Schrödinger equation

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1: 17.17 Physical Applications

… ►See Kassel (1995). ►A substantial literature on

q

-deformed quantum-mechanical Schrödinger equations has developed recently. …

2: 18.39 Physical Applications

… ►

§18.39(i) Quantum Mechanics

►Classical OP’s appear when the time-dependent Schrödinger equation is solved by separation of variables. … ►A second example is provided by the three-dimensional time-independent Schrödinger equation …See Seaborn (1991, pp. 69-75). … ► …

3: Sidebar 22.SB1: Decay of a Soliton in a Bose–Einstein Condensate

… ►Jacobian elliptic functions arise as solutions to certain nonlinear Schrödinger equations, which model many types of wave propagation phenomena. …

4: 9.16 Physical Applications

… ►Reference to many of these applications as well as to the theory of elasticity and to the heat equation are given in Vallée and Soares (2010): a book devoted specifically to the Airy and Scorer functions and their applications in physics. ►An example from quantum mechanics is given in Landau and Lifshitz (1965), in which the exact solution of the Schrödinger equation for the motion of a particle in a homogeneous external field is expressed in terms of

\operatorname{Ai}\left(x\right)

. Solutions of the Schrödinger equation involving the Airy functions are given for other potentials in Vallée and Soares (2010). …

5: 21.9 Integrable Equations

… ►

21.9.1

4u_{t}=6uu_{x}+u_{xxx},

ⓘ

Symbols:: $u(x,y,t)$ : elevation
Permalink:: http://dlmf.nist.gov/21.9.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §21.9 and Ch.21

►and the nonlinear Schrödinger equations …

6: 20.13 Physical Applications

… ►This allows analytic time propagation of quantum wave-packets in a box, or on a ring, as closed-form solutions of the time-dependent Schrödinger equation.

7: Barry I. Schneider

… ►His current principal focus is developing novel methods for the solution of the time dependent Schrödinger equation in ultra-short, and intense laser fields. …

8: 33.22 Particle Scattering and Atomic and Molecular Spectra

… ►

§33.22(i) Schrödinger Equation

… ►The reduced mass is

m=m_{1}m_{2}/(m_{1}+m_{2})

, and at energy of relative motion

E

with relative orbital angular momentum

\ell\hbar

, the Schrödinger equation for the radial wave function

w(s)

is given by … ►The Coulomb solutions of the Schrödinger and Klein–Gordon equations are almost always used in the external region, outside the range of any non-Coulomb forces or couplings. … ►

9: 22.19 Physical Applications

… ►These include the time dependent, and time independent, nonlinear Schrödinger equations (NLSE) (Drazin and Johnson (1993, Chapter 2), Ablowitz and Clarkson (1991, pp. 42, 99)), the Korteweg–de Vries (KdV) equation (Kruskal (1974), Li and Olver (2000)), the sine-Gordon equation, and others; see Drazin and Johnson (1993, Chapter 2) for an overview. …

10: 31.17 Physical Applications

… ►Heun functions appear in the theory of black holes (Kerr (1963), Teukolsky (1972), Chandrasekhar (1984), Suzuki et al. (1998), Kalnins et al. (2000)), lattice systems in statistical mechanics (Joyce (1973, 1994)), dislocation theory (Lay and Slavyanov (1999)), and solution of the Schrödinger equation of quantum mechanics (Bay et al. (1997), Tolstikhin and Matsuzawa (2001), and Hall et al. (2010)). …