Wigner
Functions and Quantum Theory in Phase Space| Wigner
Functions and Quantum Theory in Phase Space |
Phase Space Approach to Quantum Theory > s.a. formulations
of quantum theory; Husimi
Functions; path
integrals.
* Distribution functions: Different ones are used, such as the Wigner distribution
function, the Glauber-Sudarshan
P and Q functions, the Kirkwood distribution function and
the Husimi distribution function.
@ General references: Flandrin et al PLA(84)
[properties]; Wang & O'Connell FP(88);
Kim & Wigner AJP(90);
Fairlie & Manogue
JPA(91);
Kim & Noz 91; Schroeck IJTP(94)
[advantages], 96; Stulpe 97-qp/06;
Sala et al PLA(97)
[equivalence, with singular kernel]; Ban JMP(98)
[representation of vectors]; Brif & Mann qp/98 [with
Lie symmetries]; de Gosson JPA(01)
[and the symplectic camel]; Dragoman PiO(02)qp/04 [and
classical optics]; Anastopoulos AP(03);
Campos JPA(03);
Chruscinski OSID(06)qp/04 [Berry's
phase]; de Gosson JPA(05)mp [irrep
of Heisenberg algebra]; Chaturvedi et al JPA(06)qp/05, qp/05/JPA
[new
approach];
Smith JPA(06);
Nasiri et al JMP(06)qp [general
approach]; Nha a0804 [conditions
for physical
realizability].
@ Reviews: Lee PRP(95)
[distribution functions]; Brooke & Schroeck IJTP(05)qp/06.
@ Proposals: García de Polavieja PLA(96) [causal]; de Gosson JPA(05) [Torres-Vega & Frederick equation].
Wigner Function > s.a. [quantum
mechanics]; entropy;
experiments in quantum mechanics; quantum
states.
* Idea: The distribution
function or density matrix in phase space quantization.
$ Def: If 
is
a solution of the Schrödinger equation,
it is the real function
W(x, p, t):= (
)–1
dy *(x+y, t)
(x–y,
t) exp{2i py/}
.
* Properties: It is not directly a probability distribution function, but is useful, and satisfies
dp W(x, p, t)
= |(x, t)|2, dx W(x, p, t)
= |(p, t)|2.
* Hudson's theorem:
For 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); Royer PRL(85);
Wootters AP(87);
Dragt & Habib
qp/98-in
[and symplectic maps]; Li et al PRA(04),
Revzen FP(06)
[and phase space probability density]; Khademi qp/06;
Frieden & Soffer PRA(06)
[and information theory]; Gross JMP(06), qp/07-in
[Hudson's theorem for finite-dimensional system]; Johansen a0804.
@ Intros, reviews: Zachos IJMPA(02)ht/01 [and
deformation]; Dias & Prata MPLA(05)ht-in
[and deformation]; Nassimi a0706.
@ 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 a0710 [and
symplectic connection]; Parisio a0712-JPA
[Bargmann representation].
Specific Types of Systems > s.a. Kerr
State; photon; quantum
oscillator; quantum
systems; spinning quantum particle.
@ Discrete systems: Gudder JMP(86)
[discrete phase space]; Leonhardt PRL(95), PRA(96)
[and tomography]; Atakishiyev et al JMP(98)
[finite data sets]; Takami et al PRA(01)hl/00 [lattice];
Bianucci et al PLA(02)
[e.g.,
quantum computers]; Luís
PRA(04)
[states
on
single
phase space
points]; Gibbons et al PRA(04)qp [based
on finite
fields];
Galvão PRA(05)qp/04 [and
computation]; Argüelles & Dittrich PhyA(05)qp;
Cormick et al PRA(06)qp/05 [classicality];
Chaturvedi et al Pra(05)qp [algebraic];
Franco & Penna JPA(06)qp/05 [2
qubits, entanglement]; Ruzzi et al JPA(05)
[Cahill-Glauber formalism]; Gross JMP(06)qp [non-negative];
Klimov et al qp/06-ln;
Cormick & Paz PRA(06)qp [interference];
Wootters & Sussman a0704 [and
minimum uncertainty states]; Gross & Eisert a0710 [quantum
Margulis expander
map]; Klimov et al a0806 [n qubits].
@ H atom, Coulomb potential: Holstein AJP(95);
Nouri PRA(98);
Praxmeyer et al JPA(06)qp/05.
@ Infinite potential walls: Belloni et al AJP(04),
Dong et al PS(04) [square well]; Walton qp/06.
@ Other systems: Lee PLA(90) [potential step, quantum potential]; Dias & Prata
JMP(02)qp/00 [with
boundaries]; Curtright et al PLA(02)ht/01 [N-body];
Diósi & Kiefer JPA(02)qp/01 [Markovian
open system]; Chruscinski mp/02 [damped];
Horvat & Prosen JPA(03), qp/06-in
[classically chaotic];
O'Connell
JOB(03)qp [dissipative,
decoherence]; Genovese et al qp/05 [open
boundary, failure].
@ Field theory: Ochs & Heinz AP(98)
[covariant]; > s.a. loop quantum cosmology, quantum
cosmology, quantum field theory
in curved spacetime.
Topics > s.a. huygens principle; pilot
wave;
quantum
measurement; classical limit.
@ Semiclassical states: Rios & Ozorio de Almeida JPA(02)mp/01;
Veble
et
al JPA(02);
Dittrich et al PRL(06)qp/05 [propagator];
de Gosson & de Gosson qp/06 [squeezed];
Pulvirenti
JMP(06); de
Gosson JPA(08) [and Feichtinger algebra]; > s.a.
decoherence, quantum
states.
@ And foundations: Banaszek & Wódkiewicz PRA(98)
[EPR]; Dias & Prata PLA(01),
PLA(02)qp [and
pilot wave]; 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].
@ Time evolution: Schleich et al FP(88) [and transition probabilities];
Moshinsky & Sharma
AP(00)
[and canonical
transformations]; Hashimoto et al qp/06 [and
Markov process]; Zueco & Calvo qp/06/JPA
[Bopp operators for dynamics].
@ Entanglement:
Hardy et al PS(04);
Narnhofer JPA(06);
Ozorio de Almeida qp/06-in
[in phase space].
@ Other topics: Mehta JMP(64);
Wlodarz 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]; Dias & Prata AP(04)
[admissible
states]; Ozorio de Almeida & Brodier AP(06)
[propagators for operators]; Tegmen NCB(07)mp
[simple states]; Klauder & Skagerstam JPA(07)
[generalized representations of operators]; Scott & Caves a0801 [sub-Planck structure].
Variations, Generalizations > s.a. deformation
quantization [including constrained systems].
@ Relativistic: Sonego PRA(91);
Morgan PLA(04)qp/03;
Dragoman a0803 [and localization].
@ In curved spaces: Antonsen gq/97-in; Alonso
et al JMP(02)qp [hyperboloids],
JMP(03)
[spheres].
@ Other variations: Cohen JMP(66)
[more general quasi-probability distribution functions]; Fairlie CSF(99)ht/98 [Wigner-Weyl-Moyal
approach to quantum mechanics]; Hakioglu & Tepedelenlioglu
JPA(00)
[action-angle Wigner function]; Curtright & Veitia qp/07/JMP
[quasi-hermitian].
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
11 jul 2008