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Concept
Method of quantum characteristics
Summary
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.
In Hamiltonian dynamics, classical systems with degrees of freedom are described by canonical coordinates and momenta
that form a coordinate system in the phase space. These variables satisfy the Poisson bracket relations
The skew-symmetric matrix ,
where is the identity matrix, defines nondegenerate 2-form in the phase space.
The phase space acquires thereby the structure of a symplectic manifold. The phase space is not metric space, so distance between two points is not defined. The Poisson bracket of two functions can be interpreted as the oriented area of a parallelogram whose adjacent sides are gradients of these functions.
Rotations in Euclidean space leave the distance between two points invariant.
Canonical transformations in symplectic manifold leave the areas invariant.
In quantum mechanics, the canonical variables are associated to operators of canonical coordinates and momenta
These operators act in Hilbert space and obey commutation relations
Weyl’s association rule extends the correspondence to arbitrary phase-space functions and operators.
A one-sided association rule was formulated by Weyl initially with the help of Taylor expansion of functions of operators of the canonical variables
The operators do not commute, so the Taylor expansion is not defined uniquely. The above prescription uses the symmetrized products of the operators. The real functions correspond to the Hermitian operators. The function is called Weyl's symbol of operator .
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