Phase-space formulation

The theory was fully developed by Hilbrand Groenewold in 1946 in his PhD thesis,[1] and independently by Joe Moyal,[2] each building on earlier ideas by Hermann Weyl[3] and Eugene Wigner.

Quantum mechanics in phase space is often favored in certain quantum optics applications (see optical phase space), or in the study of decoherence and a range of specialized technical problems, though otherwise the formalism is less commonly employed in practical situations.

The expectation value of the observable with respect to the phase-space distribution is[2][15] A point of caution, however: despite the similarity in appearance, W(x, p) is not a genuine joint probability distribution, because regions under it do not represent mutually exclusive states, as required in the third axiom of probability theory.

They are shielded by the uncertainty principle, which does not allow precise localization within phase-space regions smaller than ħ, and thus renders such "negative probabilities" less paradoxical.

An alternative phase-space approach to quantum mechanics seeks to define a wave function (not just a quasiprobability density) on phase space, typically by means of the Segal–Bargmann transform.

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,[16] essentially through the Fourier transform: (Thus, e.g.,[7] Gaussians compose hyperbolically: or etc.)

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.

Given the restrictions placed by the uncertainty principle on localization, Niels Bohr vigorously denied the physical existence of such trajectories on the microscopic scale.

), the solution of which involves the Laguerre polynomials as[18] introduced by Groenewold,[1] with associated ★-genvalues For the harmonic oscillator, the time evolution of an arbitrary Wigner distribution is simple.

An initial W(x, p; t = 0) = F(u) evolves by the above evolution equation driven by the oscillator Hamiltonian given, by simply rigidly rotating in phase space,[1] Typically, a "bump" (or coherent state) of energy E ≫ ħω can represent a macroscopic quantity and appear like a classical object rotating uniformly in phase space, a plain mechanical oscillator (see the animated figures).

Integrating over all phases (starting positions at t = 0) of such objects, a continuous "palisade", yields a time-independent configuration similar to the above static ★-genstates F(u), an intuitive visualization of the classical limit for large-action systems.

Suppose a particle is initially in a minimally uncertain Gaussian state, with the expectation values of position and momentum both centered at the origin in phase space.

The Wigner function for such a state propagating freely is where α is a parameter describing the initial width of the Gaussian, and τ = m/α2ħ.

However, the position and momentum become increasingly correlated as the state evolves, because portions of the distribution farther from the origin in position require a larger momentum to be reached: asymptotically, (This relative "squeezing" reflects the spreading of the free wave packet in coordinate space.)

Indeed, it is possible to show that the kinetic energy of the particle becomes asymptotically radial only, in agreement with the standard quantum-mechanical notion of the ground-state nonzero angular momentum specifying orientation independence:[24] The Morse potential is used to approximate the vibrational structure of a diatomic molecule.

Wigner function for number states a) n = 0, b) n = 1, and c) n = 19. Marginal distributions for x and p are recovered by integrating over p and x respectively.
The Wigner function of a simple harmonic oscillator at different levels of excitations. The are rescaled by in order to show that the Wigner function oscillates within that radius, and decays rapidly outside of that radius.
The Wigner function time-evolution of the Morse potential U ( x ) = 20(1 − e −0.16 x ) 2 in atomic units (a.u.). The solid lines represent level set of the Hamiltonian H ( x , p ) = p 2 /2 + U ( x ) .
The Wigner function for tunneling through the potential barrier U ( x ) = 8 e −0.25 x 2 in atomic units (a.u.). The solid lines represent the level set of the Hamiltonian H ( x , p ) = p 2 /2 + U ( x ).
The Wigner function time evolution for the potential U ( x ) = 0.1 x 4 in atomic units (a.u.). The solid lines represent the level set of the Hamiltonian H ( x , p ) = p 2 /2 + U ( x ).
Wigner function of two interfering coherent states evolving through the SHO Hamiltonian. The corresponding momentum and coordinate projections are plotted to the right and under the phase space plot.