Schrödinger Equation |
In General
> s.a. differential equations / hamilton-jacobi
theory; quantum mechanics and quantum states;
scattering.
$ Def: The equation
describing a (scalar) particle in non-relativistic quantum mechanics,
i\(\hbar\) ∂tψ = Hψ , H = −(\(\hbar\)2/2m) ∇2 + V(x) .
* Conditions:
For H to be self-adjoint on \(\cal D\)(H) ⊃
C0∞(\(\mathbb R^n\))
and positive, we impose restrictions on V.
* Non-linear generalizations: They
have been looked for partially with the motivation of explaining the "collapse
of the wave function" when carrying out observations, and the probabilistic outcome
of the latter; One adds a non-linear term for the interaction with the environment;
> s.a. non-linear quantum mechanics.
* Curved configuration space:
A proposed generalization, which arises from operator ordering arguments and
makes H self-adjoint, is
i\(\hbar\) ∂tψ = −(\(\hbar\)2/2m) gij ∇i∇j ψ + V(x)ψ −\(\hbar\)2 χR ψ , χ = constant .
@ References: Schrödinger PR(26);
Gray et al AJP(99)nov [original argument];
Granik qp/04 [from Hamilton-Jacobi theory];
Ward & Volkmer phy/06 [simple argument];
Boonserm & Visser JMP(10)-a0910 [formulation as Shabat-Zakharov system and formal solution];
Escauriaza et al BAMS(12) [uniqueness properties of solutions];
Schleich et al PNAS(13)
+ news PhysOrg(13)apr [origin of the equation].
> Online resources:
see MathWorld page.
Solutions and Approximation Methods
> s.a. green functions; Perturbation Methods;
solitons; WKB Approximation.
@ Separation of variables: Zhdanov & Zhalij JMP(99)mp;
Benenti et al JMP(02),
JMP(02) [and Hamilton-Jacobi].
@ Radial equation: Erbil qp/03;
Chadan & Kobayashi JPA(06),
JPA(06)mp [special potentials];
Tannous et al PRP(08) [canonical function method];
Khelashvili & Nadareishvili a1007 [status],
AJP(11)jun-a1009,
a1009 [boundary condition].
@ Numerical: Ceperley & Alder Sci(86)feb [Monte Carlo];
Lehtovaara et al JCP(07) [imaginary time propagation];
Márk a2004 [web-based interactive software];
> s.a. computational physics.
@ Other methods:
Praeger PRA(01) [relaxational approach];
Matzkin qp/04 [quantum phase for non-solvable V];
Amore et al JPA(04) [non-perturbative, different scales];
Robinett EJP(06)qp/05 [image method];
Sudiarta & Geldart JPA(07) [finite difference time domain, FDTD];
Kishi & Umehara a0804 [Wick rotation];
Tezcan & Sever IJTP(09)-a0807;
Pillai et al AJP(12)nov [Numerov method];
Lin a1407
[H = state-preserving + state-changing Hamiltonian];
Ajaib FP(15)-a1502 [1D, first-order equation];
Radożycki MolP(16)-a1605 [classical distributions better than WKB];
Hojman & Asenjo a2007 [based on a potential function for the wave function];
> s.a. Wegner's Flow.
@ Variational principle: Perez et al AJP(90)jun [bound states];
Bhattacharyya AJP(09)jan [bounds on ground-state energy];
Atai et al a1307.
@ Auxiliary-field method:
Buisseret et al JMP(09) [and envelope theory];
Semay & Silvestre-Brac JPA(10)-a1001 [eigenstates];
Silvestre-Brac et al JPM(12)-a1101;
Semay FBS(15)-a1501 [numerical tests for few-boson systems].
@ Other approximations:
Krivec & Mandelzweig mp/04 [quasi-linearization];
Mahapatra et al IJMPA(05)qp/04;
Friedberg & Lee AP(05),
Lee JSP(05)qp [low-lying states];
Amore et al PLA(05)qp [time-dependent problems];
Friedberg et al ChP(06)qp [iterative];
> s.a. Born-Oppenheimer Approximation.
@ Other types of solutions: Garbaczewski RPMP(05)qp/04 [exotic/fractal];
Kempf & Ferreira JPA(04),
Calder & Kempf JMP(05) [superoscillating];
Karaoglu EJP(07) [large class];
Lekner EJP(08) [rotating wavepackets];
Mayer a1209
[solutions without dispersion, and inevitability of wave-packet spreading].
Other References
> s.a. formulations of quantum mechanics; physics teaching;
Propagator; quantum systems [including inverse problem].
* Generalization with a stochastic non-linear term:
It is motivated by the attempt to model state-vector collapse as a dynamical process.
@ Non-linear: Gutkin PRP(88);
Davies PhyA(90) [conservation laws];
> s.a. non-linear quantum mechanics.
@ In curved spacetime: DeWitt RMP(57);
Cheng JMP(72) [from path integral];
Ben-Abraham & Lonke JMP(73);
Benn & Tucker PLA(91);
Mannheim in(00)gq/98 [and equivalence principle];
Coelho & Amaral JPA(02)gq/01 [conical spaces];
> s.a. Bethe-Salpeter [relativistic];
modified quantum mechanics.
@ For density matrices: Mishra & Pfeifer JPA(07) [with T-dependent potential];
Shpagina et al a1812 [stationary, and applications].
@ Other generalizations:
Tomonaga PTP(46) [and special relativity];
Micu JPA(99) [q-deformed];
Kostrykin & Schrader RVMP(99)mp/00,
JPA(00)mp,
RVMP(00)mp [1D random Schrödinger operator];
Stoyanovsky in(04)mp/02 [for field theory];
Schnaid a1202
[with finite speed of wave function propagation];
Diósi PRL(14)-a1401 [diffusive stochastic Schrödinger equation];
> s.a. deformation quantization; fractional calculus;
quantum statistical mechanics [stochastic, dissipative Schrödinger equation].
@ Derivations:
De la Peña-Auerbach PLA(67) [from Markov process];
Jordan AJP(91)jul;
Granik a0801;
Field EJP(11) [from the Hamilton-Jacobi equation in Feynman's path-integral formulation];
Deriglazov & Rizzuti AJP(11)aug-a1105 [reparametrization-invariant formulation of classical mechanics];
Rusov et al AP(11)
[based on the Chetaev theorem, and pilot-wave theory];
Grinwald a1407 [from complex Gaussian propagator];
Marrocco a1705;
Izadparast & Mazharimousavi a2102 [from a Lagrangian];
> s.a. origin of quantum mechanics.
@ Related topics: Vilasi qp/97 [Hamiltonian and integrability];
Matzkin PRA(01) [amplitude-phase formulation];
Bar & Horwitz PLA(02)qp [and consistent histories];
Faraoni & Faraoni FP(02) [conformal transformation to potential-free form];
Khrennikov qp/03 [conceptual];
Maz'ya & Shubin AM(05)m.SP/03 [discreteness];
Efthimiades qp/06 [from averaged energy relation];
Flego et al PhyA(11) Legendre-transform structure];
Efthimiades a1307 [conceptual];
Arsenović et FP(14)-a1405 [Lagrangian form].
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send feedback and suggestions to bombelli at olemiss.edu – modified 8 apr 2021