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Thephase-space formulation is a formulation ofquantum mechanics that places theposition andmomentum variables on equal footing inphase space. The two key features of the phase-space formulation are that the quantum state is described by aquasiprobability distribution (instead of awave function,state vector, ordensity matrix) and operator multiplication is replaced by astar product.

The theory was fully developed byHilbrand Groenewold in 1946 in his PhD thesis,^[1] and independently byJoe Moyal,^[2] each building on earlier ideas byHermann Weyl^[3] andEugene Wigner.^[4]

In contrast to the phase-space formulation, theSchrödinger picture uses the positionor momentum representations (see alsoposition and momentum space).

The chief advantage of the phase-space formulation is that it makes quantum mechanics appear as similar toHamiltonian mechanics as possible by avoiding the operator formalism, thereby "'freeing' the quantization of the 'burden' of theHilbert space".^[5] This formulation is statistical in nature and offers logical connections between quantum mechanics and classicalstatistical mechanics, enabling a natural comparison between the two (seeclassical limit). Quantum mechanics in phase space is often favored in certainquantum optics applications (seeoptical phase space), or in the study ofdecoherence and a range of specialized technical problems, though otherwise the formalism is less commonly employed in practical situations.^[6]

The conceptual ideas underlying the development of quantum mechanics in phase space have branched into mathematical offshoots such as Kontsevich's deformation-quantization (seeKontsevich quantization formula) andnoncommutative geometry.^{[citation needed]}

The phase-space distributionf(x, p) of a quantum state is a quasiprobability 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.^[7]

There are several different ways to represent the distribution, all interrelated.^[8][9] The most noteworthy is theWigner representation,W(x, p), discovered first.^[4] Other representations (in approximately descending order of prevalence in the literature) include theGlauber–Sudarshan P,^[10][11]Husimi Q,^[12] Kirkwood–Rihaczek, Mehta, Rivier, and Born–Jordan representations.^[13][14] These alternatives are most useful when the Hamiltonian takes a particular form, such asnormal 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.

The phase-space distribution possesses properties akin to the probability density in a 2n-dimensional phase space. For example, it isreal-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 and over the position interval:

${\displaystyle \mathrm{P}[a\le X\le b]={\int }_a^b{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}W(x,p)dpdx.}$

IfÂ(x, p) is an operator representing an observable, it may be mapped to phase space asA(x,p) through theWigner transform. Conversely, this operator may be recovered by theWeyl transform.

The expectation value of the observable with respect to the phase-space distribution is^[2][15]

${\displaystyle ⟨\hat{A}⟩=\int A(x,p)W(x,p)dpdx.}$

A point of caution, however: despite the similarity in appearance,W(x, p) is not a genuinejoint probability distribution, because regions under it do not represent mutually exclusive states, as required in thethird axiom of probability theory. Moreover, it can, in general, takenegative values even for pure states, with the unique exception of (optionallysqueezed)coherent states, in violation of thefirst axiom.

Regions of such negative value are provable to be "small": they cannot extend to compact regions larger than a fewħ, and hence disappear in theclassical limit. They are shielded by theuncertainty principle, which does not allow precise localization within phase-space regions smaller thanħ, 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 ofquantum optics this equation is known as theoptical equivalence theorem. (For details on the properties and interpretation of the Wigner function, see itsmain article.)

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 theSegal–Bargmann transform. To be compatible with the uncertainty principle, the phase-space wave function cannot be an arbitrary function, or else it could be localized into an arbitrarily small region of phase space. Rather, the Segal–Bargmann transform is aholomorphic function of${\displaystyle x+ip}$. There is a quasiprobability density associated to the phase-space wave function; it is theHusimi Q representation of the position wave function.

The fundamental noncommutative binary operator in the phase-space formulation that replaces the standard operator multiplication is thestar product, represented by the symbol★.^[1] Each representation of the phase-space distribution has adifferent 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 ofleft and right derivatives. For a pair of functionsf andg, the left and right derivatives are defined as

Thedifferential 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 apower series.Additional differential relations allow this to be written in terms of a change in the arguments off andg:

It is also possible to define the★-product in a convolution integral form,^[16] essentially through theFourier 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.,^[7] Gaussians composehyperbolically:

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

or

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

etc.)

The energyeigenstate distributions are known asstargenstates,★-genstates,stargenfunctions, or★-genfunctions, and the associated energies are known asstargenvalues or★-genvalues. These are solved, analogously to the time-independentSchrödinger equation, by the★-genvalue equation,^[17][18]

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

whereH is the Hamiltonian, a plain phase-space function, most often identical to the classical Hamiltonian.

Thetime evolution of the phase space distribution is given by a quantum modification ofLiouville flow.^[2][9][19] This formula results from applying theWigner transformation to the density matrix version of thequantum Liouville equation,thevon 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 theMoyal bracket, the Wigner transform of the quantum commutator, while { , } is the classicalPoisson bracket.^[2]

This yields a concise illustration of thecorrespondence principle: this equation manifestly reduces to the classical Liouville equation in the limitħ → 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.^[2] The concept of quantum trajectory is therefore a delicate issue here.^[20] 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^[21] and themethod of quantum characteristics,^[22] although there are severe practical obstacles in both cases.

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 thestatic Wigner function then reads

Time evolution of combined ground and 1st excited state Wigner function for the simple harmonic oscillator. Note the rigid motion in phase space corresponding to the conventional oscillations in coordinate space.

Wigner function for the harmonic oscillator ground state, displaced from the origin of phase space, i.e., acoherent state. Note the rigid rotation, identical to classical motion: this is a special feature of the SHO, illustrating thecorrespondence principle. From the general pedagogy web-site.^[23]

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 (notHermite's equation!), the solution of which involves theLaguerre polynomials as^[18]

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

introduced by Groenewold,^[1] 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 initialW(x, p; t = 0) =F(u) evolves by the above evolution equation driven by the oscillator Hamiltonian given, by simplyrigidly rotating in phase space,^[1]

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

Typically, a "bump" (or coherent state) of energyE ≫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 att = 0) of such objects, a continuous "palisade", yields a time-independent configuration similar to the above static★-genstatesF(u), an intuitive visualization of theclassical limit for large-action systems.^[6]

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${\displaystyle W\star W=(2\pi \hslash {)}^{-1}W}$.

Suppose a particle is initially in a minimally uncertainGaussian 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

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

whereα is a parameter describing the initial width of the Gaussian, andτ =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 freewave 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]

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

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

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

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.

• Hamiltonian mechanics
• Symplectic geometry
• Mathematical quantization
• Foundational quantum physics

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