Phase-space formulation

Star product

The fundamental noncommutative binary operator in the phase-space formulation that replaces the standard operator multiplication is the star product, represented by the symbol ★. Each representation of the phase-space distribution has a different characteristic star product. For concreteness, we restrict this discussion to the star product relevant to the Wigner–Weyl representation.

For notational convenience, we introduce the notion of left and right derivatives. For a pair of functions f and g, the left and right derivatives are defined as

${\displaystyle \begin{array}{cc}f{\overleftarrow{\mathrm{\partial }}}_{x}g& =\left(\frac{\mathrm{\partial }f}{\mathrm{\partial }x}\right)g,\\ f{\overrightarrow{\mathrm{\partial }}}_{x}g& =f\left(\frac{\mathrm{\partial }g}{\mathrm{\partial }x}\right).\end{array}}$

The differential definition of the star product is

${\displaystyle f\star g=f\exp \left(\frac{i\hslash }{2}\left({\overleftarrow{\mathrm{\partial }}}_x{\overrightarrow{\mathrm{\partial }}}_p-{\overleftarrow{\mathrm{\partial }}}_p{\overrightarrow{\mathrm{\partial }}}_x\right)\right)g,}$

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:

${\displaystyle \begin{array}{cc}(f\star g)(x,p)& =f\left(x+\frac{i\hslash }{2}{\overrightarrow{\mathrm{\partial }}}_p,p-\frac{i\hslash }{2}{\overrightarrow{\mathrm{\partial }}}_x\right)\cdot g(x,p)\\ & =f(x,p)\cdot g\left(x-\frac{i\hslash }{2}{\overleftarrow{\mathrm{\partial }}}_p,p+\frac{i\hslash }{2}{\overleftarrow{\mathrm{\partial }}}_x\right)\\ & =f\left(x+\frac{i\hslash }{2}{\overrightarrow{\mathrm{\partial }}}_p,p\right)\cdot g\left(x-\frac{i\hslash }{2}{\overleftarrow{\mathrm{\partial }}}_p,p\right)\\ & =f\left(x,p-\frac{i\hslash }{2}{\overrightarrow{\mathrm{\partial }}}_x\right)\cdot g\left(x,p+\frac{i\hslash }{2}{\overleftarrow{\mathrm{\partial }}}_x\right).\end{array}}$

It is also possible to define the ★-product in a convolution integral form, essentially through the Fourier transform:

${\displaystyle (f\star g)(x,p)=\frac{1}{{\pi }^2{\hslash }^2}\int f(x+x^{\prime },p+p^{\prime })g(x+x^{″},p+p^{″})\exp \left(\frac{2i}{\hslash }(x^{\prime }{p}^{″}-x^{″}{p}^{\prime })\right)d{x}^{\prime }d{p}^{\prime }d{x}^{″}d{p}^{″}.}$

(Thus, e.g., Gaussians compose hyperbolically:

${\displaystyle \exp (-a(x^2+p^2))\star \exp (-b(x^2+p^2))=\frac{1}{1+{\hslash }^{2}ab}\exp \left(-\frac{a+b}{1+{\hslash }^{2}ab}(x^2+p^2)\right),}$

or

${\displaystyle \delta (x)\star \delta (p)=\frac{2}{h}\exp \left(2i\frac{xp}{\hslash }\right),}$

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,

${\displaystyle H\star W=E\cdot W,}$

where H 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

${\displaystyle \frac{\mathrm{\partial }f}{\mathrm{\partial }t}=-\frac{1}{i\hslash }\left(f\star H-H\star f\right),}$

or, for the Wigner function in particular,

${\displaystyle \frac{\mathrm{\partial }W}{\mathrm{\partial }t}=-\{\{W,H\}\}=-\frac{2}{\hslash }W\sin \left(\frac{\hslash }{2}({\overleftarrow{\mathrm{\partial }}}_x{\overrightarrow{\mathrm{\partial }}}_p-{\overleftarrow{\mathrm{\partial }}}_p{\overrightarrow{\mathrm{\partial }}}_x)\right)H=-\{W,H\}+O({\hslash }^2),}$

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

${\displaystyle H=\frac{1}{2}m{\omega }^2{x}^2+\frac{p^2}{2m}.}$

The ★-genvalue equation for the static Wigner function then reads

${\displaystyle \begin{array}{cc}H\star W& =\left(\frac{1}{2}m{\omega }^2{x}^2+\frac{p^2}{2m}\right)\star W\\ & =\left(\frac{1}{2}m{\omega }^2{\left(x+\frac{i\hslash }{2}{\overrightarrow{\mathrm{\partial }}}_p\right)}^2+\frac{1}{2m}{\left(p-\frac{i\hslash }{2}{\overrightarrow{\mathrm{\partial }}}_x\right)}^2\right)W\\ & =\left(\frac{1}{2}m{\omega }^2\left(x^2-\frac{{\hslash }^2}{4}{\overrightarrow{\mathrm{\partial }}}_p^2\right)+\frac{1}{2m}\left(p^2-\frac{{\hslash }^2}{4}{\overrightarrow{\mathrm{\partial }}}_x^2\right)\right)W\\ & +\frac{i\hslash }{2}\left(m{\omega }^{2}x{\overrightarrow{\mathrm{\partial }}}_p-\frac{p}{m}{\overrightarrow{\mathrm{\partial }}}_x\right)W\\ & =E\cdot W.\end{array}}$

Consider, first, the imaginary part of the ★-genvalue equation,

${\displaystyle \frac{\hslash }{2}\left(m{\omega }^{2}x{\overrightarrow{\mathrm{\partial }}}_p-\frac{p}{m}{\overrightarrow{\mathrm{\partial }}}_x\right)\cdot W=0}$

This implies that one may write the ★-genstates as functions of a single argument:

${\displaystyle W(x,p)=F\left(\frac{1}{2}m{\omega }^2{x}^2+\frac{p^2}{2m}\right)\equiv F(u).}$

With this change of variables, it is possible to write the real part of the ★-genvalue equation in the form of a modified Laguerre equation (not Hermite's equation!), the solution of which involves the Laguerre polynomials as

${\displaystyle F_n(u)=\frac{(-1{)}^n}{\pi \hslash }{L}_n\left(4\frac{u}{\hslash \omega }\right)e^{-2u/\hslash \omega },}$

introduced by Groenewold, with associated ★-genvalues

${\displaystyle E_n=\hslash \omega \left(n+\frac{1}{2}\right).}$

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,

${\displaystyle W(x,p;t)=W\left(x\cos \omega t-\frac{p}{m\omega }\sin \omega t,p\cos \omega t+m\omega x\sin \omega t;0\right).}$

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.

The eigenfunctions can also be characterized by being rotationally symmetric (thus time-invariant) pure states. That is, they are functions of form ${\displaystyle W(x,p)=f(\sqrt{x^2+p^2})}$ that satisfy ${\displaystyle W\star W=(2\pi \hslash {)}^{-1}W}$.

Free particle angular momentum

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

${\displaystyle W(\mathbf{x},\mathbf{p};t)=\frac{1}{(\pi \hslash {)}^3}\exp \left(-{\alpha }^2{r}^2-\frac{p^2}{{\alpha }^2{\hslash }^2}\left(1+{\left(\frac{t}{\tau }\right)}^2\right)+\frac{2t}{\hslash \tau }\mathbf{x}\cdot \mathbf{p}\right),}$

where α is a parameter describing the initial width of the Gaussian, and τ = m/α²ħ.

Initially, the position and momenta are uncorrelated. Thus, in 3 dimensions, we expect the position and momentum vectors to be twice as likely to be perpendicular to each other as parallel.

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,

${\displaystyle W⟶\frac{1}{(\pi \hslash {)}^3}\exp \left[-{\alpha }^2{\left(\mathbf{x}-\frac{\mathbf{p}t}{m}\right)}^2\right].}$

(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:

${\displaystyle K_{\text{rad}}=\frac{{\alpha }^2{\hslash }^2}{2m}\left(\frac{3}{2}-\frac{1}{1+(t/\tau {)}^2}\right)}$

${\displaystyle K_{\text{ang}}=\frac{{\alpha }^2{\hslash }^2}{2m}\frac{1}{1+(t/\tau {)}^2}.}$

Morse potential

The Morse potential is used to approximate the vibrational structure of a diatomic molecule.

Quantum tunneling

Tunneling is a hallmark quantum effect where a quantum particle, not having sufficient energy to fly above, still goes through a barrier. This effect does not exist in classical mechanics.

Quartic potential

Schrödinger cat state