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The density matrix and the partition function of a system of free particles

1. What is the goal?

We have:

  • \(N\) identical, non-interacting quantum particles
  • inside a box of volume \(V\)
  • at temperature \(T\) (so \(\beta = 1/kT\))

We want:

  1. The density matrix \(\rho = e^{-\beta H}/Q_N\) for this gas.
  2. From it, the partition function \(Q_N(V,T)\).
  3. To see when this quantum gas behaves classically, and what quantum correlations look like when it doesn't.

2. Density matrix in coordinates

The matrix element in position space is

\[ \langle r_1,\dots,r_N|\rho|r'_1,\dots,r'_N\rangle = \frac{1}{Q_N(\beta)} \langle r_1,\dots,r_N|e^{-\beta H}|r'_1,\dots,r'_N\rangle \]
  • This is just: density matrix = Boltzmann factor of Hamiltonian, normalized.
  • \(Q_N(\beta) = \mathrm{Tr}\, e^{-\beta H}\) is the partition function.

To handle \(e^{-\beta H}\), they insert a complete set of energy eigenstates \(|E\rangle\):

\[ \langle r|\, e^{-\beta H}\, |r'\rangle = \sum_E e^{-\beta E}\, \psi_E(r_1,\dots,r_N)\, \psi_E^*(r'_1,\dots,r'_N) \]

So everything reduces to knowing the many-particle eigenfunctions \(\psi_E\).

3. Single-particle states: plane waves

For non-interacting particles in a box with periodic boundary conditions:

  • Single-particle eigenfunctions are plane waves:
\[ u_{\mathbf k}(\mathbf r) = V^{-1/2} e^{i \mathbf k \cdot \mathbf r} \]
  • Their energies are
\[ \varepsilon_{\mathbf k} = \frac{\hbar^2 k^2}{2m} \]

For \(N\) particles, the total energy is just the sum of single-particle energies:

\[ E = \frac{\hbar^2}{2m} (k_1^2 + \dots + k_N^2) \]

So a many-particle state is basically labelled by the set of wave-vectors \(\mathbf K = (\mathbf k_1,\dots,\mathbf k_N)\).

4. Many-particle wavefunction for identical particles

Because the particles are identical:

  • For bosons the wavefunction must be symmetric under exchange.
  • For fermions it must be antisymmetric.

So they build \(\psi_{\mathbf K}(1,\dots,N)\) from the product of plane waves but summed over all permutations \(P\) of the particle labels:

\[ \psi_{\mathbf K}(1,\dots,N) = (N!)^{-1/2} \sum_P \delta_P\, P[u_{k_1}(1)\dots u_{k_N}(N)] \]
  • \(P[\dots]\) means "permute the labels 1,…,N inside".
  • \(\delta_P = +1\) for bosons, and \(\delta_P = \pm 1\) for fermions depending on whether \(P\) is even or odd. This gives the correct symmetry.

This is the standard (anti)symmetrized wavefunction.

5. Putting it together in the density matrix

When you plug these \(\psi_{\mathbf K}\) into the sum

\[ \sum_{\mathbf K} e^{-\beta E_{\mathbf K}} \psi_{\mathbf K}(r_1,\dots,r_N) \psi_{\mathbf K}^*(r'_1,\dots,r'_N) \]

you get:

  • A sum over all sets of wave vectors \(\mathbf k_1,\dots,\mathbf k_N\).
  • For each set, you have products of plane waves and their complex conjugates.
  • You also have sums over permutations \(P\) and \(\tilde P\) with factors \(\delta_P \delta_{\tilde P}\).

After some combinatorics, the result simplifies so that the sum over \(\mathbf k_i\) factorizes into \(N\) similar sums (since particles don't interact), and you are left with something of the form

\[ \langle 1,\dots,N|e^{-\beta H}|1',\dots,N'\rangle \propto \sum_P \delta_P f(P r_1 - r'_1)\dots f(P r_N - r'_N) \]

where \(f\) will turn out to be a Gaussian function.

6. Doing the \(\mathbf k\)-sums → Gaussian and thermal wavelength

Because the box is large, sums over \(\mathbf k\) become integrals:

\[ \sum_{\mathbf k} \to \frac{V}{(2\pi)^3} \int d^3k \]

Each integral is Gaussian:

\[ \int d^3k \; \exp\!\left[-\beta \frac{\hbar^2 k^2}{2m} + i \mathbf k\cdot(\mathbf r - \mathbf r')\right] \]

This gives a Gaussian in the distance \(|\mathbf r - \mathbf r'|\):

\[ f(\xi) = \exp\!\left(-\frac{m}{2\beta \hbar^2} \xi^2\right) \]

They rewrite the coefficient using the thermal de Broglie wavelength

\[ \lambda = \frac{h}{\sqrt{2\pi m kT}} \]

so that

\[ f(r) = \exp\left(-\pi \frac{r^2}{\lambda^2}\right) \]

This \(\lambda\) is the typical quantum "spread" of a thermal particle: if particles are much farther apart than \(\lambda\), their wavefunctions barely overlap.

With this, the diagonal elements (set \(r_i' = r_i\)) of the density matrix become

\[ \langle r_1,\dots,r_N|\rho|r_1,\dots,r_N\rangle \propto \sum_P \delta_P f(r_{P1}-r_1) \dots f(r_{PN}-r_N) \]

7. Classical limit and condition \(n\lambda^3 \ll 1\)

To get the partition function, you integrate the diagonal over all coordinates:

\[ Q_N(V,T) = \int d^{3N}r\; \langle r_1,\dots,r_N|\;e^{-\beta H}\;|r_1,\dots,r_N\rangle \]

You must understand the sum over permutations

\[ \sum_P \delta_P f_{P1 1}\dots f_{PN N} \]

where \(f_{ij} = f(r_i - r_j)\).

They argue:

  1. The leading term is the one where no particles are exchanged (identity permutation). In that term \(f(0)=1\).

  2. The next group of terms are those where only one pair of coordinates is exchanged; a typical factor looks like \(f(r_j-r_i) f(r_i-r_j)=f_{ij}^2\).

  3. More complicated permutations give products involving three or more pairs.

So they expand

\[ \sum_P (\text{stuff}) \approx 1 \pm \sum_{i<j} f_{ij} + \dots \]

The crucial physical point:

  • \(f_{ij} = f(r_i-r_j)\) is a sharply peaked function of \(r_i-r_j\) with width \(\sim \lambda\).
  • If the average distance between particles is much larger than \(\lambda\), then for most configurations \(f_{ij}\approx 0\).

The average interparticle distance is about \(n^{-1/3}\) with number density \(n = N/V\). So the "no-overlap" condition is

\[ n^{-1/3} \gg \lambda \quad\Rightarrow\quad n\lambda^3 \ll 1 \]

This is the famous degeneracy parameter:

  • \(n\lambda^3 \ll 1\): gas is classical (nondegenerate).
  • \(n\lambda^3 \gtrsim 1\): gas is quantum-degenerate.

In the classical limit, all those exchange terms are negligible and

\[ \sum_P (\dots) \approx 1 \]

Then the integral is easy and you get

\[ Q_N(V,T) \approx \frac{1}{N!} \left(\frac{V}{\lambda^3}\right)^N \]

This is exactly the classical ideal gas partition function (with the Gibbs factor \(1/N!\) now derived from indistinguishability).

8. Density matrix & correlations in the classical limit

When \(n\lambda^3 \ll 1\), the full diagonal element factorizes:

\[ \langle r_1,\dots,r_N|\rho|r_1,\dots,r_N\rangle \approx \left(\frac{1}{V}\right)^N \]

This just says:

  • Each particle is equally likely anywhere in the box.
  • There is no spatial correlation between them — they behave like classical independent particles.

But to see quantum corrections, they keep the next term in the expansion, which introduces the factor involving \(f(r_{12})\).

For \(N=2\),

\[ \langle r_1,r_2|\rho|r_1,r_2\rangle = \frac{1}{2V^2} \left[1 \pm \exp\!\left(-2\pi \frac{r_{12}^2}{\lambda^2}\right)\right] \]

So the pair probability density is

\[ P(r_{12}) \propto 1 \pm \exp\!\left(-2\pi \frac{r_{12}^2}{\lambda^2}\right) \]

Interpretation

  • For bosons (+ sign): At small separation \(r_{12}\ll\lambda\), the factor becomes \(\sim 2\). → Double the classical probability: bosons like to bunch together ("statistical attraction").

  • For fermions (− sign): At small separation, \(1-\exp(\dots)\to 0\). → Probability goes to zero: fermions avoid each other ("statistical repulsion", essentially Pauli exclusion).

At distances \(r_{12}\gg\lambda\) the exponential is tiny and both cases go back to the classical, uncorrelated value.

9. Statistical potential

To make it look like classical particles with an effective interaction, they define a statistical potential \(v_s(r)\) by saying

\[ e^{-\beta v_s(r)} = 1 \pm \exp\!\left(-2\pi\frac{r^2}{\lambda^2}\right) \]

So

\[ v_s(r) = -kT\;\ln\!\left[1 \pm \exp\!\left(-2\pi\frac{r^2}{\lambda^2}\right)\right] \]
  • For bosons (+), \(\ln(\text{something}>1)\) → \(v_s<0\): looks like an attractive potential.
  • For fermions (−), argument of log is \(<1\) → \(v_s>0\): looks like a repulsive potential.

This isn't a real force; it's how quantum statistics alone would look if you tried to mimic it with a classical pair potential.

The plot at the end shows:

  • Bosons: negative potential near small \(r\) (downward curve).
  • Fermions: positive potential near small \(r\) (upward curve). Both go to zero for \(r \gg \lambda\).

Summary

  1. Start from the quantum density matrix of many non-interacting identical particles.
  2. Use plane-wave states and (anti)symmetrization to write it in terms of Gaussian functions of particle separations.
  3. Introduce the thermal wavelength \(\lambda\) that measures the size of a particle's wave packet at temperature \(T\).
  4. Show that if \(n\lambda^3 \ll 1\) (dilute, high-temperature), the gas behaves like a classical ideal gas and recover its partition function.
  5. Keeping next-order terms reveals spatial correlations: bosons cluster, fermions avoid each other.
  6. These correlations can be described by an effective statistical potential \(v_s(r)\).