16.2 Quantum partition functions

Quantum Partition Functions

Quantum partition functions are the backbone of quantum statistical mechanics. They connect microscopic quantum states to macroscopic thermodynamic properties, letting you calculate quantities like energy, entropy, and pressure for quantum systems.

These functions differ from their classical counterparts by accounting for discrete energy levels and inherently quantum effects like zero-point energy. The distinction matters most at low temperatures and small scales, where the classical approximation breaks down.

Quantum Partition Function Definition

The quantum partition function, denoted $Z$, describes the statistical properties of a quantum system in thermal equilibrium. It's defined as the sum of Boltzmann factors over all possible quantum states:

$Z = \sum_{i} e^{-\beta E_i}$

where $\beta = \frac{1}{k_B T}$

• $E_i$ is the energy of the $i$-th quantum state
• $k_B$ is the Boltzmann constant ($1.380649 \times 10^{-23}$ J/K)
• $T$ is the absolute temperature in Kelvin

Each term $e^{-\beta E_i}$ in the sum is a Boltzmann factor, which weights each state by how energetically accessible it is at temperature $T$. Higher-energy states contribute less to $Z$ because their Boltzmann factors are exponentially smaller.

Once you have $Z$, you can find the probability of the system occupying any particular quantum state $i$:

$P_i = \frac{e^{-\beta E_i}}{Z}$

The partition function in the denominator ensures these probabilities sum to 1. From here, you can compute ensemble averages of any physical quantity (energy, magnetization, etc.) by weighting each state's value by its probability.

Derivation for Quantum Systems

Ideal Gas

For an ideal gas of $N$ non-interacting, indistinguishable particles, the full partition function is:

$Z = \frac{1}{N!} (Z_1)^N$

Here's where each piece comes from:

1. Start with the single-particle partition function $Z_1$. For a particle in a box of volume $V$, the translational energy levels are closely spaced enough that the sum over states can be evaluated to give $Z_1 = \frac{V}{\Lambda^3}$.
2. $\Lambda$ is the thermal de Broglie wavelength: $\Lambda = \frac{h}{\sqrt{2\pi m k_B T}}$, where $h = 6.626 \times 10^{-34}$ J·s is Planck's constant and $m$ is the particle mass. This length scale tells you when quantum effects kick in: when the average interparticle spacing becomes comparable to $\Lambda$, you can no longer treat particles classically.
3. Since the $N$ particles are non-interacting, you might expect $Z = (Z_1)^N$. But identical quantum particles are indistinguishable, so you divide by $N!$ to avoid overcounting permutations of particles among states. This is the same correction Gibbs introduced to resolve the classical mixing paradox.

Quantum Harmonic Oscillator

For a single harmonic oscillator with angular frequency $\omega$, the energy levels are $E_n = \hbar \omega (n + \frac{1}{2})$ for $n = 0, 1, 2, \ldots$, where $\hbar = \frac{h}{2\pi}$.

1. Write the partition function as a sum over all quantum numbers: $Z = \sum_{n=0}^{\infty} e^{-\beta \hbar \omega (n + 1/2)}$

2. Factor out the zero-point energy term: $Z = e^{-\beta \hbar \omega / 2} \sum_{n=0}^{\infty} e^{-\beta \hbar \omega \, n}$

3. The remaining sum is a geometric series with ratio $r = e^{-\beta \hbar \omega}$. Since $r < 1$, it converges to $\frac{1}{1 - e^{-\beta \hbar \omega}}$.

4. Combining gives the closed-form result: $Z = \frac{e^{-\beta \hbar \omega / 2}}{1 - e^{-\beta \hbar \omega}} = \frac{1}{2 \sinh(\beta \hbar \omega / 2)}$

Notice that the zero-point energy $\frac{1}{2}\hbar\omega$ appears explicitly. This is a purely quantum feature with no classical analog.

Thermodynamic Property Calculations

Once you have $Z$, all standard thermodynamic quantities follow from derivatives of $\ln Z$. This is what makes the partition function so powerful.

• Average energy:

$\langle E \rangle = -\frac{\partial \ln Z}{\partial \beta}$ You differentiate $\ln Z$ with respect to $\beta$ (at constant volume). For the harmonic oscillator, this yields $\langle E \rangle = \frac{\hbar\omega}{2} + \frac{\hbar\omega}{e^{\beta\hbar\omega} - 1}$, which correctly includes the zero-point energy.

• Entropy:

$S = k_B \ln Z + k_B T \frac{\partial \ln Z}{\partial T}$ This can also be written as $S = k_B (\ln Z + \beta \langle E \rangle)$, which connects directly to the Gibbs entropy formula.

• Pressure:

$P = k_B T \frac{\partial \ln Z}{\partial V}\bigg|_T$ For the ideal gas, this recovers the familiar equation of state $PV = Nk_BT$.

• Heat capacity at constant volume:

$C_V = \frac{\partial \langle E \rangle}{\partial T} = k_B \beta^2 \frac{\partial^2 \ln Z}{\partial \beta^2}$ This involves the second derivative of $\ln Z$ with respect to $\beta$, which is also related to the variance of the energy: $C_V = \frac{\langle E^2 \rangle - \langle E \rangle^2}{k_B T^2}$.

Quantum vs Classical Partition Functions

Both quantum and classical partition functions serve the same purpose: they encode the statistical properties of a system in thermal equilibrium and let you extract thermodynamic quantities through the same derivative relations above.

The core difference is in how they count states:

• The quantum partition function sums over discrete energy levels:

$Z_{\text{quantum}} = \sum_{i} e^{-\beta E_i}$ It naturally captures quantized energy levels, zero-point energy, and particle indistinguishability.

• The classical partition function integrates over continuous phase space:

$Z_{\text{classical}} = \frac{1}{N! \, h^{3N}} \int e^{-\beta H(p,q)} \, dp \, dq$ It treats energy as a continuous variable and requires the $h^{3N}$ factor to be inserted by hand to make the result dimensionless (a hint that classical mechanics is incomplete here).

When does the classical approximation work? When the thermal de Broglie wavelength $\Lambda$ is much smaller than the average interparticle spacing. At high temperatures and low densities, $\Lambda$ shrinks and the quantum sum effectively becomes the classical integral. But at low temperatures, high densities, or for light particles (electrons, helium atoms), quantum effects dominate and the classical description fails. That's precisely where quantum partition functions become essential.