Quantum Liouville equation describes the time evolution of a quantum system in terms of the density operator rather than a wavefunction.^[1] It is the natural extension of the Schrödinger equation to statistical ensembles^[2] and is fundamental in quantum statistical mechanics and the theory of open quantum systems.^[3]

For a closed system with Hamiltonian $H$ , the density operator $ρ$ evolves according to

$i ℏ \frac{\partial ρ}{\partial t} = [H, ρ],$

where $[H, ρ] = H ρ - ρ H$ is the commutator.^[4]^[5] This equation is also known as the von Neumann equation.^[1]

Density operator formalism

The density operator provides a general description of quantum states.^[6] For a pure state,

$ρ = | ψ ⟩ ⟨ ψ | .$

For a mixed state representing an ensemble,

$ρ = \sum_{n} p_{n} | ψ_{n} ⟩ ⟨ ψ_{n} |,$

with probabilities satisfying

$p_{n} \geq 0, \sum_{n} p_{n} = 1 .$

The density operator is Hermitian, positive semi-definite, and normalized:

$T r (ρ) = 1 .$

These properties ensure that expectation values of observables can be written as $⟨ A ⟩ = T r (ρ A)$ .^[6]

Relation to the Schrödinger equation

If the system is in a pure state, substituting

$ρ = | ψ ⟩ ⟨ ψ |$

into the quantum Liouville equation reproduces the Schrödinger equation for $| ψ ⟩$ .^[4] Thus, the Liouville equation is a more general framework encompassing both pure and mixed states.

Formal solution

For a time-independent Hamiltonian, the solution can be written using the unitary time-evolution operator:

$ρ (t) = U (t) ρ (0) U^{†} (t),$

where

$U (t) = e^{- i H t / ℏ} .$

This evolution preserves trace, Hermiticity, and positivity of the density operator.^[7]

Matrix representation

In the energy eigenbasis, where

$H | n ⟩ = E_{n} | n ⟩,$

the matrix elements evolve as

$i ℏ \frac{d ρ_{m n}}{d t} = (E_{m} - E_{n}) ρ_{m n} .$

Hence,

$ρ_{m n} (t) = ρ_{m n} (0) e^{- i (E_{m} - E_{n}) t / ℏ} .$

Diagonal elements $ρ_{n n}$ represent populations, while off-diagonal elements $ρ_{m n}$ describe quantum coherences.^[7]

Connection to classical Liouville equation

The quantum Liouville equation is the operator analogue of the classical Liouville equation, which governs the evolution of a phase-space distribution function $f (q, p, t)$ .^[8] The correspondence is established via:

Classical dynamics: Poisson brackets
Quantum dynamics: commutators

In the classical limit, commutators reduce to Poisson brackets, providing a bridge between classical and quantum statistical mechanics.^[9]

Open quantum systems

For open systems interacting with an environment, the evolution is no longer purely unitary. The quantum Liouville equation is generalized to master equations such as the Lindblad equation, which include dissipative and decoherence effects.^[10]^[3]

Physics:Quantum Liouville equation

Density operator formalism

Relation to the Schrödinger equation

Formal solution

Matrix representation

Connection to classical Liouville equation

Open quantum systems

See also

Table of contents (185 articles)

Index

Full contents

References

Navigation menu

Search