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| {{Quantum mechanics|cTopic=Fundamental concepts}}
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| The '''Ehrenfest theorem''', named after [[Paul Ehrenfest]], an Austrian theoretical physicist, relates the time [[derivative]] of the [[Expectation value (quantum mechanics)|expectation value]] for a [[Quantum mechanics|quantum mechanical]] [[operator (physics)|operator]] to the expectation of the [[commutator]] of that operator with the [[Hamiltonian (quantum mechanics)|Hamiltonian]] of the system,<ref>{{cite doi|10.1007/BF01329203}}</ref> expectations which are connected to [[classical mechanics]]. | |
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| It reads<ref name="Smith">{{cite book | last=Smith | first=Henrik | year=1991 | title=Introduction to Quantum Mechanics | publisher=World Scientific Pub Co Inc |isbn=978-9810204754| pages=108–109}}</ref>
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| {{Equation box 1
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| |indent =:
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| |equation =<math>\frac{d}{dt}\langle A\rangle = \frac{1}{i\hbar}\langle [A,H] \rangle+ \left\langle \frac{\partial A}{\partial t}\right\rangle ~,</math>
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| |cellpadding= 6
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| |border
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| |border colour = #0073CF
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| |bgcolor=#F9FFF7}}
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| where ''A'' is some QM operator and <math>\langle A\rangle</math> is its [[expectation value]].
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| Ehrenfest's theorem is most apparent in the [[Heisenberg picture]] of quantum mechanics, where it is just the expectation value of the Heisenberg equation of motion. It provides mathematical support to the [[correspondence principle]].
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| The reason is that Ehrenfest's theorem is closely related to [[Liouville's theorem (Hamiltonian)|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 {{math|''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 [[Moyal bracket|approximately]], provided fluctuations are small.
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| == Derivation in the Schrödinger Picture ==
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| Suppose some system is presently in a [[quantum state]] ''Φ''. If we want to know the instantaneous time derivative of the expectation value of ''A'', that is, by definition
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| :<math> \frac{d}{dt}\langle A\rangle = \frac{d}{dt}\int \Phi^* A \Phi~dx^3 = \int \left( \frac{\partial \Phi^*}{\partial t} \right) A\Phi~dx^3 + \int \Phi^* \left( \frac{\partial A}{\partial t}\right) \Phi~dx^3 +\int \Phi^* A \left( \frac{\partial \Phi}{\partial t} \right) ~dx^3 </math>
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| :<math> = \int \left( \frac{\partial \Phi^*}{\partial t} \right) A\Phi~dx^3 + \left\langle \frac{\partial A}{\partial t}\right\rangle + \int \Phi^* A \left( \frac{\partial \Phi}{\partial t} \right) ~dx^3, </math>
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| where we are integrating over all space. If we apply the [[Schrödinger equation]], we find that
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| :<math>\frac{\partial \Phi}{\partial t} = \frac{1}{i\hbar}H\Phi</math>
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| and
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| :<math>\frac{\partial \Phi^*}{\partial t} = \frac{-1}{i\hbar}\Phi^*H^* = \frac{-1}{i\hbar}\Phi^*H.</math> <ref>In [[bra-ket notation]], <math> \frac{\partial}{\partial t}\langle \phi |x\rangle =\frac{-1}{i\hbar}\langle \phi |\hat{H}|x\rangle =\frac{-1}{i\hbar}\langle \phi |x \rangle H=\frac{-1}{i\hbar}\Phi^*H </math>, where <math>\hat{H}</math> is the Hamiltonian operator, and ''H'' is the Hamiltonian represented in coordinate space (as is the case in the derivation above). In other words, we applied the adjoint operation to the entire Schrödinger equation, which flipped the order of operations for ''H'' and ''Φ''.</ref>
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| Note <math>H=H^*</math>, because the [[Hamiltonian (quantum mechanics)|Hamiltonian]] is [[hermitian operator|Hermitian]]. Placing this into the above equation we have
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| :<math>\frac{d}{dt}\langle A\rangle = \frac{1}{i\hbar}\int \Phi^* (AH-HA) \Phi~dx^3 + \left\langle \frac{\partial A}{\partial t}\right\rangle = \frac{1}{i\hbar}\langle [A,H]\rangle + \left\langle \frac{\partial A}{\partial t}\right\rangle.</math>
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| Often (but not always) the operator A is time independent, so that its derivative is zero and we can ignore the last term.
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| == Derivation in the Heisenberg Picture ==
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| In the [[Heisenberg picture]], the derivation is trivial. The Heisenberg picture moves the time dependence of the system to operators instead of state vector. Starting with the Heisenberg equation of motion
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| <math>\frac{d}{dt}A(t) = \frac{\partial A(t)}{\partial t} + \frac{1}{i \hbar}[A(t),H],</math>
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| we can derive Ehrenfest's Theorem simply by projecting the Heisenberg Equation onto <math> |\Psi\rangle </math> from the right and <math> \langle\Psi| </math> from the left, or taking the expectation value, so | |
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| <math>\langle\Psi|\frac{d}{dt}A(t)|\Psi\rangle = \langle\Psi|\frac{\partial A(t)}{\partial t}|\Psi\rangle + \langle\Psi|\frac{1}{i \hbar}[A(t),H)]|\Psi\rangle,</math>
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| We can pull the <math>\frac{d}{dt}</math> out of the first term since the state vectors are no longer time dependent in the Heisenberg Picture. Therefore,
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| <math>\frac{d}{dt}\langle A(t)\rangle = \left\langle\frac{\partial A(t)}{\partial t}\right\rangle + \frac{1}{i \hbar}\langle[A(t),H)]\rangle</math>
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| == General example ==
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| The expectation values of the theorem, however, are the very same in the [[Schrödinger picture]] as well.
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| For the very general example of a massive [[Elementary particle|particle]] moving in a [[potential]], the Hamiltonian is simply
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| :<math> H(x,p,t) = \frac{p^2}{2m} + V(x,t) </math>
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| where {{mvar|x}} is the position of the particle.
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| Suppose we wanted to know the instantaneous change in momentum {{mvar|p}}. Using Ehrenfest's theorem, we have
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| :<math> \frac{d}{dt}\langle p\rangle = \frac{1}{i\hbar}\langle [p,H]\rangle + \left\langle \frac{\partial p}{\partial t}\right\rangle = \frac{1}{i\hbar}\langle [p,V(x,t)]\rangle ,</math>
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| since the operator {{mvar|p}} commutes with itself and has no time dependence.<ref>Although the expectation value of the momentum 〈''p'' 〉, which is a [[Real number|real-number]]-valued function of time, will have time dependence, the momentum operator itself, {{mvar|p}} does not, in this picture: Rather, the momentum operator is a constant [[linear operator]] on the [[Hilbert space]] of the system. The time dependence of the expectation value, in this picture, is due to the [[time evolution]] of the wavefunction for which the expectation value is calculated. An [[Ad hoc]] example of an operator which does have time dependence is 〈''xt''<sup>2</sup>〉, where {{mvar| x}} is the ordinary position operator and {{mvar|t}} is just the (non-operator) time, a parameter.
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| </ref> By expanding the right-hand-side, replacing ''p'' by <math>-i\hbar \nabla</math>, we get
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| :<math> | |
| \frac{d}{dt}\langle p\rangle = \int \Phi^* V(x,t)\nabla\Phi~dx^3 - \int \Phi^* \nabla (V(x,t)\Phi)~dx^3.
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| </math>
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| After applying the [[product rule]] on the second term, we have
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| :<math> | |
| \begin{align}
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| \frac{d}{dt}\langle p\rangle =& \int \Phi^* V(x,t)\nabla\Phi~dx^3 - \int \Phi^* (\nabla V(x,t))\Phi ~dx^3 - \int \Phi^* V(x,t)\nabla\Phi~dx^3 \\
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| =& - \int \Phi^* (\nabla V(x,t))\Phi ~dx^3 \\
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| =& \langle -\nabla V(x,t)\rangle = \langle F \rangle,
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| \end{align}
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| </math>
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| but we recognize this as [[Newton's second law]]. This is an example of the [[correspondence principle]]: the result manifests as Newton's second law in the case of having so many excitations superposed in the wavefunction that the net motion is given by the expectation value simulating a classical particle.
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| Similarly we can obtain the instantaneous change in the position expectation value.
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| :<math>
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| \begin{align}
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| \frac{d}{dt}\langle x\rangle =& \frac{1}{i\hbar}\langle [x,H]\rangle + \left\langle \frac{\partial x}{\partial t}\right\rangle \\
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| =& \frac{1}{i\hbar}\langle [x,\frac{p^2}{2m} + V(x,t)]\rangle + 0 = \frac{1}{i\hbar}\langle [x,\frac{p^2}{2m}]\rangle \\
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| =& \frac{1}{i\hbar}\langle [x,\frac{p^2}{2m}]\rangle = \frac{1}{i\hbar 2 m}\langle [x,p] \frac{d}{dp} p^2\rangle \\
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| =& \frac{1}{i\hbar 2 m}\langle i \hbar 2 p\rangle = \frac{1}{m}\langle p\rangle
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| \end{align}
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| </math>
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| This result is again in accord with the classical equation.
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| == Derivation of the Schrödinger equation from the Ehrenfest theorems ==
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| It was established above that the Ehrenfest theorems are consequences of the [[Schrödinger equation]]. However, the converse is also true: the Schrödinger equation can be inferred from the Ehrenfest theorems.<ref name=Bondar2012>{{Cite doi|10.1103/PhysRevLett.109.190403}}</ref> We begin from
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| :<math>
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| \begin{align}
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| m\frac{d}{dt} \langle \Psi(t) | \hat{x} | \Psi(t) \rangle &= \langle \Psi(t) | \hat{p} | \Psi(t) \rangle, \\
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| \frac{d}{dt} \langle \Psi(t) | \hat{p} | \Psi(t) \rangle &= \langle \Psi(t) | -V'(\hat{x}) | \Psi(t) \rangle.
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| \end{align}
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| </math>
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| Applications of the [[product rule]] leads to
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| :<math>
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| \begin{align}
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| \langle d\Psi/dt | \hat{x} | \Psi \rangle + \langle \Psi | \hat{x} | d\Psi/dt \rangle &= \langle \Psi | \hat{p}/m | \Psi \rangle, \\
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| \langle d\Psi/dt | \hat{p} | \Psi \rangle + \langle \Psi | \hat{p} | d\Psi/dt \rangle & = \langle \Psi | -V'(\hat{x}) | \Psi \rangle,
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| \end{align}
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| </math>
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| into which we substitute a consequence of [[Stone's theorem on one-parameter unitary groups|Stone's theorem]]
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| :<math>
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| i\hbar | d \Psi(t)/dt \rangle = \hat{H} | \Psi(t) \rangle,
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| </math>
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| where <math>\hbar</math> was introduced as a normalization constant to the balance dimensionality. Since these identities must be valid for any initial state, the averaging can be dropped and the system of commutator equations for the unknown quantum generator of motion <math>\hat{H}</math> are derived
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| :<math>
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| im [\hat{H}, \hat{x}] = \hbar \hat{p}, \qquad i [\hat{H}, \hat{p}] = -\hbar V'(\hat{x}).
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| </math>
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| Assuming that observables of the coordinate and momentum obey the [[canonical commutation relation]] <math>[ \hat{x}, \hat{p} ] = i\hbar</math>. Setting <math>\hat{H} = H(\hat{x}, \hat{p})</math>, the commutator equations can be converted into the differential equations
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| <ref name=Bondar2012/><ref name=Transtrum2005>{{Cite doi|10.1063/1.1924703}}</ref>
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| :<math>
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| m \frac{\partial H (x,p)}{\partial p} = p, \qquad \frac{\partial H(x,p)}{\partial x} = V'(x),
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| </math>
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| whose solution is the familiar [[Hamiltonian (quantum mechanics)|quantum Hamiltonian]]
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| :<math>
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| \hat{H} = \frac{\hat{p}^2}{2m} + V(\hat{x}).
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| </math>
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| Whence, the [[Schrödinger equation]] was derived from the Ehrenfest theorems by assuming the canonical commutation relation between the coordinate and momentum. If one assumes that the coordinate and momentum commute, the same computational method leads to the [[Koopman–von Neumann classical mechanics]], which is the [[Hilbert space]] formulation of [[classical mechanics]].<ref name=Bondar2012/> Therefore, this derivation as well as the [[Koopman–von Neumann classical mechanics#Derivation starting from operator axioms|derivation of the Koopman–von Neumann mechanics]] shows that the essential difference between quantum and classical mechanics reduces to the value of the commutator <math>[ \hat{x}, \hat{p} ]</math>.
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| == Notes ==
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| {{Reflist}}
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| [[Category:Quantum mechanics]]
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| [[Category:Theorems in quantum physics]]
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