Superoperator

Example von Neumann Equation

In quantum mechanics the Schrödinger Equation, i ℏ ∂ ∂ t Ψ = H ^ Ψ expresses the time evolution of the state vector ψ by the action of the Hamiltonian H ^ which is an operator mapping state vectors to state vectors.

In the more general formulation of John von Neumann, statistical states and ensembles are expressed by density operators rather than state vectors. In this context the time evolution of the density operator is expressed via the von Neumann equation in which density operator is acted upon by a superoperator H mapping operators to operators. It is defined by taking the commutator with respect to the Hamiltonian operator:

i ℏ ∂ ∂ t ρ = H [ ρ ]

where

H [ ρ ] = [ H ^ , ρ ] ≡ H ^ ρ − ρ H ^

As commutator brackets are used extensively in QM this explicit superoperator presentation of the Hamiltonian's action is typically omitted.

Example Derivatives of Functions on the Space of Operators

When considering an operator valued function of operators H ^ = H ^ ( P ^ ) as for example when we define the quantum mechanical Hamiltonian of a particle as a function of the position and momentum operators, we may (for whatever reason) define an “Operator Derivative” Δ H ^ Δ P ^ as a superoperator mapping an operator to an operator.

For example if H ( P ) = P 3 = P P P then its operator derivative is the superoperator defined by:

Δ H Δ P [ X ] = X P 2 + P X P + P 2 X

This “operator derivative” is simply the Jacobian matrix of the function (of operators) where one simply treats the operator input and output as vectors and expands the space of operators in some basis. The Jacobian matrix is then an operator (at one higher level of abstraction) acting on that vector space (of operators).