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{{Quantum mechanics|cTopic=Formulations}}

The '''phase space formulation of [[quantum mechanics]]''' places the [[Position (vector)|position]] ''and'' [[momentum]] variables on equal footing, in [[phase space]]. In contrast, the [[Schrödinger picture]] uses the position ''or'' momentum representations (see also [[position and momentum space]]). The two key features of the phase space formulation are that the quantum state is described by a ''[[quasiprobability distribution]]'' (instead of a [[wave function]], [[state vector]], or [[density matrix]]) and operator multiplication is replaced by a ''[[Moyal product|star product]]''.

The theory was fully detailed by [[Hilbrand J. Groenewold|Hip Groenewold]] in 1946 in his PhD thesis,<ref name="Groenewold1946">H.J. Groenewold, "On the Principles of elementary quantum mechanics", ''Physica'','''12''' (1946) pp. 405-460. {{doi|10.1016/S0031-8914(46)80059-4}}</ref> with significant parallel contributions by [[José Enrique Moyal|Joe Moyal]],<ref name="Moyal1949">J.E. Moyal, "Quantum mechanics as a statistical theory", ''Proceedings of the Cambridge Philosophical Society'', '''45''' (1949) pp. 99–124. {{doi|10.1017/S0305004100000487}}</ref> each building off earlier ideas by [[Hermann Weyl]]<ref name="Weyl1927">H.Weyl, "Quantenmechanik und Gruppentheorie", ''Zeitschrift für Physik'', '''46''' (1927) pp. 1–46, {{doi|10.1007/BF02055756}}</ref> and [[Eugene Wigner]].<ref name="Wigner1932">E.P. Wigner, "On the quantum correction for thermodynamic equilibrium", ''Phys. Rev.'' '''40''' (June 1932) 749–759. {{doi|10.1103/PhysRev.40.749}}</ref>

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]]."<ref>S. T. Ali, M. Engliš, "Quantization Methods: A Guide for Physicists and Analysts." ''Rev.Math.Phys.'', '''17''' (2005) pp. 391-490. {{doi|10.1142/S0129055X05002376}}</ref> This formulation is statistical in nature and offers logical connections between quantum mechanics and classical statistical mechanics, enabling a natural comparison between the two (cf. [[classical limit]]). 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.<ref name="cz2012">{{cite doi|10.1142/S2251158X12000069|noedit}}</ref>

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|Quasiprobability distribution|Wigner quasiprobability distribution|Wigner–Weyl transform}}
The phase space distribution {{math|''f''(''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.<ref name="ZFC2005">[[Cosmas Zachos|C. Zachos]], [[David Fairlie|D. Fairlie]], and [[Thomas Curtright|T. Curtright]], "Quantum Mechanics in Phase Space" ( World Scientific, Singapore, 2005) ISBN 978-981-238-384-6 .</ref>

There are several different ways to represent the distribution, all interrelated.<ref>{{cite doi|10.1063/1.1931206|noedit}}</ref><ref name="AgarwalWolf1970b">G. S. Agarwal and E. Wolf "Calculus for Functions of Noncommuting Operators and General Phase-Space Methods in Quantum Mechanics. II. Quantum Mechanics in Phase Space", ''Phys. Rev. D'','''2''' (1970) pp. 2187–2205. {{doi|10.1103/PhysRevD.2.2187}}</ref> The most noteworthy is the [[Wigner quasiprobability distribution|Wigner representation]], {{math|''W''(''x'',''p'')}}, discovered first.<ref name="Wigner1932" /> Other representations (in approximately descending order of prevalence in the literature) include the [[Glauber–Sudarshan P-representation|Glauber-Sudarshan P]],<ref>E. C. G. Sudarshan "Equivalence of Semiclassical and Quantum Mechanical Descriptions of Statistical Light Beams", ''Phys. Rev. Lett.'','''10''' (1963) pp. 277–279. {{doi|10.1103/PhysRevLett.10.277}}</ref><ref>R. J. Glauber "Coherent and Incoherent States of the Radiation Field", ''Phys. Rev.'','''131''' (1963) pp. 2766–2788. {{doi|10.1103/PhysRev.131.2766}}</ref> [[Husimi Q representation|Husimi Q]],<ref>Kôdi Husimi (1940). "Some Formal Properties of the Density Matrix", ''Proc. Phys. Math. Soc. Jpn.'' '''22''': 264-314 .</ref> Kirkwood-Rihaczek, Mehta, Rivier, and Born-Jordan representations.<ref>G. S. Agarwal and E. Wolf "Calculus for Functions of Noncommuting Operators and General Phase-Space Methods in Quantum Mechanics. I. Mapping Theorems and Ordering of Functions of Noncommuting Operators", ''Phys. Rev. D'','''2''' (1970) pp. 2161–2186. {{doi|10.1103/PhysRevD.2.2161}}</ref><ref>K. E. Cahill and R. J. Glauber "Ordered Expansions in Boson Amplitude Operators", ''Phys. Rev.'','''177''' (1969) pp. 1857–1881. {{doi|10.1103/PhysRev.177.1857}}; K. E. Cahill and R. J. Glauber "Density Operators and Quasiprobability Distributions", ''Phys. Rev.'','''177''' (1969) pp. 1882–1902. {{doi|10.1103/PhysRev.177.1882}}</ref> 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.

The phase space distribution possesses properties akin to the probability density in a 2''n''-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 and over the position interval:

:<math>\operatorname P [a \leq X \leq b] = \int_a^b \int_{-\infty}^{\infty} W(x, p) \, dp \, dx.</math>

If {{math|''Â''(''x'',''p'')}} is an operator representing an observable, it may be mapped to phase space as {{math|''A''(''x'', ''p'')}} through the ''[[Wigner–Weyl transform|Wigner transform]]''. Conversely, this operator may be recovered via the ''[[Wigner–Weyl transform|Weyl transform]]''.

The expectation value of the observable with respect to the phase space distribution is<ref name="Moyal1949" /><ref>M. Lax "Quantum Noise. XI. Multitime Correspondence between Quantum and Classical Stochastic Processes", ''Phys. Rev.'','''172''' (1968) pp. 350–361. {{doi|10.1103/PhysRev.172.350}}</ref>

:<math>\langle \hat{A} \rangle = \int A(x, p) W(x, p) \, dp \, dx.</math>

A point of caution, however: despite the similarity in appearance, {{math|''W''(''x'',''p'')}} is not a genuine [[joint probability distribution]], because regions under it do not represent mutually exclusive states, as required in the [[probability axioms#Third axiom|third axiom of probability theory]]. Moreover, it can, in general, take ''[[negative probability|negative values]]'' even for pure states, with the unique exception of (optionally [[squeezed coherent state|squeezed]]) [[coherent states]], in violation of the [[probability axioms#First axiom|first axiom]].

Regions of such negative value are provable to be "small": they cannot extend to compact regions larger than a few {{mvar|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 {{mvar|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 details on the properties and interpretation of the Wigner function, see its [[Wigner quasiprobability distribution|main article]].)

==Star product==
{{main|Moyal product}}
The fundamental noncommutative binary operator in the phase space formulation that replaces the standard operator multiplication is the [[Moyal product|star product]], represented by the symbol ★.<ref name="Groenewold1946" /> 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 derivative]]s. For a pair of functions ''f'' and ''g'', the left and right derivatives are defined as
:<math>f \stackrel{\leftarrow }{\partial }_x g = \frac{\partial f}{\partial x} \cdot g</math>
:<math>f \stackrel{\rightarrow }{\partial }_x g = f \cdot \frac{\partial g}{\partial x}.</math>
The differential definition of the star product is
:<math>f \star g = f \, \exp{\left( \tfrac{i \hbar}{2} (\stackrel{\leftarrow }{\partial }_x
\stackrel{\rightarrow }{\partial }_{p}-\stackrel{\leftarrow }{\partial }_{p}\stackrel{\rightarrow }{\partial }_{x}) \right)} \, g</math>
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'':
:<math>\begin{align}
f(x,p) \star g(x,p) &= f\left(x+\tfrac{i \hbar}{2} \stackrel{\rightarrow }{\partial }_{p} , p - \tfrac{i \hbar}{2} \stackrel{\rightarrow }{\partial }_{x}\right) \cdot g(x,p) \\
&= f(x,p) \cdot g\left(x -\tfrac{i \hbar}{2} \stackrel{\leftarrow }{\partial }_{p} , p + \tfrac{i \hbar}{2} \stackrel{\leftarrow }{\partial }_{x}\right) \\
&= f\left(x +\tfrac{i \hbar}{2} \stackrel{\rightarrow }{\partial }_{p} , p\right) \cdot g\left(x -\tfrac{i \hbar}{2} \stackrel{\leftarrow }{\partial }_{p} , p\right) \\
&= f\left(x , p - \tfrac{i \hbar}{2} \stackrel{\rightarrow }{\partial }_{x}\right) \cdot g\left(x , p + \tfrac{i \hbar}{2} \stackrel{\leftarrow }{\partial }_{x}\right).
\end{align}</math>
It is also possible to define the ★-product in a convolution integral form,<ref>G. Baker, “Formulation of Quantum Mechanics Based on the Quasi-probability Distribution Induced on Phase Space,” ''Physical Review'', '''109''' (1958) pp.2198–2206. {{doi|10.1103/PhysRev.109.2198}}</ref> essentially through the [[Fourier transform]]:
:<math>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(\tfrac{2i}{\hbar}(x'p''-x''p')\right)} \, dx' dp' dx'' dp'' ~.</math>
(Thus, e.g.,<ref name="ZFC2005" /> Gaussians compose [[Hyperbolic_function#Comparison_with_circular_trigonometric_functions|hyperbolically]],
: <math>
\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 ) ,
</math>
or
: <math>
\delta (q) ~ \star ~ \delta(p) = {2\over h}
\exp \left (2i{qp\over\hbar}\right ) ,
</math>
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,<ref>{{cite doi|10.1017/S0305004100038068|noedit}}</ref><ref name="CFZ1998">{{cite doi|10.1103/PhysRevD.58.025002|noedit}}</ref>
:<math>H \star W = E \cdot W,</math>
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's theorem (Hamiltonian)|Liouville flow]].<ref name="Moyal1949"/><ref name="AgarwalWolf1970b" /><ref>C. L. Mehta "Phase‐Space Formulation of the Dynamics of Canonical Variables", ''J. Math. Phys.'','''5''' (1964) pp. 677–686. {{doi|10.1063/1.1704163}}</ref> 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

:<math>\frac{\partial f}{\partial t} = - \frac{1}{i \hbar} \left(f \star H - H \star f \right),</math>

or, for the Wigner function in particular,
:<math>\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),</math>
where <nowiki>{{ , }}</nowiki> is the [[Moyal bracket]], the Wigner transform of the quantum commutator, while <nowiki>{ , }</nowiki> is the classical [[Poisson bracket]].<ref name="Moyal1949" />

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.<ref name="Moyal1949"/>
The concept of quantum trajectory is therefore a delicate issue here. (Given the restrictions placed by the uncertainty principle on localization, [[Niels Bohr]] vigorously denied the existence of physical 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<ref>M. S. Marinov, [http://www.sciencedirect.com/science/article/pii/0375960191903529 ''A new type of phase-space path integral''], Phys. Lett. A 153, 5 (1991).</ref> and the [[method of quantum characteristics]],<ref>M. I. Krivoruchenko, A. Faessler, '' Weyl's symbols of Heisenberg operators of canonical coordinates and momenta as quantum characteristics'', J. Math. Phys. 48, 052107 (2007) {{doi|10.1063/1.2735816}}.</ref> although there are practical obstacles in both cases.

==Examples==

===Simple harmonic oscillator===
{{main|quantum harmonic oscillator}}
[[File:Wigner functions.jpg|right|thumb|The Wigner quasiprobability distribution {{math|''Fn''(''u'')}} for the simple harmonic oscillator with a) ''n'' = 0, b) ''n'' = 1, and c) ''n'' = 5.]]
The Hamiltonian for the simple harmonic oscillator in one spatial dimension in the Wigner-Weyl representation is
:::<math>H=\frac{1}{2}m \omega^2 x^2 + \frac{p^2}{2m}.</math>

The ★-genvalue equation for the ''static'' Wigner function then reads
:<math>\begin{align}
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{align}</math>

{{multiple image
| position = center
| footer = (Click to animate.)
| width1 = 230
| image1 = TwoStateWF.gif
| caption1 = 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. | width2 = 125
| width2 = 230
| image2 = DisplacedGaussianWF.gif
| caption2 = Wigner function for the harmonic oscillator ground state, displaced from the origin of phase space, i.e., a [[coherent state]]. Note the rigid rotation, identical to classical motion: this is a special feature of the SHO, illustrating the [[correspondence principle]]. From the general pedagogy web-site.<ref>Curtright, T.L. [http://server.physics.miami.edu/~curtright/TimeDependentWignerFunctions.html Time-dependent Wigner Functions]</ref>
}}

Consider first the imaginary part of the ★-genvalue equation.
:::<math>\frac{\hbar}{2} \left(m \omega^2 x \stackrel{\rightarrow }{\partial }_{p} - \frac{p}{m} \stackrel{\rightarrow }{\partial }_{x}\right) \cdot W=0</math>
This implies that one may write the ★-genstates as functions of a single argument,
::<math>W(x,p)=F\left(\frac{1}{2} m \omega^2 x^2 + \frac{p^2}{2m}\right)\equiv F(u).</math>
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 equation (not [[Hermite polynomials]]), the solution of which involves the [[Laguerre polynomials]] as<ref name="CFZ1998" />
:<math>F_n(u) = \frac{(-1)^n}{\pi \hbar} L_n\left(4\frac{u}{\hbar \omega}\right) e^{-2u/\hbar \omega} ~,</math>
introduced by Groenewold in his paper,<ref name="Groenewold1946" /> with associated ★-genvalues
:::<math>E_n = \hbar \omega \left(n+\frac{1}{2}\right)~.</math>

For the harmonic oscillator, the time evolution of an arbitrary Wigner distribution is simple. An initial {{math|''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'',<ref name="Groenewold1946" />

:<math>W(x,p;t)=W(m\omega x \cos \omega t - p \sin \omega t , ~ p \cos \omega t + \omega m x \sin \omega t ;0) ~.</math>

Typically, a "bump" (or coherent state) of energy {{math|''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.<ref name="cz2012"/>

===Free particle angular momentum===
Suppose a particle is initially in a minimally uncertain [[Wave_packet#Gaussian_wavepackets_in_quantum_mechanics|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
:<math>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)} ~,</math>
where ''α'' is a parameter describing the initial width of the Gaussian, and {{math|''τ'' {{=}} ''m''/''α''2''ħ''}}.

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,
:<math>W \longrightarrow \frac{1}{(\pi\hbar)^3}\exp\left[-\alpha^2\left(\mathbf{x}-\frac{\mathbf{p}t}{m}\right)^2\right]\,.</math>

(This relative [[Squeezed coherent state|"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:<ref>J. P. Dahl and [[Wolfgang P. Schleich|W. P. Schleich]], "Concepts of radial and angular kinetic energies", ''Phys. Rev. A'','''65''' (2002). {{doi|10.1103/PhysRevA.65.022109}}</ref>
:<math>K_{rad}=\frac{\alpha^2 \hbar^2}{2m}\left(\frac{3}{2} - \frac{1}{1+(t/\tau)^2}\right)</math>
:<math>K_{ang}=\frac{\alpha^2 \hbar^2}{2m}\frac{1}{1+(t/\tau)^2}~.</math>

===Morse potential===

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

[[File:Wigner function propagation for morse potential.ogv|400px|center|thumb|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 mechanics|Hamiltonian]] ''H''(''x'', ''p'') = ''p''2/2 + ''U''(''x'')''.]]

===Quantum tunneling===

[[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.

[[File:Wigner function for tunnelling.ogv|400px|center|thumb|The [[Wigner function]] for [[Quantum tunneling|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 mechanics|Hamiltonian]] ''H''(''x'', ''p'') = ''p''2/2 + ''U''(''x'').]]

===Quartic potential===

[[File:Wigner function for quartic potential.ogv|400px|center|thumb|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 mechanics|Hamiltonian]] ''H''(''x'', ''p'') = ''p''2/2 + ''U''(''x'').]]

==References==
<references/>

[[Category:Quantum mechanics]]
[[Category:Hamiltonian mechanics]]
[[Category:Symplectic geometry]]
[[Category:Mathematical quantization]]
[[Category:Foundational quantum physics]]

Congruence coefficient - Revision history

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en>Victor Chmara: reshuffling