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Nonlinear Schrödinger equation

From Biocrawler, the free encyclopedia.

In theoretical physics, the nonlinear Schrödinger equation is a nonlinear version of Schrödinger's equation in two dimensions. It can be considered as a classical equation, or a second quantized bosonic theory. It is an example of an integrable model.

Classically, we have a complex field ψ satisfying the partial differential equation

$i\partial_t\psi=-{1\over 2}\partial^2_x\psi+\kappa|\psi|^2 \psi$

It is described by the Hamiltonian

$H=\int dx \left[{1\over 2}|\partial_x\psi|^2+{\kappa \over 2}|\psi|^4\right]$

with the Poisson brackets

{ψ(x),ψ(y)} = {ψ * (x),ψ * (y)} = 0

{ψ * (x),ψ(y)} = i δ(x - y)

To get the quantized version, simply replace the Poisson brackets by commutators

[ψ(x),ψ(y)] = [ψ * (x),ψ * (y)] = 0

[ψ * (x),ψ(y)] = - δ(x - y)

and normal order the Hamiltonian

$H=\int dx \left[{1\over 2}\partial_x\psi^\dagger\partial_x\psi+{\kappa \over 2}\psi^\dagger\psi^\dagger\psi\psi\right]$

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External links

Nonlinear Schrodinger Equation with a Cubic Nonlinearity (http://eqworld.ipmnet.ru/en/solutions/npde/npde1401.pdf) at EqWorld: The World of Mathematical Equations.
Nonlinear Schrodinger Equation with a Power-Law Nonlinearity (http://eqworld.ipmnet.ru/en/solutions/npde/npde1402.pdf) at EqWorld: The World of Mathematical Equations.
Nonlinear Schrodinger Equation of General Form (http://eqworld.ipmnet.ru/en/solutions/npde/npde1403.pdf) at EqWorld: The World of Mathematical Equations.

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