Schrödinger
Equation| 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
t
= H , H = –{2/2m}
2 + V(x) .
* Conditions: For H to
be self-adjoint on 
(H)
C0infty(Rn)
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 t = –(2/2m)
gijij + V(x) – 2
R , =
constant .
@ References: Gray et al AJP(99)nov [original argument]; Granik qp/04 [from Hamilton-Jacobi theory]; Ward & Volkmer phy/06 [simple argument]; Boonserm & Visser a0910 [formulation as Shabat-Zakharov system and formal solution].
Solutions and Approximation Methods > s.a. green
functions; Perturbation
Methods.
* WKB approximation:
(Wentzel, Kramers & Brillouin) Consists in
writing
the solution of the Schrödinger equation in the form =
A exp(iS), with S real, rapidly varying wrt A;
Then S satisfies
the classical Hamilton-Jacobi equation, with any of whose solutions we associate
a
family
of classical
trajectories in configuration space; Sometimes equivalent
to
the stationary-phase or one-loop approximation; Fails in a neighborhood of the
boundary
between
the classically allowed and forbidden regions.
@ Separation of variables: Zhdanov & Zhalij JMP(99)mp;
Benenti
et al JMP(02), JMP(02)
[and Hamilton-Jacobi].
@ Solution of radial equation: Erbil qp/03;
Chadan
& Kobayashi JPA(06), JPA(06)mp [special
potentials]; Tannous et al PRP(08) [canonical function method].
@ 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; > s.a. Wegner's
Flow.
@ WKB approximation: Lindblom & Robiscoe JMP(91);
Bronzan PRA(96)
[modified];
Romanovski & Robnik JPA(00)
[convergence, examples]; Hyouguchi
et al PRL(02),
AP(04)
[divergence-free
modification]; Sergeenko qp/02 [0th-order];
Friedrich & Trost
PRP(04)
[far from semiclassical limit]; Voros mp/04-in
[1D, overview]; Fityo et al JPA(06)
[with minimal length]; Bracken mp/06 [time-dependent
version, for tunneling]; Carles CMP(07)
[for non-linear quantum mechanics].
@ Variational principle:
Perez et al AJP(90)jun
[bound states]; Bhattacharyya AJP(09)jan [bounds on ground-state energy].
@ Other approximation methods: 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 qp/06 [iterative];
Buisseret et al JMP(09)
[auxiliary-field method and envelope theory]; > 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].
References > s.a. formulations of
quantum mechanics; Propagator; quantum
systems.
@ Non-linear: Gutkin PRP(88);
Davies PhyA(90)
[conservation laws].
@ 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.
@ 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
mp/02/FAA
[for field theory]; Mishra & Pfeifer JPA(07)
[for density matrix, with T-dependent potential]; > s.a. analysis [fractional]; deformation
quantization.
@ Derivations: De la Peña-Auerbach PLA(67)
[from Markov process]; Jordan AJP(91)jul;
Granik a0801; > 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].
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send feedback and suggestions to bombelli at olemiss.edu – modified 17
oct 2009