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: 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. …

3: 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 $\mathrm{Ai}\left(x\right)$. Solutions of the Schrödinger equation involving the Airy functions are given for other potentials in Vallée and Soares (2010). …

4: 18.39 Applications in the Physical Sciences

… ►These eigenfunctions are the orthonormal eigenfunctions of the time-independent Schrödinger equation … ►argument b) The Morse Oscillator … ►c) A Rational SUSY Potential argument ►The Schrödinger equation with potential … ►

Other Analytically Solved Schrödinger Equations

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5: 21.9 Integrable Equations

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21.9.1 $$4u_t=6u{u}_x+u_{xxx},$$

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Symbols:

$u(x,y,t)$: elevation

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http://dlmf.nist.gov/21.9.E1

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Annotations for §21.9 and Ch.21

►and the nonlinear Schrödinger equations …

6: 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. …

7: 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.

8: 33.22 Particle Scattering and Atomic and Molecular Spectra

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§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 $\mathrm{\ell }\mathrm{\hslash }$, 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: 18.38 Mathematical Applications

… ►Schneider et al. (2016) discuss DVR/Finite Element solutions of the time-dependent Schrödinger equation. … ►The solved Schrödinger equations of §18.39(i) involve shape invariant potentials, and thus are in the family of supersymmetric or SUSY potentials. … ►Hermite EOP’s appear in solutions of a rationally modified Schrödinger equation in §18.39. …