Ehrenfest theorem information

The Ehrenfest theorem, named after Austrian theoretical physicist Paul Ehrenfest, relates the time derivative of the expectation values of the position and momentum operators x and p to the expectation value of the force $F=-V'(x)$ on a massive particle moving in a scalar potential $V(x)$ ,^[1]

$m{\frac {d}{dt}}\langle x\rangle =\langle p\rangle ,\;\;{\frac {d}{dt}}\langle p\rangle =-\left\langle V'(x)\right\rangle ~.$

The Ehrenfest theorem is a special case of a more general relation between the expectation of any quantum mechanical operator and the expectation of the commutator of that operator with the Hamiltonian of the system ^[2]^[3]

${\frac {d}{dt}}\langle A\rangle ={\frac {1}{i\hbar }}\langle [A,H]\rangle +\left\langle {\frac {\partial A}{\partial t}}\right\rangle ~,$

where $A$ is some quantum mechanical operator and $⟨ A ⟩$ is its expectation value.

It is most apparent in the Heisenberg picture of quantum mechanics, where it amounts to just the expectation value of the Heisenberg equation of motion. It provides mathematical support to the correspondence principle.

The reason is that Ehrenfest's theorem is closely related to Liouville's theorem of Hamiltonian mechanics, which involves the Poisson bracket instead of a commutator. Dirac's rule of thumb suggests that statements in quantum mechanics which contain a commutator correspond to statements in classical mechanics where the commutator is supplanted by a Poisson bracket multiplied by $iħ$ . This makes the operator expectation values obey corresponding classical equations of motion, provided the Hamiltonian is at most quadratic in the coordinates and momenta. Otherwise, the evolution equations still may hold approximately, provided fluctuations are small.

^ Hall 2013 Section 3.7.5
^ Ehrenfest, P. (1927). "Bemerkung über die angenäherte Gültigkeit der klassischen Mechanik innerhalb der Quantenmechanik". Zeitschrift für Physik. 45 (7–8): 455–457. Bibcode:1927ZPhy...45..455E. doi:10.1007/BF01329203. S2CID 123011242.
^ Smith, Henrik (1991). Introduction to Quantum Mechanics. World Scientific Pub Co Inc. pp. 108–109. ISBN 978-9810204754.

[1] Hall 2013 Section 3.7.5

[2] Ehrenfest, P. (1927). "Bemerkung über die angenäherte Gültigkeit der klassischen Mechanik innerhalb der Quantenmechanik". Zeitschrift für Physik. 45 (7–8): 455–457. Bibcode:1927ZPhy...45..455E. doi:10.1007/BF01329203. S2CID 123011242.

[Smith-3] Smith, Henrik (1991). Introduction to Quantum Mechanics. World Scientific Pub Co Inc. pp. 108–109. ISBN 978-9810204754.

Ehrenfest theorem information

and 22 Related for: Ehrenfest theorem information

Ehrenfest theorem

Paul Ehrenfest

Matrix mechanics

Virial theorem

Ehrenfest

BEST theorem

Tatyana Ehrenfest

Heisenberg picture

Ehrenfest model

Classical limit

Ehrenfest paradox

List of Dutch discoveries

Force

Classical physics

List of equations in quantum mechanics

Rashba effect

Adiabatic theorem

Good quantum number

Angular momentum operator

Uncertainty principle

Lindbladian

History of gravitational theory

Quantum mechanics
Part of a series of articles about
$i\hbar {\frac {d}{dt}}\|\Psi \rangle ={\hat {H}}\|\Psi \rangle$ Schrödinger equation
Introduction Glossary History
Background Classical mechanics Old quantum theory Bra–ket notation Hamiltonian Interference
Fundamentals Complementarity Decoherence Entanglement Energy level Measurement Nonlocality Quantum number State Superposition Symmetry Tunnelling Uncertainty Wave function Collapse
Experiments Bell's inequality Davisson–Germer Double-slit Elitzur–Vaidman Franck–Hertz Leggett–Garg inequality Mach–Zehnder Popper Quantum eraser Delayed-choice Schrödinger's cat Stern–Gerlach Wheeler's delayed-choice
Formulations Overview Heisenberg Interaction Matrix Phase-space Schrödinger Sum-over-histories (path integral)
Equations Dirac Klein–Gordon Pauli Rydberg Schrödinger
Interpretations Bayesian Consistent histories Copenhagen de Broglie–Bohm Ensemble Hidden-variable Local Superdeterminism Many-worlds Objective collapse Quantum logic Relational Transactional Von Neumann–Wigner
Advanced topics Relativistic quantum mechanics Quantum field theory Quantum information science Quantum computing Quantum chaos EPR paradox Density matrix Scattering theory Quantum statistical mechanics Quantum machine learning
Scientists Aharonov Bell Bethe Blackett Bloch Bohm Bohr Born Bose de Broglie Compton Dirac Davisson Debye Ehrenfest Einstein Everett Fock Fermi Feynman Glauber Gutzwiller Heisenberg Hilbert Jordan Kramers Lamb Landau Laue Moseley Millikan Onnes Pauli Planck Rabi Raman Rydberg Schrödinger Simmons Sommerfeld von Neumann Weyl Wien Wigner Zeeman Zeilinger
v t e