 Method of quantum characteristics

In quantum mechanics, quantum characteristics are phasespace trajectories that arise in the deformation quantization through the WeylWigner transform of Heisenberg operators of canonical coordinates and momenta. These trajectories obey the Hamilton’s equations in quantum form and play the role of characteristics in terms of which timedependent Weyl's symbols of quantum operators can be expressed. In classical limit, quantum characteristics turn to classical trajectories. The knowledge of quantum characteristics is equivalent to the knowledge of quantum dynamics.
Contents
WeylWigner association rule
In Hamiltonian dynamics classical systems with n degrees of freedom are described by 2n canonical coordinates and momenta
that form a coordinate system in the phase space. These variables satisfy the Poisson bracket relations
The skewsymmetric matrix ,
where is the identity matrix, defines nondegenerate 2form 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
The Weyl’s association rule ^{[1]} extends the correspondence to arbitrary phasespace functions and operators.
Taylor expansion
A onesided 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 .
Under the reverse association , the density matrix turns to Wigner function. ^{[2]} Wigner functions have numerous applications in quantum manybody physics, kinetic theory, collision theory, quantum chemistry.
A refined version of the WeylWigner association rule is proposed by Stratonovich. ^{[3]}
Stratonovich basis
The set of operators acting in the Hilbert space is closed under multiplication of operators by cnumbers and summation. Such a set constitutes a vector space . The association rule formulated with the use of the Taylor expansion preserves operations on the operators. The correspondence can be illustrated with the following diagram:
Here, and are functions and and are the associated operators.
The elements of basis of are labelled by canonical variables . The commonly used Stratonovich basis looks like
The WeylWigner twosided association rule for function and operator has the form
The function provides coordinates of the operator in the basis . The basis is complete and orthogonal:
Alternative operator bases are discussed also. ^{[4]} The freedom in choice of the operator basis is better known as operator ordering problem.
Starproduct
The set of operators is closed under the multiplication of operators. The vector space is endowed thereby with an associative algebra structure. Given two functions
one can construct a third function
called product ^{[5]} ^{[6]} or Moyal product. It is given explicitly by
where
is the Poisson operator. The product splits into symmetric and skewsymmetric parts
The product is not associative. In the classical limit product becomes the dotproduct. The skewsymmetric part is known under the name of Moyal bracket. This is the Weyl's symbol of commutator. In the classical limit Moyal bracket becomes Poisson bracket. Moyal bracket is quantum deformation of Poisson bracket.
Quantum characteristics
The correspondence shows that coordinate transformations in the phase space are accompanied by transformations of operators of the canonical coordinates and momenta and vice versa. Let be the evolution operator,
and is Hamiltonian. Consider the following scheme:
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 phasespace coordinates that correspond to new operators in the old basis are given by
with the initial conditions
The functions define quantum phase flow. In the general case, it is canonical to first order in τ. ^{[7]}
Starfunction
The set of operators of canonical variables is complete in the sense that any operator can be represented as a function of operators . Transformations
induce under the Wigner association rule transformations of phasespace functions:
Using the Taylor expansion, the transformation of function under the evolution can be found to be
Composite function defined in such a way is called function. The composition law differs from the classical one. However, semiclassical expansion of around is formally well defined and involves even powers of only. This equation shows that, given quantum characteristics are constructed, physical observables can be found without further addressing to Hamiltonian. The functions play the role of characteristics ^{[8]} similarly to classical characteristics used to solve classical Liouville equation.
Quantum Liouville equation
Wigner transform of the evolution equation for the density matrix in the Schrödinger representation leads quantum Liouville equation for the Wigner function. Wigner transform of the evolution equation for operators in the Heisenberg representation,
leads to the same equation with the opposite (plus) sign in the righthand side:
function solves this equation in terms of quantum characteristics:
Similarly, the evolution of the Wigner function in the Schrödinger representation is given by
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 righthand side is calculated like in the classical mechanics. The composite function is, however, function. The product violates canonicity of the phase flow beyond the first order in τ.
Conservation of Moyal bracket
The antisymmetrized products of even number of operators of canonical variables are cnumbers as a consequence of the commutation relations. These products are left invariant by unitary transformations and, in particular,
Phasespace transformations induced by the evolution operator preserve the Moyal bracket and do not preserve the Poisson bracket, so the evolution map
is not canonical. ^{[8]} Transformation properties of canonical variables and phasespace 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 starcomposition law
which is nonlocal and is distinct from the dotcomposition law of classical mechanics.
Energy conservation
The energy conservation implies
 ,
where
is Hamilton's function. In the usual geometric sense, is not conserved along quantum characteristics.
Summary
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 WeylWigner transform of the von Neumann evolution equation for the density matrix in Schrödinger representation. The quantum Hamilton's equations are the WeylWigner transforms of the evolution equations for operators of the canonical coordinates and momenta in Heisenberg representation.
In classical systems, characteristics satisfy usually firstorder ODE, e.g., classical Hamilton's equations, and solve firstorder PDE, e.g., classical Liouville equation. Functions are characteristics also, despite both and obey infiniteorder PDE.










Liouville equation Finiteorder PDE Infiniteorder PDE Hamilton's equations Finiteorder ODE Infiniteorder PDE Initial conditions Initial conditions Composition law composition law Conservation of Poisson bracket Conservation of Moyal bracket Energy conservation Energy conservation Solutions to Liouville equation









The quantum phase flow contains entire information on the quantum evolution. Semiclassical expansion of quantum characteristics and functions of quantum characteristics in power series in allows calculation of the average values of timedependent physical observables by solving a finiteorder coupled system of ODE for phase space trajectories and Jacobi fields.^{[9]}^{[10]} The order of the system of ODE depends on truncation of the power series. The tunneling effect is nonperturbative in and is not captured by the expansion. Quantum characteristics are distinct from trajectories of the de Broglie  Bohm theory. ^{[11]}
See also
 Weyl quantization
 Wigner distribution function
 Modified Wigner distribution function
 Negative probability
 Method of characteristics
References
 ^ H. Weyl, Z. Phys. 46, 1 (1927).
 ^ E. P. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev. 40, 749 (1932).
 ^ R. L. Stratonovich, Sov. Phys. JETP 4, 891 (1957).
 ^ C. L. Mehta, J. Math. Phys. 5, 677 (1964).
 ^ H. J. Groenewold, On the Principles of elementary quantum mechanics, Physica, 12, 405 (1946).
 ^ J. E. Moyal, Quantum mechanics as a statistical theory, Proceedings of the Cambridge Philosophical Society, 45, 99 (1949).
 ^ P. A. M. Dirac, The Principles of Quantum Mechanics, First Edition (Oxford: Clarendon Press, 1930).
 ^ ^{a} ^{b} M. I. Krivoruchenko, A. Faessler, Weyl's symbols of Heisenberg operators of canonical coordinates and momenta as quantum characteristics, J. Math. Phys. 48, 052107 (2007).
 ^ M. I. Krivoruchenko, C. Fuchs, A. Faessler, Semiclassical expansion of quantum characteristics for manybody potential scattering problem, Annalen der Physik 16, 587 (2007).
 ^ S. Maximov, On a special picture of dynamical evolution of nonlinear quantum systems in the phasespace representation, Physica D238, 1937 (2009).
 ^ P. R. Holland, The Quantum Theory of Motion, (Cambridge Uni. Press, Cambridge, 1993).
Textbooks
 H. Weyl, The Theory of Groups and Quantum Mechanics, (Dover Publications, New York Inc., 1931).
 V. I. Arnold, Mathematical Methods of Classical Mechanics, (2nd ed. SpringerVerlag, New York Inc., 1989).
 M. V. Karasev and V. P Maslov, Nonlinear Poisson Brackets, (Nauka, Moscow, 1991).
Categories: Partial differential equations

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