.4 Does entropy increase in quantum systems? We saw in Exercise 5.7 that in classical Hamiltonian systems the non-equilibrium entropy $S_{nonequil} = −k_B\int\rho \ln \rho$ is constant in a classical mechanical Hamiltonian system. Here you will show that in the microscopic evolution of an isolated quantum system, the entropy is also time-independent, even for general, time-dependent density matrices $\rho(t)$.

Using the evolution law (eqn 7.19) $\partial \rho/\partial t= [H, \rho]/(i\hbar)$, prove that $S=-k_B Tr(\rho\log \rho)$ is time-independent, where $\rho$ is any density matrix. (Two approaches: (1) Go to an orthonormal basis $\psi_i$ which diagonalizes $\rho$. Show that $\psi_i(t)$ is also orthonormal, and take the trace in that basis. (2) Let $U(t)=exp(-iHt/\hbar)$ be the unitary operator that time evolves the wave function $\psi(t)$. Show that $\rho(t) = U(t)\rho(0)U^\dagger(t)$. Write $S(t)$ as a formal power series in $\rho(t)$. Show, term by- term in the series, that $S(t) = U(t)S(0)U^\dagger(t)$. Then use the cyclic invariance of the trace.)

Pb 7.5 Photon density matrices.

Write the density matrix for a vertically polarized photon $|V\rangle$ in the basis where $|V\rangle=(1,\; 0)^T$ and a horizontal photon $|H\rangle=(0,\; 1)^T$. Write the density matrix for a diagonally polarized photo $(1/\sqrt{2} ,\; 1/\sqrt{2})^T$, and the density matrix for unpolarized light (note 2 on p. 180). Calculate $Tr(\rho), Tr(\rho^2)$, and $S=-k_B Tr(\rho\log \rho)$. Interpret the values of the three traces physically. (Hint: One is a check for pure states, one is a measure of information, and one is a normalization.)

Pb7.11 Phonons on a string. A continuum string of length $L$ with mass per unit length $\mu$ under tension $\tau$ has a vertical, transverse displacement $u(x,t)$. The kinetic energy density is $(\mu/2)(\partial u/\partial t)^2$ and the potential energy density is $(\tau/2)(\partial u/partial x)^2$. The string has fixed boundary conditions at $x = 0$ and $x = L$.

Write the kinetic energy and the potential energy in new variables, changing from $u(x, t)$ to normal modes $q_k(t)$ with $u(x, t) = \sum_n q_{k_n}(t) \sin(k_n x)$, $k_n = n\pi/L$. Show in these variables that the system is a sum of decoupled harmonic oscillators. Calculate the density of normal modes per unit frequency $g(\omega)$ for a long string $L$. Calculate the specific heat of the string $c(T)$ per unit length in the limit $L\rightarrow \infty$, treating the oscillators quantum mechanically. What is the specific heat of the classical string? (Hint: $\int_0^\infty x/(e^x-1)dx=\pi^2/6$.)

Almost the same calculation, in three dimensions, gives the low-temperature specific heat of crystals.