6.1.3 Quantum Statistics

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?

  • For a single mode, derive the partition function and average occupation number for bosons. How does this connect to identical particle statistics from Chapter 2?

  • How does the \(\pm 1\) in the denominator 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 \(\langle n \rangle\) 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 \(Z = \text{Tr}(\mathrm{e}^{-\beta \hat{H}})\) 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 \(r = \mathrm{e}^{-\beta \varepsilon} < 1\):

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

Derivation: Geometric Series for Partition Function

For \(|r| < 1\), the geometric series is:

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

With \(r = \mathrm{e}^{-\beta\varepsilon}\), we have \(r < 1\) (since \(\beta\varepsilon > 0\)), so:

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

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

Average Occupation Number#

The average occupation is the thermal expectation value:

\[ \langle n \rangle = \text{Tr}(\hat{\rho} \hat{n}) = \frac{\text{Tr}(\mathrm{e}^{-\beta \hat{H}} \hat{n})}{Z}\]

where \(\hat{n} = \hat{b}^\dagger \hat{b}\) is the number operator.

Expanding in the Fock basis:

\[ \text{Tr}(\mathrm{e}^{-\beta \hat{H}} \hat{n}) = \sum_{n=0}^{\infty} n \mathrm{e}^{-\beta n \varepsilon} = \mathrm{e}^{-\beta\varepsilon} \frac{\mathrm{d}}{\mathrm{d}(\mathrm{e}^{-\beta\varepsilon})} Z\]

Taking the derivative:

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

After computation:

Bose-Einstein Distribution

The average occupation number of a single bosonic mode is:

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

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 \(n\)).

Physical Meaning#

  • High temperature (\(\beta\varepsilon \ll 1\)): \(\langle n \rangle \approx 1/(\beta\varepsilon) \to \infty\) — many particles populate the mode.

  • Low temperature (\(\beta\varepsilon \gg 1\)): \(\langle n \rangle \approx \mathrm{e}^{-\beta\varepsilon} \to 0\) — 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 operator \(\hat{c}^\dagger \hat{c}\) 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:

(117)#\[ Z = \mathrm{e}^{-\beta \cdot 0} + \mathrm{e}^{-\beta\varepsilon} = 1 + \mathrm{e}^{-\beta\varepsilon}\]

Simple and elegant.

Average Occupation Number#

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

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

Fermi-Dirac Distribution

The average occupation number of a single fermionic mode is:

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

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

Physical Meaning#

  • High temperature (\(\beta\varepsilon \ll 1\)): \(\langle n \rangle \approx 1/2\) — the state is equally likely occupied or empty.

  • Low temperature (\(\beta\varepsilon \gg 1\)): \(\langle n \rangle \approx \mathrm{e}^{-\beta\varepsilon} \to 0\) — state mostly empty, but exponentially suppressed.

  • Pauli exclusion: Occupation is always \(0 \leq \langle n \rangle \leq 1\). 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

\(Z = 1/(1 - \mathrm{e}^{-\beta\varepsilon})\)

\(Z = 1 + \mathrm{e}^{-\beta\varepsilon}\)

\(\langle n \rangle\)

\(1/(\mathrm{e}^{\beta\varepsilon} - 1)\)

\(1/(\mathrm{e}^{\beta\varepsilon} + 1)\)

Denominator

\((\cdots - 1)\)

\((\cdots + 1)\)

Low \(T\)

\(\langle n \rangle \approx \mathrm{e}^{-\beta\varepsilon}\)

\(\langle n \rangle \approx \mathrm{e}^{-\beta\varepsilon}\)

High \(T\)

\(\langle n \rangle \to \infty\)

\(\langle n \rangle \to 1/2\)

Physics

Bunching; condensation

Exclusion; saturation

Discussion

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 Chapter 2?

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 temperature is high or the energy \(\varepsilon\) is low, the quantum effects become negligible and both distributions should reduce to classical statistical mechanics.

Condition: \(\mathrm{e}^{\beta\varepsilon} \gg 1\)#

In this regime:

Bosons:

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

Fermions:

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

Boltzmann Distribution#

Both reduce to:

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

This is the Boltzmann distribution: the average energy fraction of a classical harmonic oscillator or any mode at energy \(\varepsilon\) in thermal equilibrium.

Physical Interpretation: At high temperature, the quantum statistical nature (BE vs FD) becomes invisible. The system behaves classically. This is the correspondence principle: quantum and classical statistics agree when quantum effects are negligible (\(\hbar \omega \ll k_B T\)).

Summary#

  • Bose-Einstein distribution: \(\langle n \rangle = 1/(\mathrm{e}^{\beta\varepsilon} - 1)\) from a single bosonic mode with unbounded occupation.

  • Fermi-Dirac distribution: \(\langle n \rangle = 1/(\mathrm{e}^{\beta\varepsilon} + 1)\) 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.

See Also

Homework#

1. Bose-Einstein Partition Function

A single bosonic mode has energy \(\varepsilon = k_B T\) (mode energy equals thermal energy).

(a) Compute \(\beta\varepsilon\).

(b) Write out the first four terms of the partition function \(Z = \sum_{n=0}^{\infty} \mathrm{e}^{-\beta n \varepsilon}\).

(c) Using the geometric series formula, compute \(Z\) exactly.

(d) Compute \(\langle n \rangle\) using the BE distribution. What is the physical interpretation?

2. Fermi-Dirac Occupation

A single fermionic mode has \(\varepsilon = 2k_B T\).

(a) Compute \(\beta\varepsilon\).

(b) Write out the partition function \(Z = 1 + \mathrm{e}^{-\beta\varepsilon}\) numerically.

(c) Compute \(\langle n \rangle\) using the FD distribution.

(d) What is the probability that the state is occupied? Empty?

3. Comparison: BE vs FD at Fixed Temperature

Both a bosonic and fermionic mode have the same energy \(\varepsilon = 0.1 k_B T\).

(a) Compute \(\beta\varepsilon\).

(b) For the boson: compute \(Z\) and \(\langle n \rangle_{\text{BE}}\).

(c) For the fermion: compute \(Z\) and \(\langle n \rangle_{\text{FD}}\).

(d) Compare the two occupations. Why is the boson’s occupation larger?

(e) Sketch \(\langle n \rangle\) vs \(\beta\varepsilon\) for both on the same plot (range \(0.1 \leq \beta\varepsilon \leq 5\)).

4. Classical Limit Verification

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

(a) For the BE distribution, show: \(\frac{1}{\mathrm{e}^{\beta\varepsilon} - 1} \approx \mathrm{e}^{-\beta\varepsilon}\) when \(\beta\varepsilon \gg 1\).

(b) For the FD distribution, show: \(\frac{1}{\mathrm{e}^{\beta\varepsilon} + 1} \approx \mathrm{e}^{-\beta\varepsilon}\) when \(\beta\varepsilon \gg 1\).

(c) Physically, why does quantum statistics become irrelevant at high temperature?

5. High-Temperature Behavior: BE

In the high-temperature limit (\(\beta\varepsilon \to 0\)), approximate \(\mathrm{e}^{\beta\varepsilon} \approx 1 + \beta\varepsilon\) for small \(\beta\varepsilon\).

(a) Show that the BE distribution becomes: \(\langle n \rangle \approx \frac{1}{\beta\varepsilon} = \frac{k_B T}{\varepsilon}\).

(b) Interpret: What does this divergence represent? Why is it unphysical for a real many-body system?

(c) [Hint for future chapters: In a many-body system with fixed particle number, this divergence is cured by a chemical potential \(\mu\). The distribution becomes \(1/(\mathrm{e}^{\beta(\varepsilon - \mu)} - 1)\), which remains finite.]

6. Pauli Exclusion Constraint

For a fermionic mode, prove that the FD occupation is always bounded: \(0 \leq \langle n \rangle \leq 1\).

(a) At what temperature is \(\langle n \rangle = 1/2\) exactly?

(b) Show that as \(T \to 0\), either \(\langle n \rangle \to 0\) or \(\langle n \rangle \to 1\) depending on the sign of \(\varepsilon - \mu\) (note: here we set \(\mu = 0\), so the sign of \(\varepsilon\) determines it).

(c) Sketch \(\langle n \rangle\) vs \(T\) for a fermionic mode. How does the shape differ from a bosonic mode?

7. Partition Function Ratio

For a fixed temperature and single-particle energy \(\varepsilon\), compute the ratio:

\[ \frac{Z_{\text{boson}}}{Z_{\text{fermion}}} = \frac{1/(1-\mathrm{e}^{-\beta\varepsilon})}{1 + \mathrm{e}^{-\beta\varepsilon}}\]

(a) Simplify this expression.

(b) In the low-temperature limit (\(\beta\varepsilon \gg 1\)), what is the ratio?

(c) In the high-temperature limit (\(\beta\varepsilon \ll 1\)), what is the ratio?

(d) Interpret: Why is the boson partition function larger?

8. Energy and Heat Capacity

For a single bosonic mode, the average energy is:

\[ \langle E \rangle = \mathrm{e}^{-\beta\varepsilon} + 2\mathrm{e}^{-2\beta\varepsilon} + 3\mathrm{e}^{-3\beta\varepsilon} + \cdots\]

or equivalently: \(\langle E \rangle = \varepsilon \langle n \rangle\).

(a) Verify: \(\langle E \rangle = \varepsilon / (\mathrm{e}^{\beta\varepsilon} - 1)\).

(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) In the high-temperature limit (\(\beta\varepsilon \ll 1\)), show \(C \approx k_B\) (equipartition).

(d) In the low-temperature limit (\(\beta\varepsilon \gg 1\)), show \(C \propto \mathrm{e}^{-\beta\varepsilon}\) (exponentially suppressed).

9. Conceptual: Why Does the Sign Matter?

The only algebraic difference between BE and FD distributions is the \(\pm 1\):

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

Write a 2–3 sentence explanation of how the \(+1\) (vs \(-1\)) comes from Pauli exclusion and fermionic anticommutation relations. How does this single sign difference lead to fundamentally different thermodynamics?

10. Challenge: Fermi Energy at \(T = 0\)

For a non-interacting fermion gas in a potential well, all states up to the Fermi energy \(E_F\) are occupied at \(T = 0\), and all states above are empty.

(a) Sketch \(\langle n(\varepsilon) \rangle\) vs \(\varepsilon\) for the FD distribution at \(T = 0\) and \(T > 0\) (on the same plot).

(b) At finite temperature, the distribution broadens over a region of width \(\sim k_B T\) around \(E_F\). Explain why.

(c) For an electron gas with \(E_F = 5\) eV and \(T = 300\) K, estimate the thermal broadening \(k_B T\) in eV. Is quantum degeneracy pressure still significant?