Wigner Distribution

The Wigner distribution of a square-integrable complex function is the time-frequency function

(1)

Its marginal integrals, interpreted weakly when necessary, recover the position and frequency energy densities,

			(2)
			(3)

where is the Fourier transform with kernel . If , both marginals are probability density functions. Under this normalization, the real-valued Wigner distribution is a quasiprobability distribution, since it need not be nonnegative and therefore need not be a joint probability density function.

For a normalized wave function , the momentum convention used in quantum mechanics is

(4)

where is momentum and is the reduced Planck constant (Case 2008, p. 938). The factor ensures that the integral of over phase space is 1.

For the quantum harmonic oscillator with Hamiltonian

(5)

where and , the normalized stationary wave functions are scaled Hermite functions. Their Wigner functions are

(6)

where , 1, 2, , is a Laguerre polynomial, , and (Curtright et al. 1998). The ground state gives a positive Gaussian function, whereas the first excited state has . A coherent state, obtained by displacing the ground state to position and momentum , has the Gaussian function (Case 2008, pp. 940-942)

(7)

For a time-independent Hamiltonian with a polynomial of degree at most 2, the exact quantum evolution of the Wigner function obeys the classical transport equation

(8)

where is the Poisson bracket. The quantum corrections vanish because they contain derivatives of of order at least 3 (Case 2008, pp. 941-942). If is the flow determined by Hamilton's equations, then

(9)

For the harmonic oscillator this is a rigid clockwise rotation in the scaled coordinates with angular speed , for every initial state. In particular, a coherent state's Gaussian function retains its shape while its center follows the classical orbit.