Quantum Statistical Mechanics: Advanced Topics

The density matrix (or density operator) provides the most general description of a quantum system, encompassing both pure and mixed states:

$\hat{\rho} = \sum_i p_i\,|\psi_i\rangle\langle\psi_i|$

Where $p_i$ is the probability of being in state $|\psi_i\rangle$.

Properties:

• $\text{Tr}(\hat{\rho}) = 1$ (normalisation)
• $\hat{\rho}^\dagger = \hat{\rho}$ (hermiticity)
• $\hat{\rho}^2 = \hat{\rho}$ if and only if the state is pure
• $\text{Tr}(\hat{\rho}^2) \leq 1$ with equality for pure states

Ensemble averages: $\langle \hat{A} \rangle = \text{Tr}(\hat{\rho}\hat{A})$.

Canonical ensemble: $\hat{\rho} = \frac{1}{Z}\exp(-\beta\hat{H})$ where $Z = \text{Tr}[\exp(-\beta\hat{H})]$.

Von Neumann entropy: $S = -k_B\text{Tr}(\hat{\rho}\ln\hat{\rho})$.

For a pure state: $S = 0$. For a thermal state: $S > 0$ (maximum for the maximally mixed state).

Time evolution. The von Neumann equation governs the density matrix:

$i\hbar\frac{\partial\hat{\rho}}{\partial t} = [\hat{H}, \hat{\rho}]$

This is the quantum analogue of Liouville”s equation. For a closed system, the von Neumann entropy is constant (unitary evolution preserves eigenvalues of $\hat{\rho}$).

For a system of non-interacting quantum particles, the grand canonical partition function is:

$\ln\mathcal{Z} = \pm\sum_{\mathbf{k}}\ln(1 \mp e^{-\beta(\epsilon_{\mathbf{k}} - \mu)})$

Where $+$ is for fermions and $-$ for bosons.

The thermodynamic quantities follow from:

$N = \sum_{\mathbf{k}}\frac{1}{e^{\beta(\epsilon_{\mathbf{k}} - \mu)} \pm 1}, \quad E = \sum_{\mathbf{k}}\frac{\epsilon_{\mathbf{k}}}{e^{\beta(\epsilon_{\mathbf{k}} - \mu)} \pm 1}$

$\Omega = -PV = \mp k_BT\sum_{\mathbf{k}}\ln(1 \mp e^{-\beta(\epsilon_{\mathbf{k}} - \mu)})$

In the continuum limit:

$\Omega = \mp k_BT\int_0^\infty g(\epsilon)\ln(1 \mp e^{-\beta(\epsilon - \mu)})\,d\epsilon$

Fermi—Dirac distribution: $f(\epsilon) = \frac{1}{e^{\beta(\epsilon - \mu)} + 1}$

Bose—Einstein distribution: $f(\epsilon) = \frac{1}{e^{\beta(\epsilon - \mu)} - 1}$

Classical limit. When $\beta(\epsilon - \mu) \gg 1$, both distributions reduce to the Maxwell—Boltzmann distribution: $f(\epsilon) \approx e^{-\beta(\epsilon - \mu)}$.

Below the Bose—Einstein condensation temperature $T_c$, the chemical potential is pinned at $\mu = \epsilon_0$ (the ground state energy, taken as zero). The integral for $N$ splits into condensate and excited fractions:

$N = N_0 + N_{\text{ex}} = N_0 + \int_0^\infty \frac{g(\epsilon)}{e^{\beta\epsilon} - 1}\,d\epsilon$

For a 3D gas: $g(\epsilon) = \frac{V(2m)^{3/2}}{4\pi^2\hbar^3}\sqrt{\epsilon}$.

The critical temperature:

$k_BT_c = \frac{2\pi\hbar^2}{m}\left(\frac{N}{2.612\,V}\right)^{2/3}$

The excited fraction: $N_{\text{ex}}/N = (T/T_c)^{3/2}$.

Condensate fraction: $N_0/N = 1 - (T/T_c)^{3/2}$.

Low-$T$ properties of the condensate:

• Ground state energy: $E_0 = 0$ (no kinetic energy)
• Heat capacity: $C_V \propto T^3$ (from excited states only)
• The condensate does not contribute to $C_V$ (all particles in the ground state have fixed energy)
• Superfluidity: the condensate flows without viscosity below $T_c$

Experimental realisation. BEC was first achieved in dilute alkali gases (Rb, Na, Li) in 1995 (Cornell, Wieman, Ketterle — Nobel Prize 2001). Key requirement: $n\lambda_{\text{dB}}^3 \gtrsim 2.612$ (where $\lambda_{\text{dB}} = h/\sqrt{2\pi mk_BT}$ is the thermal de Broglie wavelength).

At $T = 0$, all states up to the Fermi energy $E_F = \mu(0)$ are filled:

$E_F = \frac{\hbar^2}{2m}(3\pi^2 n)^{2/3}$

Low-temperature expansion. The Sommerfeld expansion gives:

$\mu(T) \approx E_F\left[1 - \frac{\pi^2}{12}\left(\frac{k_BT}{E_F}\right)^2\right]$

$C_V = \frac{\pi^2}{2}Nk_B\frac{T}{T_F} \quad \text{(linear in $T$)}$

The linear specific heat is a signature of degenerate fermions and is observed in metals (electronic contribution) and white dwarf stars.

Pauli paramagnetism. The spin susceptibility of a degenerate Fermi gas:

$\chi = \mu_0\mu_B^2 g(E_F)$

is independent of temperature (Pauli limit), in contrast to the Curie law $\chi \propto 1/T$ for classical spins.

In a magnetic field $\mathbf{B} = B\hat{z}$, the energy levels of a free electron gas become quantised into Landau levels:

$\epsilon_n = \left(n + \frac{1}{2}\right)\hbar\omega_c + \frac{\hbar^2 k_z^2}{2m_e}, \quad \omega_c = \frac{eB}{m_e}$

The density of states becomes a series of peaks (van Hove singularities) at each Landau level.

de Haas—van Alphen effect. The magnetisation oscillates as a function of $1/B$ with period:

$\Delta\left(\frac{1}{B}\right) = \frac{2\pi e}{\hbar A_{\text{FS}}}$

Where $A_{\text{FS}}$ is the extremal cross-sectional area of the Fermi surface. This is used to map the Fermi surface topology of metals.

Shubnikov—de Haas effect. The resistivity oscillates similarly, used for Fermi surface measurements in semiconductors.

Photons are massless bosons with $\mu = 0$ (not conserved). The Planck distribution:

$\langle n_\omega \rangle = \frac{1}{e^{\hbar\omega/(k_BT)} - 1}$

gives the mean number of photons per mode. The energy density:

$u(\omega) = \frac{\hbar\omega^3}{\pi^2 c^3}\frac{1}{e^{\hbar\omega/(k_BT)} - 1}$

integrates to the Stefan—Boltzmann law: $u = aT^4$ where $a = \pi^2 k_B^4/(15\hbar^3 c^3)$.

Phonons are quantised lattice vibrations, also bosons with $\mu = 0$. At low $T$: $C_V \propto T^3$ (Debye model), consistent with the experimental $T^3$ law for insulators.

1. Confusing chemical potential for bosons and fermions. For bosons, $\mu < \epsilon_0$ (bounded above by the ground state energy). For fermions, $\mu$ can be positive and equals $E_F$ at $T = 0$.

2. Applying Bose—Einstein statistics to photons without setting $\mu = 0$. Photons are not conserved, so $\mu = 0$. Using $\mu \neq 0$ gives incorrect results.

3. Forgetting the 2.612 factor in $T_c$. The critical temperature for BEC includes the Riemann zeta function value $\zeta(3/2) \approx 2.612$. Omitting this gives the wrong condensation temperature.

4. Assuming the Sommerfeld expansion is valid at all temperatures. It requires $k_BT \ll E_F$. Near $T_c$ or for classical gases, the full distribution must be used.

$g(\epsilon) = \frac{eB}{2\pi^2\hbar}\sum_n \frac{1}{\sqrt{\epsilon - (n + 1/2)\hbar\omega_c}}$

Shubnikov—de Haas oscillations: As $B$ is varied, Landau levels pass through the Fermi energy, causing oscillations in the resistivity with period:

$\Delta\!\left(\frac{1}{B}\right) = \frac{2\pi e}{\hbar A_{\text{ext}}}$

Where $A_{\text{ext}}$ is the extremal cross-sectional area of the Fermi surface perpendicular to $\mathbf{B}$.

de Haas—van Alphen oscillations: Similar oscillations in the magnetisation (and hence the susceptibility). These provide the most precise tool for mapping Fermi surface geometry.