Wigner Functions| Wigner Functions |
In General > s.a. quantum mechanics; formulations
of quantum theory; phase space / entropy;
experiments in quantum mechanics; quantum states.
* Idea: The distribution
function or density matrix in phase space quantization; Its time evolution
is governed by Moyal's equation.
$ Def: For a solution
ψ of the Schrödinger equation, the Wigner function
is the real function
W(x, p, t):= (π\(\hbar\))−1 ∫ dy ψ*(x+y, t) ψ(x−y, t) exp{2i py/\(\hbar\)} .
* Properties: It is not directly a probability distribution function, but it is useful, and satisfies
∫ dp W(x, p, t) = |ψ(x, t)|2, ∫ dx W(x, p, t) = |ψ(p, t)|2.
* Hudson's theorem:
For non-relativistic continuous variable systems, the Wigner function of a pure state has no
negative values if and only if the state is Gaussian.
@ General references:
Wigner PR(32) [proposal];
Tatarskii SPU(83);
Narcowich & Fulling ed;
Royer PRL(85);
Wootters AP(87);
Dragt & Habib qp/98-proc [and symplectic maps];
Li et al PRA(04),
Revzen FP(06) [and phase-space probability density];
Khademi qp/06;
Nassimi a0706;
Johansen a0804;
Case AJP(08)oct [and Weyl transforms, for pedestrians];
Surhone et al ed-10;
O'Connell in(09)-a1009 [rev];
Bauke & Itzhak a1101;
Bednorz & Belzig PRA(11),
Bondar et al PRA(13)-a1202 [negativity];
Schroeck JPA(12) [probability];
Steuernagel et al PRD(12) [flow and topological order in quantum dynamics];
Bondar et al PRA(13) [as a wave function];
Giese et al proc(14)-a1402 [intro];
Blass & Gurevich a1502
[and marginal distributions of x and p];
Rakotoson et al a1707 [phase-space representations];
Schwonnek & Werner a1802 [for arbitrary sets of observables];
Perepelkin et al a1904
[new representation, and universal density matrix].
@ Propagator: Dittrich et al PRL(06)qp/05 [semiclassical];
Ozorio de Almeida & Brodier AP(06);
Sels et al JPA(13)-a1207 [path-integral approach];
Cabrera et al PRA(15)-a1212 [effective numerical propagation].
@ And information theory:
Frieden & Soffer PRA(06);
Bernardini & Bertolami a1901-conf [continuity equations for quantum information flux].
@ Hudson's theorem: Gross JMP(06),
APB(07)qp-proc [for finite-dimensional system];
Mandilara et al PRA(09)-a0808 [for mixed states].
@ Relationships:
Leavens & Sala Mayato PLA(01) [and wave function];
Bracken RPMP(06)qp/05 [vs Hilbert space, and superposition];
Praxmeyer & Wodkiewicz LP(05)phy [and spectrum, for light];
Isidro IJGMP(08)-a0710 [and symplectic connection];
Parisio JPA(08)-a0712 [Bargmann representation];
Lieb & Ostrover JMP(10)-a1007 [Gaussians and localization in phase space];
> s.a. quantum chaos.
Related topics:
see specific systems and generalizations.
Related Topics
> s.a. huygens principle; pilot-wave interpretation;
quantum correlations; quantum measurement;
classical limit; Wigner Transform.
@ Semiclassical states: Rios & Ozorio de Almeida JPA(02)mp/01;
Veble et al JPA(02);
de Gosson & de Gosson qp/06 [squeezed];
Pulvirenti JMP(06);
de Gosson JPA(08) [and Feichtinger algebra];
Dechoum et al PRA(10)-a1107 [two-mode entangled state];
Song & Fan IJTP(12) [squeezed];
Kalligiannaki & Makrakis a1402 [perturbative analysis];
Giannopoulou & Makrakis a1705 [approximate series solution];
> s.a. decoherence; quantum states.
@ Other states: Tegmen NCB(07)mp [simple states];
Vanbever a2104
[vacuum, as majorizing mixtures of Fock states].
@ And pilot-wave interpretation: Dias & Prata PLA(01)qp,
PLA(02)qp;
Hiley FP(10).
@ And foundations: Banaszek & Wódkiewicz PRA(98) [EPR];
Franco qp/07 [EPR].
@ Calculation: Hug et al JPA(98);
Samson JPA(00) [coherent state path integral],
JPA(03)qp [phase-space path integral];
Curtright et al JMP(01) [generating functions];
Sels et al PhyA(13) [propagator for complex dynamical systems, path integral approach];
Kakofengitis et al PRA(17)-a1611 [integral form];
Gozzi et al a2004 [Marinov path interal and response field].
@ Time evolution:
Schleich et al FP(88) [and transition probabilities];
Moshinsky & Sharma AP(00) [and canonical transformations];
Hashimoto et al JPA(07)qp/06 [and Markov process];
Zueco & Calvo JPA(07)qp/06 [Bopp operators for dynamics];
Lewis-Swan et al a1503
[interpretation of individual phase-space trajectories].
@ Entanglement: Hardy et al PS(04);
Narnhofer JPA(06);
Ozorio de Almeida LNP(09)qp/06 [in phase space].
@ Other topics: Mehta JMP(64);
Włodarz PLA(88) [averaging and positivity];
Zavialov & Malokostov TMP(99)ht/98;
Lougovski et al PRL(03)qp/02 [operational def];
Dias & Prata JMP(04)ht/03 [t-dependent transformations];
Oliveira et al AP(04) [from star product];
Klauder & Skagerstam JPA(07) [generalized representations of operators];
Scott & Caves AP(08)-a0801 [sub-Planck structure];
Kube et al JCP(09) [Monte Carlo sampling];
Pennini & Plastino PLA(10) [thermal effects];
Mari & Eisert PRL(12)-a1208 [Wigner-function positivity and efficient classical simulation];
Przanowski & Brzykcy a1512 [number-phase Wigner function];
Hernández & Riedel a2103 [rapidly decaying Wigner functions];
> s.a. complexity;
non-extensive statistical mechanics.
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