Quantum Oscillators| Quantum Oscillators |
Quantum Harmonic Oscillator
> s.a. feynman propagator; formulations [combinatorial];
oscillators; Polymer Representation;
time in quantum theory.
* Propagator: The Feynman
propagator for the quantum harmonic oscillator is (Δt:=
t2−t1)
DF(x2, t2; x1, t1) = [mω / (2πi sin(ωΔt))1/2] exp{i mω/2 [(x22 + x12) · (ωΔt) − 2 x2 x1 csc(ωΔt)] } .
@ General references:
Jauch & Hill PR(40);
Leubner et al AJP(88)dec [vs classical];
Boya et al IJMPA(98) [inequivalent];
Muñoz AJP(98)mar [and integral equations];
Jordan AJP(01)oct [simple solution];
Baker et al AJP(02)may [applications, numerical];
Kastrup AdP(07)qp/06 [new look];
Plimak & Stenholm AP(08) [response properties];
Marsiglio AJP(09)mar [and bound states of short-range potentials];
Andrews AJP(16)apr-a1509 [evolution of wave functions];
Zubizarreta et al a1607 [structure of the Hilbert space];
Rushka & Freericks a1912/AJP [algebraic solution].
@ 2D: Chen & Huang JPA(03) [coherent states, vortex structure];
Montesinos & Torres del Castillo PRA(04),
comment Latimer PRA(07)qp/06 [different symplectic structures];
Doll & Ingold AJP(07)mar [semiclassical, Lissajous curves];
Borondo et al JPA(09)-a0907 [and Bohmian trajectories].
@ Entanglement:
Kim & Iafrate FPL(04) [coupled];
Rios JPA(07) [non-interacting];
Jost et al Nat(09)jun-a0901 [non-interacting, demonstration].
@ Different approaches: Donoghue & Holstein AJP(88)mar [path integral];
Koikawa PTP(01)ht,
PTP(02)ht/01 [Moyal];
Shiri-Garakani & Finkelstein JMP(06) [general quantization];
Finkelstein & Shiri-Garakani IJTP(11)-qp/06 [as model for spacetime decondensation];
Nettel & Quevedo mp/06-proc [topological quantization];
Vicary IJTP(08) [categorical framework];
Pimentel & de Castro EJP(13)-a1211 [Laplace-transform approach];
Ahmadzadegan et al PRA(16)-a1510 [Koopman and Moyal formalisms, classical and quantum aspects];
Nagao & Nielsen a1902 [future-included];
Belmonte & Cuéllar MPAG(20)-a2001 [Weyl quantization];
Bojowald et al a2012 [algebraic derivation of eigenvalues];
Mostowski & Pietraszewicz a2104 [Wigner function, classical limit];
> s.a. path integrals [non-standard].
@ Green function:
Khrebtukov & Macek JPA(98);
Shao AJP(16)oct [elementary derivation];
> s.a. feynman propagator.
@ Coupled oscillators: Maxson ht/03,
ht/03,
ht/03,
ht/03 [correlated];
Colosi a0706 [general relativistic, two-point function];
Bhattacharya et al AJP(13)apr.
@ Coupled to a heat bath: Ford & O'Connell PhyE(05)qp/06;
Isar & Scheid PhyA(07)-a0705 [decoherence and classical correlations];
Ford & O'Connell PRA(14)-a1408 [two oscillators];
Kheirandish PLA(18)-a1806 [exact quantum propagator and density matrix].
@ Other modifications: Fernández a2007 [rotating harmonic oscillator];
Aremua & Gouba JPComm(21)-a2012 [on the half line, affine quatization].
@ Related topics: de la Peña & Cetto JMP(79) [and stochastic electrodynamics];
Dattoli & Torre NCB(95) [phase space, coherent states];
Lorente PLA(01)qp/04 [discrete model];
Morigi et al PRA(02) [irreversibility];
Leshem & Gat PRL(09) [violation of macroscopic realism];
Betz & Castrigiano CMP(11) [coupled to a photon field, density of states];
Wang a1303 [new approach];
Suslov PS(13) [variant of Berry's phase];
Chowdhury et al PRL(20)-a1907 [squeezed states, signature].
> States: see fock space;
coherent states; hilbert space [inverted];
representations of quantum mechanics [Bargmann].
> Other topics: see approaches to quantum
mechanics; Born-Jordan Quantization; Coarse-Graining;
deformation quantization; geometric quantization;
quantum-classical coupling; resonance;
statistical-mechanical systems.
Anharmonic Oscillators
@ Cubic:
Ferreira & Sesma JPA(14) [eigenstates].
@ Quartic:
Pesquera & Claverie JMP(82) [in stochastic electrodynamics];
Matamala & Maldonado PLA(03),
Liverts et al JMP(06) [spectrum and eigenfunctions, analytic];
Mutuk a1811 [energy levels, neural network approach];
Blinder a1903 [eigenvalues].
@ Other / non-linear / perturbed: Pathak JPA(00)mp/02;
Speliotopoulos JPA(00) [spectrum];
Calogero & Graffi PLA(03);
Calogero PLA(03);
Bhattacharyya & Bhattacharjee PLA(04)
[subharmonic, V ∝ |x|r];
Gómez & Sesma in(04)qp [bound states],
JPA(05)qp [new approach];
Cariñena et al AP(07) [solvable];
Liverts & Mandelzweig PS(08) [approximate solution];
Fernández a0804 [eigenvalues, upper and lower bounds];
Midya & Roy JPA(09) [exactly solvable, quasi-exactly solvable et al];
Wang & Liu IJTP(09);
Nachtergaele et al RVMP(10) [anharmonic oscillator lattice systems];
Tosto a1105 [simple quantum model];
Wójcik APPB-a1210 [numerical renormalization group procedure];
Fernández & García CEJP(14)-a1310 [V(x,
y) = x2y2; eigenvalues
and eigenfunctions; the spectrum seems to be discrete];
Fernández & García APol(17)-a1511 [accurate calculation of eigenvalues];
> s.a. coherent states.
@ Semiclassical states: Matzkin & Lombardi JPA(05)-a0706 [quantum and semiclassical phase functions];
Moncrief et al JMP(12)-a1201 [semiclassical approach];
Brizuela PRD(14)-a1411 [dynamical evolution of classical and quantum probability distributions in terms of moments].
@ Perturbation methods: Cicuta ht/97;
Amore mp/04-proc [classical and quantum];
Voros mp/06-proc;
> s.a. Perturbation Methods.
Damped Oscillator
> s.a. Lindblad Equation; quantum probability.
* Idea: A dissipative
system; > s.a. dissipation.
@ General references:
Pedrosa & Baseia PRD(84) [+ oscillator bath];
Milburn & Holmes PRL(96);
Isar & Sandulescu RJP(92)qp/06 [rev];
de Brito et al NCB(98);
Um et al PRP(02);
Banerjee & Mukherjee JPA(02)qp/01 [canonical approach];
Blasone et al qp/03-conf;
Montesinos PRA(03) [Heisenberg picture];
Latimer JPA(05)qp/04;
López & López IJTP(06)qp/05;
Endo et al IJGMP(08)-a0710,
Fujii & Suzuki IJGMP(09)-a0806 [general solution];
Cordero-Soto et al a0905;
Baldiotti et al PLA(11)-a1005;
Philbin NJP(12) [with a continuum of oscillators as reservoir];
Barnett et al a1508 [strongly damped];
Ahmed et al JMP-a1902;
Bagarello et al PLA(19)-a1906,
reply to comment a1910
[no quantization using the Bateman lagrangian].
@ Bateman's dual system: (A damped simple harmonic oscillator coupled to its time-reversed image)
Blasone & Jizba AP(04)qp/01.
@ Related topics: Kheirandish & Amooshahi MPLA(05)qp [radiation reaction];
Chruściński & Jurkowski AP(06) [resonances];
Isar O&S(07)qp/06-conf [decoherence, Lindblad theory].
@ In alternative approaches:
Vandyck JPA(94) [pilot-wave theory];
Dito & Turrubiates PLA(06)qp/05 [deformation quantization];
Fujii qp/07-conf [complex time].
Other Types and Effects > s.a. decoherence;
Dirac Oscillator; histories formulation;
non-commutative systems; Perturbation Methods.
@ Relativistic: Guerrero & Aldaya
MPLA(99) [perturbative];
Toyama & Nogami PRA(99);
Bars PRD(09)-a0810;
Arbab a1709 [quaternionic].
@ Inverted: Blume-Kohout & Zurek PRA(03)qp/02 [upside down, decoherence];
Chruściński JMP(04)mp/03;
Yuce et al PS(06);
Golovinski a1905 [forced]; Bhattacharyya et al SciPost(21)-a2007 [chaos and complexity].
@ On the sphere and hyperbolic space: Cariñena et al AP(07)-a0709 [2D];
Mardoyan a0708-proc [in d dimensions];
Quesne PLA(15)-a1411.
@ Forced, time-dependent:
Dodonov PLA(96) [kicked];
Graffi & Yajima CMP(00)mp [forced];
Kim & Yee PRA(02)ht;
Moya-Cessa & Fernández-Guasti PLA(03)qp [sudden change, coherent states];
Adler JPA(05)qp/04 [stochastic collapse and decoherence];
Gómez & Villaseñor AP(09) [and quantum field theory];
Velasco-Martínez et al a1409 [unitary approach];
> s.a. stochastic quantization; Wigner Transform.
@ With gup, minimal length: Chang et al PRD(02)ht/01;
Nouicer PLA(06);
Gemba et al a0712
[algebraic solution, deformed su(1,1) algebra];
Fakel & Merad JMP(09);
Lewis & Takeuchi PRD(11);
Valtancoli MPLA(12)-a1205 [with a minimal uncertainty in position];
Valtancoli a1306 [with a minimal length];
Das et al CJP(16)-a1412 [in phase space];
Quintela et al BJP(16)-a1510 [classical limit].
@ Other deformed oscillators:
Man'ko et al qp/97-proc;
De Freitas & Salamó ht/99;
Gruver PLA(99);
Sogami & Koizumi PTP(02)mp/01;
Isar & Scheid PhyA(02)qp/07,
PhyA(04)qp/07 [in dissipative environment];
Albanese & Lawi JPA(04)ht/03;
Narayana Swami qp/04 [and intermediate statistics];
Jafarov et al JPA(07)mp [Wigner function];
Batouli & El Baz FP(14) [classical interpretation];
Sadurní & Rivera-Mociños JPA(15)-a1504 [with fractal position spectrum];
> s.a. modified coherent states [including Grassmann].
@ Pais-Uhlenbeck oscillator: (An example of higher-derivative theory)
Mannheim & Davidson PRA(05)ht/04 [Dirac quantization];
Andrzejewski et al PTP(11)-a0904 [Euclidean path-integral approach].
@ Supersymmetric:
Thienel JPA(96) [Bargmann representation];
> s.a. coherent states.
@ Coupled oscillators: Bender et al PRA(14)-a1406 [PT-symmetric];
Bruschi et al a1912 [time evolution].
@ Other types: Dragovich TMP(94)ht/04,
IJMPA(95)ht/04 [adelic];
Banerjee & Ghosh JPA(98) [chiral];
Badescu & Landsberg JPA(02) [τ-oscillator];
Kim & Page qp/02 [generalized];
Blasone et al PLA(03)qp/02 [group contraction];
Guido a1208 [Intrinsic Quantum Oscillator (IQuO)];
Valtancoli PTEP(13)-a1306 [in a Snyder geometry];
Belenchia et al CQG(19)-a1901 [non-local, Hamiltonian formulation];
Giardino EPJP(21)-a2101 [quaternionic];
> s.a. stochastic quantization [Fermi oscillator].
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