Schrödinger operators
E924202
Schrödinger operators are a class of differential operators fundamental in quantum mechanics and spectral theory, used to describe the energy and dynamics of quantum systems.
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
differential operator
ⓘ
linear operator ⓘ self-adjoint operator (typically) ⓘ unbounded operator ⓘ |
| actsOn |
Hilbert space L^2(R^n)
ⓘ
wave functions ⓘ |
| associatedWith |
Hamiltonian operator
ⓘ
time-independent Schrödinger equation NERFINISHED ⓘ unitary time evolution ⓘ |
| domain | Sobolev spaces (e.g., H^2(R^n)) ⓘ |
| field |
functional analysis
ⓘ
mathematical physics ⓘ partial differential equations ⓘ quantum mechanics ⓘ spectral theory ⓘ |
| governs | time evolution via i∂_t ψ = Hψ ⓘ |
| hasComponent |
kinetic energy term
ⓘ
potential energy term ⓘ |
| hasForm | H = -Δ + V(x) in suitable units ⓘ |
| hasProperty |
spectrum depends on potential V
ⓘ
spectrum may be continuous ⓘ spectrum may be discrete ⓘ spectrum may have absolutely continuous part ⓘ spectrum may have singular continuous part ⓘ |
| namedAfter | Erwin Schrödinger NERFINISHED ⓘ |
| relatedTo |
Dirichlet boundary conditions
NERFINISHED
ⓘ
Feynman–Kac formula NERFINISHED ⓘ Neumann boundary conditions NERFINISHED ⓘ Robin boundary conditions ⓘ semigroup e^{-tH} ⓘ |
| requiresCondition |
boundary conditions on domain
ⓘ
self-adjointness for physical interpretation ⓘ |
| specialCase |
discrete Schrödinger operator on lattices
ⓘ
free Schrödinger operator (V = 0) ⓘ magnetic Schrödinger operator ⓘ multi-particle Schrödinger operator ⓘ periodic Schrödinger operator ⓘ random Schrödinger operator ⓘ |
| studiedIn |
Anderson localization theory
ⓘ
quantum chemistry ⓘ solid-state physics ⓘ |
| usedFor |
analyzing bound states
ⓘ
analyzing scattering states ⓘ describing dynamics of quantum systems ⓘ describing energy of quantum systems ⓘ modeling non-relativistic quantum particles ⓘ studying localization phenomena ⓘ studying quantum tunneling ⓘ studying spectra of quantum Hamiltonians ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.