Ehrenfest theorem
E735839
The Ehrenfest theorem is a fundamental result in quantum mechanics that links the time evolution of expectation values of quantum observables to the corresponding classical equations of motion, thereby bridging quantum and classical physics.
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf | quantum mechanics theorem ⓘ |
| appliesTo |
expectation values
ⓘ
general Hermitian operators ⓘ momentum operator ⓘ position operator ⓘ quantum observables ⓘ |
| assumes |
self-adjoint operators for observables
ⓘ
sufficiently well-behaved wavefunctions ⓘ |
| bridges | quantum mechanics and classical mechanics ⓘ |
| canBeGeneralizedTo | relativistic quantum mechanics ⓘ |
| category | theorems in quantum mechanics ⓘ |
| clarifies | relationship between operator equations and classical laws ⓘ |
| concerns | expectation value dynamics ⓘ |
| describes | time evolution of expectation values of observables ⓘ |
| field | quantum mechanics ⓘ |
| hasDomain | nonrelativistic quantum mechanics ⓘ |
| hasFormula | d⟨A⟩/dt = (i/ħ)⟨[H,A]⟩ + ⟨∂A/∂t⟩ ⓘ |
| historicalPeriod | early 20th century ⓘ |
| holdsIn |
Heisenberg picture
NERFINISHED
ⓘ
Schrödinger picture NERFINISHED ⓘ |
| implies |
d⟨p⟩/dt = −⟨∂V/∂x⟩ for potential V(x)
ⓘ
d⟨x⟩/dt = ⟨p⟩/m for a particle of mass m ⓘ |
| importance | fundamental for understanding quantum–classical correspondence ⓘ |
| involves |
Hamiltonian operator
ⓘ
commutators ⓘ |
| isUsedFor |
connecting quantum dynamics to classical trajectories
ⓘ
interpretation of quantum motion ⓘ semiclassical analysis ⓘ |
| mathematicallyExpressedIn |
Hilbert space framework
ⓘ
operator formalism ⓘ |
| mayFailToGive | exact classical motion for highly nonclassical states ⓘ |
| namedAfter | Paul Ehrenfest NERFINISHED ⓘ |
| relatedTo |
Heisenberg equation of motion
NERFINISHED
ⓘ
Heisenberg uncertainty principle NERFINISHED ⓘ Schrödinger equation of motion NERFINISHED ⓘ correspondence principle ⓘ |
| relates |
quantum expectation values to classical equations of motion
ⓘ
time derivative of expectation value to commutator with Hamiltonian ⓘ |
| shows | classical equations emerge from quantum mechanics in expectation values ⓘ |
| specialCase | Newton’s second law in expectation value form ⓘ |
| typeOf | dynamical theorem ⓘ |
| usedIn |
atomic physics
ⓘ
condensed matter physics ⓘ molecular dynamics with quantum effects ⓘ quantum chemistry ⓘ |
| uses |
Heisenberg equation of motion
NERFINISHED
ⓘ
Schrödinger equation NERFINISHED ⓘ |
| validUnder | smooth potentials ⓘ |
Referenced by (1)
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