of the star product is where the argument of the exponential function can be interpreted as a power series. Additional differential relations allow this to be written in terms of a change in the arguments of f and g: It is also possible to define the ★-product in a convolution integral form, essentially through the Fourier transform: (Thus, e.g., Gaussians compose hyperbolically: or etc.) The energy eigenstate distributions are known as stargenstates, ★-genstates, stargenfunctions, or ★-genfunctions, and the associated energies are known as stargenvalues or ★-genvalues. These are solved, analogously to the time-independent Schrödinger equation, by the ★-genvalue equation, where is the Hamiltonian, a plain phase-space function, most often identical to the classical Hamiltonian. Time evolution The time evolution of the phase space distribution is given by a quantum modification of Liouville flow. This formula results from applying the Wigner transformation to the density matrix version of the quantum Liouville equation, the von Neumann equation. In any representation of the phase space distribution with its associated star product, this is or, for the Wigner function in particular, where {{ , }} is the Moyal bracket, the Wigner transform of the quantum commutator, while { , } is the classical Poisson bracket. This yields a concise illustration of the correspondence principle: this equation manifestly reduces to the classical Liouville equation in the limit ħ → 0. In the quantum extension of the flow, however, the density of points in phase space is not conserved; the probability fluid appears "diffusive" and compressible. The concept of quantum trajectory is therefore a delicate issue here. See the movie for the Morse potential, below, to appreciate the nonlocality of quantum phase flow. N.B. Given the restrictions placed by the uncertainty principle on localization, Niels Bohr vigorously denied the physical existence of such trajectories on the microscopic scale. By means of formal phase-space trajectories, the time evolution problem of the Wigner function can be rigorously solved using the path-integral method and the method of quantum characteristics, although there are severe practical obstacles in both cases. Examples Simple harmonic oscillator The Hamiltonian for the simple harmonic oscillator in one spatial dimension in the Wigner–Weyl representation is The ★-genvalue equation for the static Wigner function then reads Consider, first, the imaginary part of the ★-genvalue equation, This implies that one may write the ★-genstates as functions of a single argument: With this change of variables, it is possible