Coherent solutions of the classical Liouville equation for the rigid rotator are presented as positive phase-space distributions localized on the Lagrangian submanifolds of Hamilton-Jacobi theory. These solutions become Wigner-type quasiprobability distributions by a formal discretization of the left-invariant vector fields from their Fourier transform in angular momentum. The results are consistent with the usual quantization of the anisotropic rotator, but the expected value of the Hamiltonian contains a finite “zero point” energy term. It is shown that during the time when a quasiprobability distribution evolves according to the Liouville equation, the related quantum wave function should satisfy the time-dependent Schrödinger equation.
Coherent solutions of the classical Liouville equation for the rigid rotator are presented as positive phase-space distributions localized on the Lagrangian submanifolds of Hamilton-Jacobi theory. These solutions become Wigner-type quasiprobability distributions by a formal discretization of the left-invariant vector fields from their Fourier transform in angular momentum. The results are consistent with the usual quantization of the anisotropic rotator, but the expected value of the Hamiltonian contains a finite “zero point” energy term. It is shown that during the time when a quasiprobability distribution evolves according to the Liouville equation, the related quantum wave function should satisfy the time-dependent Schrödinger equation.
We derive a classical Schrodinger type equation from the classical Liouville equation in phase space. The derivation is based on a Wigner type Fourier transform of the classical phase space probability distribution, which depends on an arbitrary constant α with dimension of action. In order to achieve this goal two requirements are necessary: 1) It is assumed that the classical probability amplitude Ψ(x,t) can be expanded in a complete set of functions Φn(x) defined in the configuration space; 2) the classical phase space distribution W(x,p,t) obeys the Liouville equation and is a real function of the position, the momentum and the time. We show that the constant α appearing in the Fourier transform of the classical phase space distribution, and also in the classical Schrodinger type equation, has its origin in the spectral distribution of the vacuum zero-point radiation, and is identified with the Planck's constant ℏ.
Correct use of the Wigner representation of quantum mechanics, which is realized with joint distributions of quasiprobabilities in phase space, requires the use of certain specific rules and attention to a number of properties which distinguish the quasiprobability from a true probability. This paper is mainly concerned with these problems. In the Wigner representation the quantum Liouville equation appears instead of the Schroedinger equation. The solution may have no physical meaning unless it is subjected to a necessary and sufficient condition which selects an allowed class of distributions which describe quantum-mechanical pure states. This condition contains Planck's constant and imposes, besides the uncertainty relations, severe restrictions on the possible form of the Wigner function. When this condition is satisfied, one can reconstruct the wave function from the Wigner function. In the case of an oscillator, the quantization condition for the energies of the stationary states does not follow from the Liouville equation, but from this supplementary condition. Unlike the true probability density, any Wigner function (except Gaussian ones) of a pure state takes on negative values. Another important peculiarity is that the quasiprobability is not concentrated on certain hypersurfaces, but is smeared out over the entire phase space. These and other features considered in this paper should be kept in mind when using the Wigner representation in quantum-mechanical problems.
We consider the quantum Liouville equation and give a characterization of the solutions which satisfy the Heisenberg uncertainty relation. We analyze three cases. Initially we consider a particular solution of the quantum Liouville equation: the Wigner transform f(x,v,t) of a generic solution ψ(x;t) of the Schrödinger equation. We give a representation of ψ(x, t) by the Hermite functions. We show that the values of the variances of x and v calculated by using the Wigner function f(x,v,t) coincide, respectively, with the variances of position operator and conjugate momentum operator obtained using the wave function ψ(x,t). Then we consider the Fourier transform of the density matrix ρ(z,y,t) = ψ∗(z,t)ψ(y,t). We find again that the variances of x and v obtained by using ρ(z, y,t) are respectively equal to the variances of and calculated in ψ(x,t). Finally we introduce the matrix and we show that a generic square‐integrable function g(x,v,t) can be written as Fourier transform of a density matrix, provided that the matrix is diagonalizable.
The authors contrast the classical and quantum dynamics of SU(1,1) coherent states by the use of a positive-definite quasiprobability distribution, a Q-function, defined over these states. For Hamiltonians that are linear in the SU(1,1) generators, therefore coherence preserving, the quantum and classical Liouville equations are identical. This is illustrated for a degenerate parametric amplifier. For an anharmonic oscillator, the quantum Liouville equation contains additional terms containing second-order derivatives with non-positive-definite coefficients. These terms give rise to the quantum recurrences not seen in the classical evolution of this system. It is pointed out that the quantum equation for the Q-function goes over to the classical equation as the Bargmann index, k, becomes large, in agreement with previous semiclassical considerations.
We propose a Wigner transformation of the density matrix and operators of a system with bounded coordinates and discrete momentum spectra (typically, a particle in a box) and prove that all requirements for such a transformation are satisfied. The phase-space mapping adopted here is employed to compute the quasiprobability distributions associated with the pure states of a quantum rotator and the image of the angular momentum operator. The mapping of the dynamics of these objects, both in the reversible and irreversible frame, is constructed as well, and the conditions for the existence of a classical limit in the form of a Fokker-Planck equation of motion are investigated.
A canonical structure compatible with the action of the Lorentz group can be obtained considering the energy and time as conjugate variables of an extended phase space. Scalar probability waves, describing free relativistic particles, are associated with functional coherent states for an extended Liouville equation. Relativistic action waves are provided by distributions localized in the momentum space, evolving according to the continuity and Hamilton-Jacobi equations. Presuming the existence of minimum space and time intervals, the action distributions take the form of relativistic Wigner functions. The nonrelativistic quantum dynamics is retrieved approximating the time distribution function by a Gaussian wave packet.