Phase-space formulation

The phase space formulation of quantum mechanics is a physical theory, equivalent to the usual Schrödinger picture of quantum mechanics, that places position and momentum on equal footing, in phase space. The two key features of the phase space formulation are that the quantum state is described by a quasi-probability distribution (instead of a wave function, state vector, or density matrix) and operator multiplication is replaced by a star product.

The theory was fully detailed by Hip Groenewold in 1946 in his PhD thesis, with significant parallel contributions by Joe Moyal,, each building off earlier ideas by Hermann Weyl and Eugene Wigner.

The chief advantage of the phase space formulation is that it makes quantum mechanics appear as similar to Hamiltonian mechanics as possible by avoiding the operator formalism, thereby "'freeing' the quantization of the 'burden' of the Hilbert space." 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 conceptual ideas underlying the development of quantum mechanics in phase space have branched into mathematical offshoots such as deformation theory (cf. Kontsevich quantization formula) and noncommutative geometry.

Phase space distribution

Main article: Quasi-probability distribution

Main article: Wigner quasi-probability distribution

Main article: Wigner–Weyl transform

The phase space distribution $f(x,p)$ of a quantum state is a quasi-probability distribution. In the phase space formulation, the phase-space distribution may be treated as the fundamental, primitive description of the quantum system, without any reference to wave functions or density matrices.

There are several different ways to represent the distribution, all interrelated. The most noteworthy is the Wigner representation, $W(x,p)$, discovered first. Other representations (in approximately descending order of prevalence in the literature) include the Glauber-Sudarshan P, Husimi Q, Kirkwood-Rihaczek, Mehta, Rivier, and Born-Jordan representations. These alternatives are most useful when the Hamiltonian takes a particular form, such as normal order for the Glauber–Sudarshan P-representation. Since the Wigner representation is the most common, this article will usually stick to it, unless otherwise specified.

By construction, the phase space distribution possesses properties akin to the probability density in a 2n-dimensional phase space. For example, it is real-valued, unlike the generally complex-valued wave function. We can understand the probability of lying within a position interval, for example, by integrating the Wigner function over all momenta:

$\operatorname {P} [a\leq X\leq b]=\int _{a}^{b}\int _{-\infty }^{\infty }W(x,p)\,dp\,dx.$

If ${\hat {A}}(x,p)$ is an operator representing an observable, we may map it into the phase space as $A(x,p)$ using the Wigner transform. Conversely, the operator may be restored via the Weyl transform. We may take the expectation value of the observable with respect to the phase space distribution:

$\langle A\rangle =\int A(x,p)W(x,p)\,dp\,dx.$

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. Moreover, it can in general take negative values even for pure states, with the unique exception of (optionally squeezed) coherent states, in violation of the first axiom. Regions of such negative value are provable to be "small": they cannot extend to compact regions larger than a few ħ, and hence disappear in the classical limit. 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. If the left side of the equation is to be interpreted as an expectation value in the Hilbert space with respect to an operator, then in the context of quantum optics this equation is known as the optical equivalence theorem. For more details on the properties and interpretation of the Wigner function, see its main article.

Star product

Main article: Moyal 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

$f{\stackrel {\leftarrow }{\partial }}_{x}g={\frac {\partial f}{\partial x}}\cdot g$

$f{\stackrel {\rightarrow }{\partial }}_{x}g=f\cdot {\frac {\partial g}{\partial x}}.$

The differential definition of the star product is

$f\star g=f\,\exp {\left({\frac {i\hbar }{2}}({\stackrel {\leftarrow }{\partial }}_{x}{\stackrel {\rightarrow }{\partial }}_{p}-{\stackrel {\leftarrow }{\partial }}_{p}{\stackrel {\rightarrow }{\partial }}_{x})\right)}\,g$

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:

${\begin{aligned}f(x,p)\star g(x,p)&=f\left(x+{\frac {i\hbar }{2}}{\stackrel {\rightarrow }{\partial }}_{p},p-{\frac {i\hbar }{2}}{\stackrel {\rightarrow }{\partial }}_{x}\right)\cdot g(x,p)\\&=f(x,p)\cdot g\left(x-{\frac {i\hbar }{2}}{\stackrel {\leftarrow }{\partial }}_{p},p+{\frac {i\hbar }{2}}{\stackrel {\leftarrow }{\partial }}_{x}\right)\\&=f\left(x+{\frac {i\hbar }{2}}{\stackrel {\rightarrow }{\partial }}_{p},p\right)\cdot g\left(x-{\frac {i\hbar }{2}}{\stackrel {\leftarrow }{\partial }}_{p},p\right)\\&=f\left(x,p-{\frac {i\hbar }{2}}{\stackrel {\rightarrow }{\partial }}_{x}\right)\cdot g\left(x,p+{\frac {i\hbar }{2}}{\stackrel {\leftarrow }{\partial }}_{x}\right).\end{aligned}}$

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

$f(x,p)\star g(x,p)={\frac {1}{\pi ^{2}\hbar ^{2}}}\,\int f(x+x',p+p')\,g(x+x'',p+p'')\,\exp {\left({\frac {2i}{\hbar }}(x'p''-x''p')\right)}\,dx'dp'dx''dp''~.$

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

$\exp \left(-{a}(q^{2}+p^{2})\right)~\star ~\exp \left(-{b}(q^{2}+p^{2})\right)={1 \over 1+\hbar ^{2}ab}\exp \left(-{a+b \over 1+\hbar ^{2}ab}(q^{2}+p^{2})\right),$

or

$\delta (q)~\star ~\delta (p)={2 \over h}\exp \left(2i{qp \over \hbar }\right),$

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 fin, analogously to the time-independent Schrödinger equation, by the ★-genvalue equation

$H\star W=E\cdot W,$

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 can be obtained by 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

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

or, for the Wigner function in particular,

${\frac {\partial W}{\partial t}}=-\{\{W,H\}\}=-{\frac {2}{\hbar }}W\sin \left({{\frac {\hbar }{2}}({\stackrel {\leftarrow }{\partial }}_{x}{\stackrel {\rightarrow }{\partial }}_{p}-{\stackrel {\leftarrow }{\partial }}_{p}{\stackrel {\rightarrow }{\partial }}_{x})}\right)\ H=-\{W,H\}+O(\hbar ^{2}),$

where $\{\{\cdot ,\cdot \}\}$ is the Moyal bracket and $\{\cdot ,\cdot \}$ is the classical Poisson bracket.

This yields a concise interpretation of the correspondence principle, in that this equation manifestly reduces to the classical Liouville theorem in the limit ħ→0. In the quantum extension of the flow, however, the density of points in phase space is not conserved, so the notion of a quantum trajectory is hardly defined: the probability fluid is "diffusive". (The time evolution of the Wigner function may be formally solved for, among other ways, by the method of quantum characteristics.)

Examples

Simple harmonic oscillator

The Wigner quasi-probability distribution for a simple harmonic oscillator with a) n=0, b) n =1, and c) n =5.

Main article: quantum harmonic oscillator

The Hamiltonian for the simple harmonic oscillator in one spatial dimension in the Wigner-Weyl representation is

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

The ★-genvalue equation reads

${\begin{aligned}H\star W&=\left({\frac {1}{2}}m\omega ^{2}x^{2}+{\frac {p^{2}}{2m}}\right)\star W\\&=\left({\frac {1}{2}}m\omega ^{2}\left(x+{\frac {i\hbar }{2}}{\stackrel {\rightarrow }{\partial }}_{p}\right)^{2}+{\frac {1}{2m}}\left(p-{\frac {i\hbar }{2}}{\stackrel {\rightarrow }{\partial }}_{x}\right)^{2}\right)~W\\&=\left({\frac {1}{2}}m\omega ^{2}\left(x^{2}-{\frac {\hbar ^{2}}{4}}{\stackrel {\rightarrow }{\partial }}_{p}^{2}\right)+{\frac {1}{2m}}\left(p^{2}-{\frac {\hbar ^{2}}{4}}{\stackrel {\rightarrow }{\partial }}_{x}^{2}\right)\right)~W\\&\,\,\,\,\,+{\frac {i\hbar }{2}}\left(m\omega ^{2}x{\stackrel {\rightarrow }{\partial }}_{p}-{\frac {p}{m}}{\stackrel {\rightarrow }{\partial }}_{x}\right)~W\\&=E\cdot W.\end{aligned}}$

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

${\frac {\hbar }{2}}\left(m\omega ^{2}x{\stackrel {\rightarrow }{\partial }}_{p}-{\frac {p}{m}}{\stackrel {\rightarrow }{\partial }}_{x}\right)\cdot W=0$

This implies that we may write the ★-genstates as functions of a single argument,

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

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, the solution of which involves the Laguerre polynomials as

$F_{n}(u)={\frac {(-1)^{2}}{\pi \hbar }}L_{n}\left({\frac {u}{\hbar \omega }}\right)e^{-u/\hbar \omega }~,$

introduced by Groenewold in his paper, with associated ★-genvalues

$E_{n}=\hbar \omega \left(n+{\frac {1}{2}}\right)~.$

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 is

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

where α is a parameter describing the initial width of the Gaussian and $\tau =m/\alpha ^{2}\hbar$. 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. On the other hand, the position and momentum should become increasingly correlated as the state evolves because portions of the distribution far from the origin in position require a large amount of momentum to be reached. Indeed, it is possible to show that the kinetic energy of the particle becomes asymptotically radial only, in agreement with the usual quantum mechanical notion of the nonzero ground state angular momentum's orientation independence:

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

$K_{ang}={\frac {\alpha ^{2}\hbar ^{2}}{2m}}{\frac {1}{1+(t/\tau )^{2}}}$