Quantum Statistical Mechanics

Classical statistical mechanics deals with phase-space points $(q_i, p_i)$. In the quantum realm, physical states are vectors in a complex Hilbert space, and physical observables are Hermitian operators.

1. Pure States versus Mixed States

A **pure state** is a system described by a single state vector $|\Psi\rangle$ with complete quantum coherence. The expectation value of an observable $\hat{A}$ is $\langle \hat{A} \rangle = \langle \Psi | \hat{A} | \Psi \rangle$. A **mixed state** represents a statistical ensemble where the system has probability $p_n$ of being in state $|\psi_n\rangle$. There is two-fold uncertainty:

1. Fundamental quantum indeterminacy (wavefunction collapse).
2. Classical statistical ignorance of which state the system occupies.

2. The Postulate of Equal A Priori Probabilities (Quantum Formulation)

For an isolated quantum system in thermodynamic equilibrium with energy in shell $[E, E + \delta E]$, all accessible orthonormal eigenstates $|n\rangle$ of the Hamiltonian $\hat{H}$ are equally likely: $$p_n = \begin{cases} \frac{1}{\Omega(E)} & \text{if } E \le E_n \le E + \delta E \\ 0 & \text{otherwise} \end{cases}$$

3. The Postulate of Random Phases

When expanding a general state in energy eigenstates $|\Psi\rangle = \sum_n c_n e^{i\theta_n} |n\rangle$, the quantum phases $\theta_n$ are completely uncorrelated and uniformly distributed over $[0, 2\pi]$: $$\overline{c_n^* c_m} = |c_n|^2 \delta_{nm}$$ All off-diagonal interference terms vanish on statistical average. The density matrix is diagonal in the energy representation for any equilibrium system.

How does quantum statistical mechanics reproduce classical physics in the macroscopic limit?

1. The Phase Cell Volume and Planck's Constant

In classical mechanics, the number of microstates in phase volume $\Delta \Gamma$ was computed with an arbitrary volume constant $h_0$: $\Omega = \Delta \Gamma / h_0$. Quantum mechanics reveals that the elementary volume of a single quantum state in 6-dimensional single-particle phase space is precisely Planck's constant cubed: $$h_0 = h^3 = (6.626 \times 10^{-34}\text{ J}\cdot\text{s})^3$$ For an $N$-particle system: $$h_0^{(N)} = N! \, h^{3N}$$ where $N!$ accounts for quantum indistinguishability.

2. The Quantum Degeneracy Criterion

The transition between classical and quantum regimes is governed by the ratio of the thermal de Broglie wavelength $\lambda_{\text{th}}$ to the average interparticle spacing $\bar{r} = (V/N)^{1/3}$: $$\lambda_{\text{th}} = \frac{h}{\sqrt{2\pi m k_B T}}$$ We define the dimensionless **quantum degeneracy parameter** $\xi$: $$\xi \equiv n \lambda_{\text{th}}^3 = \frac{N}{V} \left( \frac{h}{\sqrt{2\pi m k_B T}} \right)^3$$

• Classical Regime ($\xi \ll 1$): When $n \lambda_{\text{th}}^3 \ll 1$ (high temperature, low density, large particle mass), individual particle wavepackets do not overlap. Quantum exchange effects vanish, and the system obeys classical Maxwell-Boltzmann statistics.
• Quantum Degenerate Regime ($\xi \gtrsim 1$): When $n \lambda_{\text{th}}^3 \ge 1$ (low temperature, high density, light particles such as electrons in metals or helium at $2\text{ K}$), wavepackets overlap substantially. Quantum symmetry dominates, requiring Fermi-Dirac or Bose-Einstein statistics.

In classical physics, identical particles can always be distinguished by tracking their continuous trajectories. In quantum mechanics, the uncertainty principle makes tracing continuous trajectories impossible: identical particles are **fundamentally indistinguishable**.

1. The Permutation Operator ($\hat{P}_{12}$)

Consider a system of two identical particles. Let $|\Psi(1, 2)\rangle = |\Psi(\mathbf{r}_1, \sigma_1; \mathbf{r}_2, \sigma_2)\rangle$ be the state vector, where $\mathbf{r}_i$ is position and $\sigma_i$ is spin. Define the particle exchange (permutation) operator $\hat{P}_{12}$: $$\hat{P}_{12} |\Psi(1, 2)\rangle = |\Psi(2, 1)\rangle$$ Because the particles are identical, exchanging them cannot alter any physical observable. Thus, the Hamiltonian commutes with the permutation operator: $$[\hat{H}, \hat{P}_{12}] = 0$$ Furthermore, applying $\hat{P}_{12}$ twice returns the original state: $$\hat{P}_{12}^2 = \hat{I} \implies \text{eigenvalues of } \hat{P}_{12} \text{ are } \lambda = \pm 1$$

2. Symmetric and Antisymmetric States

Every physical state of identical particles in nature belongs strictly to one of two one-dimensional representations:

• Symmetric States ($\lambda = +1$): $$\Psi(2, 1) = +\Psi(1, 2)$$
• Antisymmetric States ($\lambda = -1$): $$\Psi(2, 1) = -\Psi(1, 2)$$

States of mixed symmetry are never observed in the physical universe.

The relationship between particle intrinsic spin and wavefunction symmetry is one of the deepest theorems in theoretical physics.

1. The Spin-Statistics Theorem

Proved by Wolfgang Pauli (1940) within relativistic quantum field theory under the axioms of Lorentz invariance, microcausality, and positive-definite energy:

Spin-Statistics Theorem:

1. Particles possessing integer intrinsic spin ($s = 0, 1, 2, \dots$) are BOSONS. Their many-body state vectors are strictly symmetric under particle exchange.
2. Particles possessing half-odd-integer intrinsic spin ($s = 1/2, 3/2, 5/2, \dots$) are FERMIONS. Their many-body state vectors are strictly antisymmetric under particle exchange.

Examples:

• Fermions: Electrons ($s=1/2$), protons ($s=1/2$), neutrons ($s=1/2$), quarks ($s=1/2$), $^3\text{He}$ atoms.
• Bosons: Photons ($s=1$), gluons ($s=1$), $W^\pm, Z^0$ bosons ($s=1$), Higgs boson ($s=0$), $^4\text{He}$ atoms ($s=0$).

2. The Pauli Exclusion Principle

For a system of $N$ non-interacting identical fermions, the total antisymmetric wavefunction is given by the **Slater Determinant**: $$\Psi(\mathbf{x}_1, \dots, \mathbf{x}_N) = \frac{1}{\sqrt{N!}} \begin{vmatrix} \phi_1(\mathbf{x}_1) & \phi_1(\mathbf{x}_2) & \dots & \phi_1(\mathbf{x}_N) \\ \phi_2(\mathbf{x}_1) & \phi_2(\mathbf{x}_2) & \dots & \phi_2(\mathbf{x}_N) \\ \vdots & \vdots & \ddots & \vdots \\ \phi_N(\mathbf{x}_1) & \phi_N(\mathbf{x}_2) & \dots & \phi_N(\mathbf{x}_N) \end{vmatrix}$$ If two fermions attempt to occupy the identical single-particle quantum state ($\phi_i = \phi_j$), two rows of the determinant become identical, so $\Psi \equiv 0$.

Pauli Exclusion Principle: No two identical fermions can simultaneously occupy the same quantum state. The occupation number of any single-particle quantum state $i$ is strictly restricted to: $$n_i \in \{0, 1\}$$

For bosons, no such restriction exists: $n_i \in \{0, 1, 2, 3, \dots\}$.

Wavefunction symmetry introduces a purely quantum mechanical effective force between particles known as the **Exchange Interaction**.

1. Two-Particle Spatial Wavefunction

Consider two non-interacting identical particles in spatial states $\phi_a(\mathbf{r})$ and $\phi_b(\mathbf{r})$. The properly symmetrized / antisymmetrized spatial wavefunctions are: $$\psi_S(\mathbf{r}_1, \mathbf{r}_2) = \frac{1}{\sqrt{2}}\left[ \phi_a(\mathbf{r}_1)\phi_b(\mathbf{r}_2) + \phi_b(\mathbf{r}_1)\phi_a(\mathbf{r}_2) \right] \quad (\text{Bosons})$$ $$\psi_A(\mathbf{r}_1, \mathbf{r}_2) = \frac{1}{\sqrt{2}}\left[ \phi_a(\mathbf{r}_1)\phi_b(\mathbf{r}_2) - \phi_b(\mathbf{r}_1)\phi_a(\mathbf{r}_2) \right] \quad (\text{Fermions})$$

2. Probability Density and Quantum Statistical Force

The probability density of finding particle 1 at $\mathbf{r}_1$ and particle 2 at $\mathbf{r}_2$ is: $$P_{\pm}(\mathbf{r}_1, \mathbf{r}_2) = |\psi_{\pm}(\mathbf{r}_1, \mathbf{r}_2)|^2 = \frac{1}{2}|\phi_a(\mathbf{r}_1)|^2 |\phi_b(\mathbf{r}_2)|^2 + \frac{1}{2}|\phi_b(\mathbf{r}_1)|^2 |\phi_a(\mathbf{r}_2)|^2 \pm \operatorname{Re}[\phi_a(\mathbf{r}_1)\phi_b^*(\mathbf{r}_1) \phi_b(\mathbf{r}_2)\phi_a^*(\mathbf{r}_2)]$$ Notice what happens as the two particles approach the same point in space ($\mathbf{r}_1 \to \mathbf{r}_2 = \mathbf{r}$):

• For Fermions ($-$) with parallel spins: $$P_A(\mathbf{r}, \mathbf{r}) = 0$$ Identical fermions avoid each other in space, creating an effective **Pauli Exclusion Hole**. This statistical repulsion stabilizes atoms, prevents white dwarf collapse, and explains the periodic table.
• For Bosons ($+$): $$P_S(\mathbf{r}, \mathbf{r}) = 2 |\phi_a(\mathbf{r})|^2 |\phi_b(\mathbf{r})|^2 > P_{\text{classical}}$$ Identical bosons tend to clump or bunch together in the same spatial quantum state. This statistical attraction drives **Bose-Einstein Condensation** and laser coherence!

The **density matrix** $\hat{\rho}$, introduced independently by John von Neumann and Lev Landau in 1927, is the central mathematical object of quantum statistical mechanics.

1. Definition of the Density Operator

For a mixed state ensemble consisting of states $|\psi_n\rangle$ with statistical probabilities $p_n$ ($p_n \ge 0, \sum_n p_n = 1$), the density operator $\hat{\rho}$ is defined as: $$\hat{\rho} \equiv \sum_n p_n |\psi_n\rangle \langle \psi_n|$$

2. Fundamental Properties of $\hat{\rho}$

1. Hermiticity: $\hat{\rho}^\dagger = \hat{\rho}$.
2. Unit Trace (Normalization): $\operatorname{Tr}(\hat{\rho}) = \sum_k \langle k | \hat{\rho} | k \rangle = \sum_n p_n \sum_k |\langle k | \psi_n \rangle|^2 = \sum_n p_n = 1$.
3. Positive Semi-Definite: For any state $|\phi\rangle$, $\langle \phi | \hat{\rho} | \phi \rangle = \sum_n p_n |\langle \phi | \psi_n \rangle|^2 \ge 0$.
4. Purity Criterion: $$\operatorname{Tr}(\hat{\rho}^2) \le 1$$ $$\operatorname{Tr}(\hat{\rho}^2) = 1 \iff \text{Pure State}, \quad \operatorname{Tr}(\hat{\rho}^2) < 1 \iff \text{Mixed State}$$

3. Expectation Value of an Observable $\hat{A}$

The ensemble average of any quantum observable $\hat{A}$ is given cleanly by the trace: $$\langle \hat{A} \rangle = \sum_n p_n \langle \psi_n | \hat{A} | \psi_n \rangle = \sum_n p_n \sum_k \langle \psi_n | k \rangle \langle k | \hat{A} | \psi_n \rangle = \sum_k \langle k | \hat{A} \left( \sum_n p_n |\psi_n\rangle \langle \psi_n| \right) | k \rangle$$ $$\langle \hat{A} \rangle = \operatorname{Tr}(\hat{\rho} \hat{A})$$

The time evolution of the density operator represents the quantum analog of classical Liouville dynamics.

1. Derivation of the von Neumann Equation

Differentiating $\hat{\rho}(t) = \sum_n p_n |\psi_n(t)\rangle \langle \psi_n(t)|$ with respect to time: $$i\hbar \frac{\partial \hat{\rho}}{\partial t} = i\hbar \sum_n p_n \left( \frac{\partial |\psi_n\rangle}{\partial t} \langle \psi_n| + |\psi_n\rangle \frac{\partial \langle \psi_n|}{\partial t} \right)$$ Using the time-dependent Schrödinger equation $i\hbar \frac{\partial |\psi_n\rangle}{\partial t} = \hat{H} |\psi_n\rangle$ and its adjoint $-i\hbar \frac{\partial \langle \psi_n|}{\partial t} = \langle \psi_n| \hat{H}$: $$i\hbar \frac{\partial \hat{\rho}}{\partial t} = \sum_n p_n \left( \hat{H} |\psi_n\rangle \langle \psi_n| - |\psi_n\rangle \langle \psi_n| \hat{H} \right) = \hat{H} \hat{\rho} - \hat{\rho} \hat{H}$$ $$i\hbar \frac{\partial \hat{\rho}}{\partial t} = [\hat{H}, \hat{\rho}]$$ This is the **von Neumann Equation** (or quantum Liouville equation).

2. Classical Correspondence

Under Dirac's correspondence rule $[\hat{A}, \hat{B}] \longleftrightarrow i\hbar \{A, B\}_{\text{PB}}$: $$\frac{\partial \hat{\rho}}{\partial t} = \frac{1}{i\hbar} [\hat{H}, \hat{\rho}] \longleftrightarrow -\{\rho, H\}_{\text{PB}}$$ The von Neumann equation matches Liouville's theorem $\frac{\partial \rho}{\partial t} = -\{\rho, H\}$.

3. Stationary Equilibrium State

For a quantum system in statistical equilibrium, $\frac{\partial \hat{\rho}}{\partial t} = 0$, requiring: $$[\hat{H}, \hat{\rho}] = 0$$ The equilibrium density operator must commute with the Hamiltonian. Consequently, $\hat{\rho}$ must be diagonal in the basis of energy eigenstates.

We now express the fundamental thermodynamic ensembles in terms of the quantum density operator and derive the three quantum distribution functions.

1. Quantum Density Operators for the Three Ensembles

• Quantum Microcanonical Ensemble: $$\hat{\rho} = \frac{1}{\Omega} \sum_{E \le E_n \le E + \delta E} |n\rangle \langle n|, \quad S = -k_B \operatorname{Tr}(\hat{\rho} \ln \hat{\rho}) = k_B \ln \Omega$$
• Quantum Canonical Ensemble: $$\hat{\rho} = \frac{e^{-\beta \hat{H}}}{Z}, \quad Z = \operatorname{Tr}(e^{-\beta \hat{H}}) = \sum_n e^{-\beta E_n}$$ $$F = -k_B T \ln Z$$
• Quantum Grand Canonical Ensemble: $$\hat{\rho} = \frac{e^{-\beta (\hat{H} - \mu \hat{N})}}{\Xi}, \quad \Xi = \operatorname{Tr}(e^{-\beta (\hat{H} - \mu \hat{N})})$$ $$\Phi_G = -k_B T \ln \Xi = -PV$$

2. Derivation of Unified Mean Occupation Numbers

For non-interacting quantum particles with single-particle energy levels $\epsilon_i$, the grand partition function factorizes over independent single-particle states: $$\Xi = \prod_i \Xi_i, \quad \Xi_i = \sum_{n_i} e^{-\beta (\epsilon_i - \mu) n_i}$$

• For Fermions (Fermi-Dirac): $n_i \in \{0, 1\}$ due to Pauli exclusion: $$\Xi_i^{\text{FD}} = 1 + e^{-\beta(\epsilon_i - \mu)}$$ $$\bar{n}_i^{\text{FD}} = -\frac{1}{\beta}\frac{\partial \ln \Xi_i}{\partial \epsilon_i} = \frac{e^{-\beta(\epsilon_i - \mu)}}{1 + e^{-\beta(\epsilon_i - \mu)}} = \frac{1}{e^{\beta(\epsilon_i - \mu)} + 1}$$
• For Bosons (Bose-Einstein): $n_i \in \{0, 1, 2, \dots\}$ (requires $\mu < \epsilon_0$ for convergence): $$\Xi_i^{\text{BE}} = \sum_{n=0}^\infty \left(e^{-\beta(\epsilon_i - \mu)}\right)^n = \frac{1}{1 - e^{-\beta(\epsilon_i - \mu)}}$$ $$\bar{n}_i^{\text{BE}} = -\frac{1}{\beta}\frac{\partial \ln \Xi_i}{\partial \epsilon_i} = \frac{1}{e^{\beta(\epsilon_i - \mu)} - 1}$$
• Classical Maxwell-Boltzmann Limit: When $e^{\beta(\epsilon_i - \mu)} \gg 1$: $$\bar{n}_i^{\text{MB}} = \frac{1}{e^{\beta(\epsilon_i - \mu)}}$$

3. The Master Master Distribution Formula

All three statistics are captured by a single unified equation: $$\bar{n}_i = \frac{1}{e^{\beta(\epsilon_i - \mu)} + a}, \quad a = \begin{cases} +1 & \text{Fermi-Dirac (Fermions)} \\ -1 & \text{Bose-Einstein (Bosons)} \\ 0 & \text{Maxwell-Boltzmann (Classical)} \end{cases}$$