 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 = ,    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) gijij 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]; > 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]; > 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; 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; > 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].