We humans find great joy in being able to comprehend complex ideas in simple terms. Our sensory and intellectual faculties severely limit the range of our experiences as well as their understanding. Thus, the exercise of simplification enables us to see the beauty we’d not have seen otherwise. In this sense, statistical mechanics serves as a lens that enables us to appreciate incomprehensible microscopic complexity by bringing it to relatively simpler statistical terms that we are able to appreciate.

It is common in statistical mechanics to describe the equilibrium behavior of a system of particles by using a statistical ensemble. For example, a microcanonical ensemble is used to describe an isolated system i.e. the one that doesn’t exchange energy or particles with the surrounding environment. Regardless of the initial state, such a system settles down to a typical state without the memory of where it began. A microcanonical ensemble is an equal probability distribution over a constant energy surface. For a system consisting of simple harmonic oscillators, we can think of a microcanonical ensemble as a spherical shell of thickness $$\delta E$$ at a radius $$E$$ from the center (see Fig. 1). In classical systems, ergodicity and Louiville’s theorem provide the link between the time-averaging and microcanonical averaging.

Fig 1. A microcanonical ensemble i.e. a statistical description of an isolated system in equilibrium is defined as an equal probability distribution over a constant energy shell.

In our statistical mechanics courses, we borrow the microcanonical idea, and apply it to isolated quantum systems. Nevertheless, we also learn that the dynamics of a quantum state $$\ket{\psi_0}$$ is described by a unitary evolution i.e. $$\ket{\psi_t} = e^{\frac{-iHt}{\hbar}} \ket{\psi_0}$$. A natural question then arises- how can a reversible process be described by a statistical ensemble that completely ignores the history of the state? John von Neuman, one of the founding father of quantum mecahnics, attempted to answer this question with his quantum ergodic theorem (QET) [1], the main topic of this article.

Let’s start by introducing some notations. We use the density matrix formalism to describe the classical probability distributions of quantum wavefunctions. If the initial state belongs to a constant energy subspace $$\mathcal{H}$$ of the Hilbert space $$\mathcal{H}_{total}$$, then the resulting equilibrium state can be described by the microcanonical ensemble given by $$\rho_{mc} = \frac{1}{D} \mathrm{I}_{D},$$ where $$D = \text{dim} (\mathcal{H})$$. Notice how all the states in the constant energy subspace have equal classical probability of $$\frac{1}{D}$$. Thus, the equilibrium expectation value of an observable $$O$$ is

$$\langle O \rangle = \text{Tr}(\rho_{mc} O) = \frac{1}{D} \text{Tr}(O).$$

When we say that a state is equivalent to a microcanonical ensemble, we mean it in the macroscopic sense. A macroscopic observable $$O$$ partitions $$\mathcal{H}$$ into ‘macro-spaces’ $$H_\nu$$’s such that (see Fig. 2 ): $$\mathcal{H} = \bigoplus_\nu H_\nu.$$ Two density matrices $$\rho$$ and $$\rho'$$ are said to be macroscopically equivalent (with respect to a macroscopic observable $$O$$) if and only if $$\label{eqn:macroscopic_equivalence} \text{Tr}(\rho P_\nu) \approx \text{Tr}(\rho’ P_{\nu}),$$ where $$P_\nu$$’s are the projections onto the $$d_\nu$$ dimensional subspace $$H_\nu$$. Thus, the macroscopic equivalence of $$\ket{\psi} \bra{\psi}$$ and $$\rho_{mc}$$ requires from Eqn \eqref{eqn:macroscopic_equivalence}: $$\label{eqn:projection_dimension} \text{Tr}(\ket{\psi}\bra{\psi} P_\nu) \approx \text{Tr}(\rho_{mc} P_{\nu}) \implies |P_{\nu} \ket{\psi} |^2 \approx\frac{d_\nu}{D},$$ for all $$\nu$$. A state $$\ket{\psi}$$ that satisfies this property is called normal.

The QET provides conditions under which a system is normal for every initial state vector $$\psi_0$$. Consider a Hamiltonian $$H$$ with non-degenerate energy levels $$E_0, E_1, \cdots, E_D$$ as well as non-degenerate gaps i.e. $$E_m - E_n \neq E_m' - E_n'$$, unless $$m=m'$$ and $$m \neq m'$$, or $$m=m'$$ and $$n=n'$$. QET states that such a system holds normal typicality with respect to most macroscopic observables for every initial wave function $$\ket{\psi_0} \in \mathcal{H}$$ such that $$<\psi_0|\psi_0> =1$$. In late 1950’s numerous papers dismissing the validity of QET appeared, and it was later realized that the papers were instead considering a weaker version of QET rather than von Neumann’s statement. One might ask- at what time does a state start being normal. QET doesn’t answer this question; von Neumann made noted that the equilibration time could be longer than the age of the universe. When a state reaches normal typicality, it is said to be thermalized. We note that thermalization is a stronger condition that equilibration. Equilibration means that we will be able to express time-averages with a diagonal ensemble[2] i.e.

$$\langle O \rangle_T = \frac{1}{T}\int_0^T dt \bra{\psi_t} O \ket{\psi_t} = \text{Tr}(O \rho_D),$$

where $$\rho_D = \overline{\ket{\psi_t}\bra{\psi_t}}$$. Thermalization requires an additional condition i.e.

$$\text{Tr}(O \rho_D) = \text{Tr}(O \rho_{mc}) = <O_{mc}>,$$

which requires Eqn. \eqref{eqn:macroscopic_equivalence} to hold true.

Fig 2. A macroscopic observable $$O$$ partitions the constant energy subspace $$\mathcal{H}$$ into subspaces $$\{ \mathcal{H}_\nu \}$$ such that $$\sum_\nu d_\nu =D$$, where $$d_\nu = \text{dim} (\mathcal{H}_\nu)$$, and $$D = \text{dim} (\mathcal{H})$$.

[1] John von Neumann. Proof of the ergodic theorem and the h-theorem in quantum mechanics. The European Physical Journal H, 35(2):201–237, 2010.

[2] Masahito Ueda. Quantum equilibration, thermalization and prethermalization in ultracold atoms. Nature Reviews Physics, 2(12):669–681, 2020