Density Matrices and Statistical Ensembles

1. Equilibrium Density Matrices: The General Forms

For a system in thermal equilibrium, the state is stationary. The condition $[\hat{H},\hat{\rho }]=0$ tells us that the equilibrium density matrix must be a function of the Hamiltonian. If other quantities like particle number $\hat{N}$ are conserved, $\hat{\rho }$ will commute with them as well. The specific functional form of $\hat{\rho }$ depends on the physical constraints imposed on the system, which defines the ensemble.

The following resource provides a concise summary of the results we will spend this lesson deriving and understanding. It's useful to see the final forms up front.

2. The Microcanonical Ensemble (N, V, E)

The microcanonical ensemble describes an isolated system with a fixed number of particles $N$, volume $V$, and energy $E$. The fundamental postulate, which you'll recall from our discussions on classical statistical mechanics, is that of equal a priori probabilities: all accessible microstates are equally likely.

In the quantum context, the "microstates" are the energy eigenstates of the system. If the system's energy is precisely $E$, the only accessible states are the eigenstates $|n⟩$ for which the eigenvalue $E_n$ is equal to $E$. Let's say there are $\Omega (E)$ such degenerate states. According to the postulate, the probability of finding the system in any one of these states is $p_n=1/\Omega (E)$, and the probability of finding it in any other state is zero.

Using the general definition of a mixed-state density operator, $\hat{\rho }={\sum }_n{p}_n|n⟩⟨n|$, we can construct the microcanonical density matrix:

The sum ${\sum }_{n\text{ with }{E}_n=E}|n⟩⟨n|$ is simply the projection operator, ${\hat{P}}_E$, onto the subspace of Hilbert space spanned by the eigenstates with energy $E$. Thus, we can write:

where $\Omega (E)=\text{Tr}({\hat{P}}_E)$ is the degeneracy of the energy level $E$.

A more formal, and often more useful, way to write this is using the Dirac delta function of an operator, as seen in the reading. This form, ${\hat{\rho }}_{mc}=\frac{1}{\Omega (E)}\delta (E-\hat{H})$, is the direct quantum analogue of the classical microcanonical phase-space distribution, $\rho (q,p)\propto \delta (E-H(q,p))$.

The following reading elaborates on this construction.

3. The Canonical Ensemble (N, V, T)

The canonical ensemble describes a system in thermal contact with a large heat reservoir at a fixed temperature $T$. The system can exchange energy with the reservoir, so its energy is no longer fixed.

The most rigorous way to derive the form of the canonical density matrix is by invoking the principle of maximum entropy (MaxEnt), which we first encountered in Module 1. We seek the density matrix $\hat{\rho }$ that maximizes the von Neumann entropy, $S_{vN}=-k_B\text{Tr}(\hat{\rho }\ln \hat{\rho })$, subject to the physical constraints of the ensemble.

The constraints for the canonical ensemble are:

1. Normalization: The total probability must be one. $\text{Tr}(\hat{\rho })=1$.
2. Fixed Average Energy: The ensemble average energy is a fixed value, $U$. $⟨\hat{H}⟩=\text{Tr}(\hat{\rho }\hat{H})=U$.

The following reading walks through the constrained maximization procedure using Lagrange multipliers. Given your mathematical background, you should find this derivation satisfyingly rigorous.

As the derivation shows, the unbiased density operator that maximizes entropy subject to the canonical constraints is:

Here, $\beta =1/(k_{B}T)$, and the normalization factor $Z$ is the canonical partition function, defined as the trace of the unnormalized Boltzmann factor:

This expression is the quantum mechanical centerpiece of the canonical ensemble. It is the direct analogue of the classical partition function $Z=\int e^{-\beta H(q,p)}d\Gamma$.

4. The Grand Canonical Ensemble (μ, V, T)

The grand canonical ensemble describes a system that can exchange both energy and particles with a large reservoir at fixed temperature $T$ and chemical potential $\mu$.

We can construct the density matrix using the exact same MaxEnt logic. We maximize $S_{vN}$ subject to three constraints:

1. Normalization: $\text{Tr}(\hat{\rho })=1$.
2. Fixed Average Energy: $\text{Tr}(\hat{\rho }\hat{H})=U$.
3. Fixed Average Particle Number: $\text{Tr}(\hat{\rho }\hat{N})=⟨N⟩$, where $\hat{N}$ is the particle number operator.

Introducing a third Lagrange multiplier for the particle number constraint, the procedure directly yields the grand canonical density matrix:

The normalization factor $\Xi$ is the grand partition function:

Here, the trace must be performed over a basis that spans all possible particle numbers, which is known as a Fock space. Your experience with quantum field theory and squeezed states of light will have familiarized you with this concept of a Hilbert space that accommodates a variable number of particles.

5. Representation in the Energy Eigenbasis

The operator forms we have derived are compact and general. To see what they mean in a more concrete matrix representation, it is most natural to work in the basis of energy eigenstates, $|n⟩$, where $\hat{H}|n⟩=E_n|n⟩$.

In this basis, the Hamiltonian operator $\hat{H}$ is represented by a diagonal matrix. Any well-behaved function of the Hamiltonian, such as $e^{-\beta \hat{H}}$, will also be represented by a diagonal matrix.
For the canonical ensemble, the matrix elements of ${\hat{\rho }}_c$ are:

This is a diagonal matrix. The diagonal elements, ${\rho }_{nn}=e^{-\beta E_n}/Z$, are simply the Boltzmann probabilities for the system to be found in the energy eigenstate $|n⟩$. All the off-diagonal elements (the "coherences") are zero.

This is a profound result. It tells us that a system in thermal equilibrium, when viewed in the basis of its own energy eigenstates, is a purely classical statistical mixture. The quantum "weirdness" is contained within the nature of the eigenstates themselves, but the statistical weighting among them is classical. This is very different from a general pure state, like $|\psi ⟩=c_1|E_1⟩+c_2|E_2⟩$, whose density matrix $\hat{\rho }=|\psi ⟩⟨\psi |$ would have non-zero off-diagonal elements ${\rho }_{12}=c_1{c}_2^{\ast }$.

Conclusion

In this lesson, we have constructed the explicit forms of the density operator for the three principal statistical ensembles, bridging the gap between abstract quantum formalism and concrete statistical mechanics.

Key Takeaways:

• For any system in equilibrium, its density matrix $\hat{\rho }$ is a function of its Hamiltonian $\hat{H}$ and any other conserved quantities.
• The Microcanonical density matrix represents an isolated system with fixed energy $E$, where all $\Omega (E)$ accessible states are equally probable:
  ${\hat{\rho }}_{mc}=\frac{1}{\Omega (E)}{\sum }_{n\text{ with }{E}_n=E}|n⟩⟨n|=\frac{{\hat{P}}_E}{\text{Tr}({\hat{P}}_E)}$
• The Canonical density matrix represents a system at fixed temperature $T$, and follows the Boltzmann distribution:
  ${\hat{\rho }}_c=\frac{e^{-\beta \hat{H}}}{\text{Tr}(e^{-\beta \hat{H}})}$
• The Grand Canonical density matrix represents an open system at fixed temperature $T$ and chemical potential $\mu$:
  ${\hat{\rho }}_{gc}=\frac{e^{-\beta (\hat{H}-\mu \hat{N})}}{\text{Tr}(e^{-\beta (\hat{H}-\mu \hat{N})})}$
• In the energy eigenbasis, the density matrices for the canonical and grand canonical ensembles are diagonal. The diagonal elements represent the classical probabilities of occupying each energy eigenstate.

We now have all the necessary theoretical machinery. In our next lesson, we will focus on the partition functions themselves, defining them as traces of these statistical operators. We will see how these partition functions act as generating functions for all macroscopic thermodynamic quantities, completing the fundamental link between quantum statistical mechanics and thermodynamics.