6.1.3 Quantum Statistics#

Prompts

What are the differences between Bose-Einstein and Fermi-Dirac distributions? How do they arise from the maximum entropy principle?
Why does the bosonic partition function take a geometric-series form? What sets the temperature dependence of the average occupation, and how does the unbounded occupancy connect to the statistics in the Fock space?
How does the \(\pm 1\) in the denominators of the BE/FD distributions encode the fundamental difference between bosons and fermions? What physical consequences follow?
In the classical limit where \(\beta\varepsilon \gg 1\), why do both distributions reduce to the Boltzmann distribution?
Can you sketch \(\langle \hat{n} \rangle\) vs \(\beta\varepsilon\) for both bosons and fermions? What happens as temperature increases?

Lecture Notes#

Overview#

The maximum entropy principle (§6.1.2) immediately yields the quantum distributions governing bosons and fermions. By applying the constraint of fixed average energy to a single mode, we derive the Bose-Einstein and Fermi-Dirac distributions from first principles. A single mode is sufficient to capture the essential physics: the statistics constrain thermodynamics.

Bose-Einstein Distribution#

Single Bosonic Mode. Consider a single bosonic mode with Hamiltonian:

\[ \hat{H} = \varepsilon \hat{b}^\dagger \hat{b} \]

The eigenstates are Fock states \(\vert n \rangle\) with \(n = 0, 1, 2, \ldots\) (unbounded occupation), and eigenvalues \(E_n = n\varepsilon\).

At thermal equilibrium, the system obeys the maximum entropy principle: the thermal (Gibbs) state is:

\[ \hat{\rho} = \frac{\mathrm{e}^{-\beta \hat{H}}}{Z} \]

where \(\beta = 1/(k_B T)\) and \(Z = \operatorname{Tr}(\mathrm{e}^{-\beta \hat{H}})\) is the partition function.

Partition Function. The partition function is:

\[ Z = \sum_{n=0}^{\infty} \mathrm{e}^{-\beta n \varepsilon} = \sum_{n=0}^{\infty} \left(\mathrm{e}^{-\beta \varepsilon}\right)^n \]

This is a geometric series with ratio \(r = \mathrm{e}^{-\beta \varepsilon} < 1\):

(218)#\[ Z = \frac{1}{1 - \mathrm{e}^{-\beta \varepsilon}} \]

Derivation: Geometric Series for Partition Function

The partition function is the geometric series

\[ S = \sum_{n=0}^{\infty} r^n \]

with ratio \(r = \mathrm{e}^{-\beta\varepsilon}\), which satisfies \(0 < r < 1\) since \(\beta\varepsilon > 0\). For \(|r| < 1\) the series converges, and its sum follows from a short algebraic argument. Multiply \(S\) by \(r\) and relabel the summation index:

\[ r S = \sum_{n=0}^{\infty} r^{n+1} = \sum_{n=1}^{\infty} r^n = S - 1 \]

The shifted sum is the original series with the \(n = 0\) term (\(r^0 = 1\)) removed, so \(rS = S - 1\). Collecting the \(S\) terms gives \((1 - r)\,S = 1\), hence:

\[ S = \sum_{n=0}^{\infty} r^n = \frac{1}{1 - r} \]

Substituting \(r = \mathrm{e}^{-\beta\varepsilon}\) gives the partition function:

\[ Z = \frac{1}{1 - \mathrm{e}^{-\beta\varepsilon}} \]

This partition function is valid for all \(T > 0\) (no phase transition, no Bose-Einstein condensation in a single mode).

Average Occupation Number. The mean occupation is the thermal expectation value of the number operator \(\hat{n} = \hat{b}^\dagger \hat{b}\). Expanding the trace in the Fock basis \(\vert n \rangle\):

\[ \langle \hat{n} \rangle = \operatorname{Tr}(\hat{\rho}\,\hat{n}) = \frac{1}{Z}\sum_{n=0}^{\infty} n\,\mathrm{e}^{-\beta n \varepsilon} \]

Each factor of \(n\) in this sum can be generated by differentiating the Boltzmann factor \(\mathrm{e}^{-\beta n\varepsilon}\) with respect to \(\beta\), so the occupation is fixed by the partition function alone:

\[ \langle \hat{n} \rangle = -\frac{1}{\varepsilon Z}\frac{\mathrm{d}Z}{\mathrm{d}\beta} = -\frac{1}{\varepsilon}\frac{\mathrm{d}\ln Z}{\mathrm{d}\beta} \]

Substituting \(Z = 1/(1 - \mathrm{e}^{-\beta\varepsilon})\) and evaluating the derivative yields the Bose-Einstein distribution.

Bose-Einstein Distribution

The average occupation number of a single bosonic mode is:

(219)#\[ \langle \hat{n} \rangle = \frac{1}{\mathrm{e}^{\beta\varepsilon} - 1} \]

where \(\beta = 1/(k_B T)\) and \(\varepsilon\) is the single-particle energy.

This is the Bose-Einstein (BE) distribution. Bosons can accumulate in the same state (no limit on \(n\)).

Derivation: Average Occupation from the Partition Function

Differentiating the Boltzmann factor \(\mathrm{e}^{-\beta n\varepsilon}\) with respect to \(\beta\) brings down a factor \(-n\varepsilon\):

\[ \frac{\mathrm{d}}{\mathrm{d}\beta}\,\mathrm{e}^{-\beta n\varepsilon} = -n\varepsilon\,\mathrm{e}^{-\beta n\varepsilon} \]

So each factor of \(n\) in the occupation sum can be traded for a \(\beta\)-derivative. Summing over \(n\) and exchanging the sum with the derivative:

\[ \sum_{n=0}^{\infty} n\,\mathrm{e}^{-\beta n\varepsilon} = -\frac{1}{\varepsilon}\frac{\mathrm{d}}{\mathrm{d}\beta}\sum_{n=0}^{\infty}\mathrm{e}^{-\beta n\varepsilon} = -\frac{1}{\varepsilon}\frac{\mathrm{d}Z}{\mathrm{d}\beta} \]

Dividing by \(Z\) expresses the occupation through the partition function, using \(Z^{-1}\,\mathrm{d}Z/\mathrm{d}\beta = \mathrm{d}\ln Z/\mathrm{d}\beta\):

\[ \langle \hat{n} \rangle = \frac{1}{Z}\sum_{n=0}^{\infty} n\,\mathrm{e}^{-\beta n\varepsilon} = -\frac{1}{\varepsilon Z}\frac{\mathrm{d}Z}{\mathrm{d}\beta} = -\frac{1}{\varepsilon}\frac{\mathrm{d}\ln Z}{\mathrm{d}\beta} \]

Insert \(\ln Z = -\ln(1 - \mathrm{e}^{-\beta\varepsilon})\) from (218). The inner derivative is \(\mathrm{d}(1 - \mathrm{e}^{-\beta\varepsilon})/\mathrm{d}\beta = \varepsilon\,\mathrm{e}^{-\beta\varepsilon}\), so the chain rule gives:

\[ \frac{\mathrm{d}\ln Z}{\mathrm{d}\beta} = -\frac{1}{1 - \mathrm{e}^{-\beta\varepsilon}}\,\frac{\mathrm{d}(1 - \mathrm{e}^{-\beta\varepsilon})}{\mathrm{d}\beta} = -\frac{\varepsilon\,\mathrm{e}^{-\beta\varepsilon}}{1 - \mathrm{e}^{-\beta\varepsilon}} \]

Substituting back, the factor \(\varepsilon\) cancels:

\[ \langle \hat{n} \rangle = -\frac{1}{\varepsilon}\left(-\frac{\varepsilon\,\mathrm{e}^{-\beta\varepsilon}}{1 - \mathrm{e}^{-\beta\varepsilon}}\right) = \frac{\mathrm{e}^{-\beta\varepsilon}}{1 - \mathrm{e}^{-\beta\varepsilon}} \]

Multiplying numerator and denominator by \(\mathrm{e}^{\beta\varepsilon}\) gives the Bose-Einstein distribution:

\[ \langle \hat{n} \rangle = \frac{1}{\mathrm{e}^{\beta\varepsilon} - 1} \]

Physical Meaning. The mean occupation behaves as follows:

High temperature (\(\beta\varepsilon \ll 1\)): \(\langle \hat{n} \rangle \approx 1/(\beta\varepsilon) \to \infty\) — many particles populate the mode.
Low temperature (\(\beta\varepsilon \gg 1\)): \(\langle \hat{n} \rangle \approx \mathrm{e}^{-\beta\varepsilon} \to 0\) — occupation suppressed, but can be any number.
No Pauli exclusion: Arbitrary occupation is allowed. Bosons “bunch”—they prefer to occupy the same state.

Fermi-Dirac Distribution#

Single Fermionic Mode. Consider a single fermionic mode with Hamiltonian:

\[ \hat{H} = \varepsilon \hat{c}^\dagger \hat{c} \]

The crucial difference: fermionic anticommutation relations impose Pauli exclusion. The number operator \(\hat{n} = \hat{c}^\dagger \hat{c}\) has only two eigenvalues: \(n = 0\) (empty) and \(n = 1\) (occupied), with energies \(E_0 = 0\) and \(E_1 = \varepsilon\).

Partition Function. The partition function has only two terms:

(220)#\[ Z = \mathrm{e}^{-\beta E_0} + \mathrm{e}^{-\beta E_1} = 1 + \mathrm{e}^{-\beta\varepsilon} \]

Simple and elegant.

Average Occupation Number. The thermal average over the two states gives:

\[ \langle \hat{n} \rangle = \frac{0\,\mathrm{e}^{-\beta E_0} + 1\,\mathrm{e}^{-\beta E_1}}{Z} = \frac{\mathrm{e}^{-\beta\varepsilon}}{1 + \mathrm{e}^{-\beta\varepsilon}} \]

Dividing numerator and denominator by \(\mathrm{e}^{-\beta\varepsilon}\):

Fermi-Dirac Distribution

The average occupation number of a single fermionic mode is:

(221)#\[ \langle \hat{n} \rangle = \frac{1}{\mathrm{e}^{\beta\varepsilon} + 1} \]

This is the Fermi-Dirac (FD) distribution. At most one fermion occupies a single state (Pauli exclusion principle).

Physical Meaning. The mean occupation behaves as follows:

High temperature (\(\beta\varepsilon \ll 1\)): \(\langle \hat{n} \rangle \approx 1/2\) — the state is equally likely occupied or empty.
Low temperature (\(\beta\varepsilon \gg 1\)): \(\langle \hat{n} \rangle \approx \mathrm{e}^{-\beta\varepsilon} \to 0\) — state mostly empty, but exponentially suppressed.
Pauli exclusion: Occupation is always \(0 \leq \langle \hat{n} \rangle \leq 1\). Fermions “exclude”—no two can occupy the same state.

Bosons vs Fermions: Side-by-Side Comparison#

Quantity	Bosons (BE)	Fermions (FD)
Max occupation	Unbounded	1 (Pauli)
Partition function	\(Z = 1/(1 - \mathrm{e}^{-\beta\varepsilon})\)	\(Z = 1 + \mathrm{e}^{-\beta\varepsilon}\)
\(\langle \hat{n} \rangle\)	\(1/(\mathrm{e}^{\beta\varepsilon} - 1)\)	\(1/(\mathrm{e}^{\beta\varepsilon} + 1)\)
Denominator	\((\cdots - 1)\)	\((\cdots + 1)\)
Low \(T\)	\(\langle \hat{n} \rangle \approx \mathrm{e}^{-\beta\varepsilon}\)	\(\langle \hat{n} \rangle \approx \mathrm{e}^{-\beta\varepsilon}\)
High \(T\)	\(\langle \hat{n} \rangle \to \infty\)	\(\langle \hat{n} \rangle \to 1/2\)
Physics	Bunching; condensation	Exclusion; saturation

Discussion: BE divergence vs FD saturation

The \(\pm 1\) in the denominators encodes a profound difference. For bosons, as temperature drops, can occupation diverge? For fermions, why is high-temperature occupation pinned at \(1/2\)? How does this connect to exchange symmetry and statistics from §2.1.3 Second quantization?

Connection to Chapter 2: The statistics (exchange symmetry of the wavefunction) directly constrains the thermodynamic behavior. Bosons, with symmetric wavefunctions, can pile up. Fermions, with antisymmetric wavefunctions, must exclude.

The Classical Limit#

When the mode energy is large compared with the thermal energy (\(\varepsilon \gg k_B T\), equivalently \(\beta\varepsilon \gg 1\)), the mode is only rarely occupied: the quantum statistical effects of bunching and exclusion become negligible, and both distributions reduce to classical statistical mechanics.

Condition. The classical limit is reached when \(\mathrm{e}^{\beta\varepsilon} \gg 1\). In this regime:

Bosons:

\[ \langle \hat{n} \rangle_{\text{BE}} = \frac{1}{\mathrm{e}^{\beta\varepsilon} - 1} \approx \frac{1}{\mathrm{e}^{\beta\varepsilon}} = \mathrm{e}^{-\beta\varepsilon} \]

Fermions:

\[ \langle \hat{n} \rangle_{\text{FD}} = \frac{1}{\mathrm{e}^{\beta\varepsilon} + 1} \approx \frac{1}{\mathrm{e}^{\beta\varepsilon}} = \mathrm{e}^{-\beta\varepsilon} \]

Boltzmann Distribution. Both reduce to:

(222)#\[ \langle \hat{n} \rangle = \mathrm{e}^{-\beta\varepsilon} \]

This is the Boltzmann distribution: the mean occupation number of a single mode at energy \(\varepsilon\) in thermal equilibrium, also called the Boltzmann occupation factor.

Physical Interpretation: In this dilute regime the mode is almost always empty (\(\langle \hat{n} \rangle \ll 1\)), so two particles rarely share it and the quantum statistical nature (BE vs FD) becomes invisible. The system behaves classically. This is the non-degenerate limit: quantum and classical statistics agree when the occupation is dilute (\(\varepsilon \gg k_B T\)).

Summary#

Bose-Einstein distribution: \(\langle \hat{n} \rangle = 1/(\mathrm{e}^{\beta\varepsilon} - 1)\) from a single bosonic mode with unbounded occupation.
Fermi-Dirac distribution: \(\langle \hat{n} \rangle = 1/(\mathrm{e}^{\beta\varepsilon} + 1)\) from a single fermionic mode with \(n \in \{0,1\}\) only.
The \(\pm 1\) difference: Encodes exchange symmetry. Bosons bunch; fermions exclude.
Partition functions: Geometric series (bosons) vs two-term sum (fermions), yet both encode the same physics.
Classical limit: When \(\mathrm{e}^{\beta\varepsilon} \gg 1\), both reduce to the Boltzmann distribution \(\mathrm{e}^{-\beta\varepsilon}\).
Connection to Chapter 2: Identical particle statistics (exchange symmetry) determine thermodynamic behavior. Quantum statistics and quantum mechanics are inseparable.

Homework#

1. Bosonic partition function. A single bosonic mode has energy \(\varepsilon\). Starting from the Fock-state sum \(Z = \sum_{n=0}^{\infty} \mathrm{e}^{-\beta n \varepsilon}\), evaluate \(Z\) as a geometric series. Differentiate \(\ln Z\) with respect to \(\beta\) to derive the Bose-Einstein distribution \(\langle \hat{n} \rangle = 1/(\mathrm{e}^{\beta\varepsilon} - 1)\).

2. Fermionic partition function. A single fermionic mode has energy \(\varepsilon\). Write the partition function using only the two allowed occupation numbers \(n = 0, 1\). Show that \(\langle \hat{n} \rangle = 1/(\mathrm{e}^{\beta\varepsilon} + 1)\) and verify that \(0 \leq \langle \hat{n} \rangle \leq 1\) for all temperatures.

3. Occupation number comparison. Consider a single mode of energy \(\varepsilon\) at temperature \(T\).

(a) Plot \(\langle \hat{n} \rangle\) versus \(\beta\varepsilon\) for both the BE and FD distributions on the same axes (range \(0.1 \leq \beta\varepsilon \leq 5\)).

(b) At which value of \(\beta\varepsilon\) does the FD occupation equal \(1/2\)? Interpret this physically.

(c) Show that \(\langle \hat{n} \rangle_{\text{BE}} > \langle \hat{n} \rangle_{\text{FD}}\) for all \(\varepsilon > 0\) and \(T > 0\). Relate this to the sign in the denominator.

4. Classical limit. Show that both the Bose-Einstein and Fermi-Dirac distributions reduce to the Boltzmann distribution \(\langle \hat{n} \rangle \approx \mathrm{e}^{-\beta\varepsilon}\) when \(\beta\varepsilon \gg 1\).

(a) Expand each distribution in powers of \(\mathrm{e}^{-\beta\varepsilon}\) and keep the leading term.

(b) Explain physically why quantum statistics becomes irrelevant at high energy or low temperature per mode.

5. Energy and heat capacity. For a single bosonic mode of energy \(\varepsilon\), the average energy is \(\langle E \rangle = \varepsilon \langle \hat{n} \rangle\).

(a) Write \(\langle E \rangle\) explicitly using the BE distribution.

(b) The heat capacity is \(C = \partial \langle E \rangle / \partial T\). Show that

\[ C = k_B (\beta\varepsilon)^2 \frac{\mathrm{e}^{\beta\varepsilon}}{(\mathrm{e}^{\beta\varepsilon}-1)^2} \]

(c) Verify that \(C \to k_B\) in the high-temperature limit (equipartition) and \(C \propto \mathrm{e}^{-\beta\varepsilon}\) in the low-temperature limit.

6. Pauli exclusion constraint. A fermionic mode has energy \(\varepsilon\) relative to the chemical potential \(\mu\) (so the effective energy is \(\varepsilon - \mu\)).

(a) At what temperature does \(\langle \hat{n} \rangle = 1/2\) exactly? Express the answer in terms of \(\varepsilon\) and \(\mu\).

(b) Sketch \(\langle \hat{n}(\varepsilon) \rangle\) versus \(\varepsilon\) at \(T = 0\) and at two finite temperatures. Describe how the step function at \(T = 0\) smooths out.

(c) For an electron gas with Fermi energy \(E_F = 5\) eV at \(T = 300\) K, estimate the thermal broadening \(k_B T / E_F\). Is the gas strongly degenerate?

7. Partition function ratio. For a single mode at fixed \(\beta\varepsilon\), define \(R = Z_{\text{BE}}/Z_{\text{FD}}\).

(a) Simplify \(R\) in closed form.

(b) Show that \(R \to 1\) as \(\beta\varepsilon \to \infty\) and \(R \to \infty\) as \(\beta\varepsilon \to 0\). Interpret each limit physically in terms of the number of thermally accessible states.