Quantum characteristics are phase-space trajectories that arise in the phase space formulation of quantum mechanics through the Wigner transform of Heisenberg operators of canonical coordinates and momenta. These trajectories obey the Hamilton equations in quantum form and play the role of characteristics in terms of which time-dependent Weyl's symbols of quantum operators can be expressed. In the classical limit, quantum characteristics reduce to classical trajectories. The knowledge of quantum characteristics is equivalent to the knowledge of quantum dynamics.
Contents
Weyl-Wigner association rule
In Hamiltonian dynamics, classical systems with
that form a coordinate system in the phase space. These variables satisfy the Poisson bracket relations
The skew-symmetric matrix
where
In quantum mechanics, the canonical variables
These operators act in Hilbert space and obey commutation relations
Weyl’s association rule extends the correspondence
Taylor expansion
A one-sided association rule
The operators
Under the reverse association
A refined version of the Weyl-Wigner association rule is proposed by Groenewold and Stratonovich.
Groenewold-Stratonovich basis
The set of operators acting in the Hilbert space is closed under multiplication of operators by
Here,
The elements of basis of
The Weyl-Wigner two-sided association rule for function
The function
Alternative operator bases are discussed also. The freedom in choice of the operator basis is better known as the operator ordering problem.
Star-product
The set of operators
one can construct a third function
called
where
is the Poisson operator. The
The
Quantum characteristics
The correspondence
and
Quantum evolution transforms vectors in the Hilbert space and, upon the Wigner association rule, coordinates in the phase space. In Heisenberg representation, the operators of the canonical variables are transformed as
The phase-space coordinates
with the initial conditions
The functions
Star-function
The set of operators of canonical variables is complete in the sense that any operator can be represented as a function of operators
induce under the Wigner association rule transformations of phase-space functions:
Using the Taylor expansion, the transformation of function
Composite function defined in such a way is called
Quantum Liouville equation
The Wigner transform of the evolution equation for the density matrix in the Schrödinger representation leads to a quantum Liouville equation for the Wigner function. The Wigner transform of the evolution equation for operators in the Heisenberg representation,
leads to the same equation with the opposite (plus) sign in the right-hand side:
Similarly, the evolution of the Wigner function in the Schrödinger representation is given by
The Liouville theorem of classical mechanics fails, to the extent that, locally, the "probability" density in phase space is not preserved in time.
Quantum Hamilton's equations
Quantum Hamilton's equations can be obtained applying the Wigner transform to the evolution equations for Heisenberg operators of canonical coordinates and momenta
The right-hand side is calculated like in the classical mechanics. The composite function is, however,
Conservation of Moyal bracket
The antisymmetrized products of even number of operators of canonical variables are c-numbers as a consequence of the commutation relations. These products are left invariant by unitary transformations and, in particular,
Phase-space transformations induced by the evolution operator preserve the Moyal bracket and do not preserve the Poisson bracket, so the evolution map
is not canonical. Transformation properties of canonical variables and phase-space functions under unitary transformations in the Hilbert space have important distinctions from the case of canonical transformations in the phase space:
Composition law
Quantum characteristics can hardly be treated visually as trajectories along which physical particles move. The reason lies in the star-composition law
which is non-local and is distinct from the dot-composition law of classical mechanics.
Energy conservation
The energy conservation implies
where
is Hamilton's function. In the usual geometric sense,
Summary
The origin of the method of characteristics can be traced back to Heisenberg’s matrix mechanics. Suppose that we have solved in the matrix mechanics the evolution equations for the operators of the canonical coordinates and momenta in the Heisenberg representation. These operators evolve according to
It is known that for any operator
This equation shows that
Table compares properties of characteristics in classical and quantum mechanics. PDE and ODE are partial differential equations and ordinary differential equations, respectively. The quantum Liouville equation is the Weyl-Wigner transform of the von Neumann evolution equation for the density matrix in Schrödinger representation. The quantum Hamilton equations are the Weyl-Wigner transforms of the evolution equations for operators of the canonical coordinates and momenta in Heisenberg representation.
In classical systems, characteristics
The quantum phase flow contains entire information on the quantum evolution. Semiclassical expansion of quantum characteristics and