The Schrödinger equation describes the motion of particles in nonrelativistic quantum mechanics, and was first written down by Erwin Schrödinger. The time-dependent Schrödinger equation is given by
![ih(partialPsi(x,y,z,t))/(partialt)=[-(h^2)/(2m)del ^2+V(x)]Psi(x,y,z,t)=H^~Psi(x,y,z,t),](/images/equations/SchroedingerEquation/NumberedEquation1.svg) |
(1)
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where 
is the reduced Planck constant 
, 
is the time-dependent wavefunction, 
is the mass of a particle, 
is the Laplacian, 
is the potential, and 
is the Hamiltonian operator. The time-independent Schrödinger
equation is
![[-(h^2)/(2m)del ^2+V(x)]psi(x,y,z)=Epsi(x,y,z),](/images/equations/SchroedingerEquation/NumberedEquation2.svg) |
(2)
|
where 
is the energy of the particle.
The one-dimensional versions of these equations are then
![ih(partialPsi(x,t))/(partialt)=[-(h^2)/(2m)(partial^2)/(partialx^2)+V(x)]Psi(x,t)=H^~Psi(x,t),](/images/equations/SchroedingerEquation/NumberedEquation3.svg) |
(3)
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and
![[-(h^2)/(2m)(d^2)/(dx^2)+V(x)]psi(x)=Epsi(x).](/images/equations/SchroedingerEquation/NumberedEquation4.svg) |
(4)
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Variants of the one-dimensional Schrödinger equation have been considered in various contexts, including the following (where 
is a suitably non-dimensionalized version of the wavefunction).
The logarithmic Schrödinger equation is given by
 |
(5)
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(Cazenave 1983; Zwillinger 1997, p. 134), the nonlinear Schrödinger equation by
 |
(6)
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(Calogero and Degasperis 1982, p. 56; Tabor 1989, p. 309; Zwillinger 1997, p. 134) or
 |
(7)
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(Infeld and Rowlands 2000, p. 126), and the derivative nonlinear Schrödinger equation by
 |
(8)
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(Calogero and Degasperis 1982, p. 56; Zwillinger 1997, p. 134).
