Schrödinger equation with point interactions
E735006
The Schrödinger equation with point interactions is a quantum-mechanical model in which particles interact via idealized zero-range potentials, typically represented mathematically by Dirac delta functions.
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
Schrödinger equation
ⓘ
differential equation with singular potential ⓘ quantum mechanical model ⓘ |
| approximationOf | short-range regular potentials in zero-range limit ⓘ |
| boundaryConditionType |
1/r behavior near interaction in 3D
ⓘ
discontinuity in derivative of wave function in 1D ⓘ logarithmic behavior near interaction in 2D ⓘ |
| dependsOnParameter |
coupling constant of each point interaction
ⓘ
positions of interaction centers ⓘ |
| describes |
bound states generated by localized interactions
ⓘ
quantum particles interacting at points ⓘ scattering by point-like impurities ⓘ |
| field |
functional analysis
ⓘ
mathematical physics ⓘ quantum mechanics ⓘ spectral theory ⓘ |
| governs | time evolution of wave functions with point-like interactions ⓘ |
| hasComponent |
free Schrödinger operator on configuration space minus interaction points
ⓘ
matching conditions for wave function at interaction points ⓘ |
| hasMathematicalForm | (-ħ^2/2m) Δψ + V ψ = E ψ with V a sum of delta functions ⓘ |
| hasProperty |
exactly solvable in many cases
ⓘ
models idealized short-range interactions ⓘ spectrum depends on coupling strengths and positions ⓘ translation invariance broken by interaction locations ⓘ |
| hasSolutionType |
bound states
ⓘ
resonant states ⓘ scattering states ⓘ |
| quantizationType | nonrelativistic ⓘ |
| relatedTo |
Bethe-Peierls boundary conditions
NERFINISHED
ⓘ
Krein resolvent formula NERFINISHED ⓘ contact interaction ⓘ delta-function potential ⓘ self-adjoint extensions of the Laplacian ⓘ |
| requires |
boundary conditions at interaction points
ⓘ
renormalization of coupling constants in higher dimensions ⓘ self-adjoint extension theory ⓘ |
| spaceDimension |
one-dimensional version
ⓘ
three-dimensional version ⓘ two-dimensional version ⓘ |
| typicalPotentialForm | V(x) = ∑_j α_j δ(x - x_j) ⓘ |
| usedFor |
modeling impurities in quantum wires
ⓘ
modeling quantum dots and point scatterers ⓘ studying bound states induced by localized perturbations ⓘ studying solvable models of scattering theory ⓘ testing renormalization methods in nonrelativistic quantum mechanics ⓘ |
| usesPotentialType |
Dirac delta potential
NERFINISHED
ⓘ
distribution-valued potential ⓘ zero-range potential ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.