View source for Ehrenfest theorem

{{Short description|Theorem in quantum mechanics}}
{{Quantum mechanics|cTopic=Fundamental concepts}}
The '''Ehrenfest theorem''', named after Austrian theoretical physicist [[Paul Ehrenfest]], relates the time [[derivative]] of the [[Expectation value (quantum mechanics)|expectation values]] of the position and momentum [[operator (physics)|operators]] ''x'' and ''p'' to the expectation value of the force <math>F=-V'(x)</math> on a massive particle moving in a scalar potential <math>V(x)</math>,<ref>{{harvnb|Hall|2013}} Section 3.7.5</ref>
{{Equation box 1
|indent =:
|equation =<math>m\frac{d}{dt}\langle x\rangle = \langle p\rangle,\;\; \frac{d}{dt}\langle p\rangle =  -\left\langle V'(x)\right\rangle  ~.</math>
|cellpadding= 6
|border
|border colour = #0073CF
|bgcolor=#F9FFF7}}The Ehrenfest theorem is a special case of a more general relation between the expectation of any [[Quantum mechanics|quantum mechanical]] [[operator (physics)|operator]] and the expectation of the  [[commutator]] of that operator with the [[Hamiltonian (quantum mechanics)|Hamiltonian]] of the system <ref>{{Cite journal | last1 = Ehrenfest | first1 = P. | title = Bemerkung über die angenäherte Gültigkeit der klassischen Mechanik innerhalb der Quantenmechanik | doi = 10.1007/BF01329203 | journal = Zeitschrift für Physik | volume = 45 | issue = 7–8 | pages = 455–457 | year = 1927 |bibcode = 1927ZPhy...45..455E | s2cid = 123011242 }}</ref><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>{{Equation box 1
|indent =:
|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>
|cellpadding= 6
|border
|border colour = #0073CF
|bgcolor=#F9FFF7}}
where {{mvar|A}} is some quantum mechanical operator and {{math|⟨''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 (Hamiltonian)|Liouville's theorem]] of [[Hamiltonian mechanics]], which involves the [[Poisson bracket]] instead of a commutator.  [[Canonical_quantization#Classical_and_quantum_brackets|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.

== Relation to classical physics ==
Although, at first glance, it might appear that the Ehrenfest theorem is saying that the quantum mechanical expectation values obey Newton’s classical equations of motion, this is not actually the case.<ref name="Wheeler">{{cite web|last=Wheeler| first=Nicholas|title=Remarks concerning the status & some ramifications of Ehrenfest's theorem|url=http://academic.reed.edu/physics/faculty/wheeler/documents/Quantum%20Mechanics/Miscellaneous%20Essays/Ehrenfest's%20Theorem.pdf}}</ref> If the pair <math>(\langle x\rangle,\langle p\rangle)</math> were to satisfy Newton's second law, the right-hand side of the second equation would have to be
<math display="block">-V'\left(\left\langle x\right\rangle\right),</math>
which is typically not the same as
<math display="block">-\left\langle V'(x)\right\rangle.</math>
If for example, the potential <math>V(x)</math> is cubic, (i.e. proportional to <math>x^3</math>), then <math>V'</math> is quadratic (proportional to <math>x^2</math>). This means, in the case of Newton's second law, the right side would be in the form of <math>\langle x\rangle^2</math>, while in the Ehrenfest theorem it is in the form of <math>\langle x^2\rangle</math>. The difference between these two quantities is the square of the uncertainty in <math>x</math> and is therefore nonzero.

An exception occurs in case when the classical equations of motion are linear, that is, when <math>V</math> is quadratic and <math>V'</math> is linear. In that special case, <math>V'\left(\left\langle x\right\rangle\right)</math> and <math>\left\langle V'(x)\right\rangle</math> do agree. Thus, for the case of a quantum harmonic oscillator, the expected position and expected momentum do exactly follow the classical trajectories.

For general systems, if the wave function is highly concentrated around a point <math>x_0</math>, then <math>V'\left(\left\langle x\right\rangle\right)</math> and <math>\left\langle V'(x)\right\rangle</math> will be ''almost'' the same, since both will be approximately equal to <math>V'(x_0)</math>. In that case, the expected position and expected momentum will ''approximately'' follow the classical trajectories, at least for as long as the wave function remains localized in position.<ref>{{harvnb|Hall|2013}} p. 78</ref>

== Derivation in the Schrödinger picture ==
Suppose some system is presently in a [[quantum state]] {{math|Φ}}. If we want to know the instantaneous time derivative of the expectation value of {{mvar|A}}, that is, by definition
<math display="block">\begin{align}
\frac{d}{dt}\langle A\rangle &= \frac{d}{dt}\int \Phi^* A \Phi \, d^3x \\
&= \int \left( \frac{\partial \Phi^*}{\partial t} \right) A\Phi\,d^3x + \int \Phi^* \left( \frac{\partial A}{\partial t}\right) \Phi \, d^3x +\int \Phi^* A \left( \frac{\partial \Phi}{\partial t} \right) \, d^3x \\
&= \int \left( \frac{\partial \Phi^*}{\partial t} \right) A\Phi\,d^3x + \left\langle \frac{\partial A}{\partial t}\right\rangle + \int \Phi^* A \left( \frac{\partial \Phi}{\partial t} \right) \, d^3x
\end{align}</math>
where we are integrating over all of space. If we apply the [[Schrödinger equation]], we find that
<math display="block">\frac{\partial \Phi}{\partial t} = \frac{1}{i\hbar}H\Phi</math>

By taking the complex conjugate we find <ref>In [[bra–ket notation]] <math>\phi^*=\langle \phi, x \rangle</math>, so<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 {{mvar|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 {{mvar|H}} and {{math|Φ}}.</ref>
<math display="block">\frac{\partial \Phi^*}{\partial t} = -\frac{1}{i\hbar}\Phi^*H^* = -\frac{1}{i\hbar}\Phi^*H.</math>

Note {{math|1=''H'' = ''H''&thinsp;<sup>∗</sup>}}, because the [[Hamiltonian (quantum mechanics)|Hamiltonian]] is [[hermitian operator|Hermitian]].  Placing this into the above equation we have

<math display="block">\frac{d}{dt}\langle A\rangle = \frac{1}{i\hbar}\int \Phi^* (AH-HA) \Phi~d^3x + \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>

Often (but not always) the operator {{mvar|A}} is time-independent so that its derivative is zero and we can ignore the last term.

== Derivation in the Heisenberg picture ==
In the [[Heisenberg picture]], the derivation is straightforward. The Heisenberg picture moves the time dependence of the system to operators instead of state vectors. Starting with the Heisenberg equation of motion,
<math display="block">\frac{d}{dt}A(t) = \frac{\partial A(t)}{\partial t} + \frac{1}{i \hbar}[A(t),H],</math>
Ehrenfest's theorem follows simply  upon 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
<math display="block">\left\langle\Psi\left|\frac{d}{dt}A(t)\right|\Psi\right\rangle = \left\langle\Psi\left|\frac{\partial A(t)}{\partial t}\right|\Psi\right\rangle + \left\langle\Psi\left|\frac{1}{i \hbar}[A(t),H]\right|\Psi\right\rangle,</math>

One may pull the {{math|{{sfrac|''d''|''dt''}}}} out of the first term, since the state vectors are no longer time dependent in the Heisenberg Picture. Therefore,
<math display="block">\frac{d}{dt}\langle A(t)\rangle = \left\langle\frac{\partial A(t)}{\partial t}\right\rangle + \frac{1}{i \hbar}\left\langle[A(t),H]\right\rangle .</math>

== General example ==
For the very general example of a massive [[Elementary particle|particle]] moving in a [[potential]], the Hamiltonian is simply
<math display="block"> H(x,p,t) = \frac{p^2}{2m} + V(x,t) </math>
where {{mvar|x}} is the position of the particle.

Suppose we wanted to know the instantaneous change in the expectation of the momentum {{mvar|p}}. Using Ehrenfest's theorem, we have
<math display="block"> \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>

since the operator {{mvar|p}} commutes with itself and has no time dependence.<ref>Although the expectation value of the momentum {{math|⟨''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 {{math|⟨''xt''<sup>2</sup>⟩}}, where {{mvar|x}} is the ordinary position operator and {{mvar|t}} is just the (non-operator) time, a parameter.</ref> By expanding the right-hand-side, replacing {{mvar|p}} by  {{math|−''iħ''∇}}, we get
<math display="block">\frac{d}{dt}\langle p\rangle = \int \Phi^* V(x,t)\frac{\partial}{\partial x}\Phi~dx - \int \Phi^* \frac{\partial}{\partial x} (V(x,t)\Phi)~dx ~.</math>

After applying the [[product rule]] on the second term, we have
<math display="block"> \begin{align}
\frac{d}{dt}\langle p\rangle &= \int \Phi^* V(x,t) \frac{\partial}{\partial x}\Phi~dx - \int \Phi^* \left(\frac{\partial}{\partial x} V(x,t)\right)\Phi ~dx - \int \Phi^* V(x,t) \frac{\partial}{\partial x}\Phi~dx \\
&= - \int \Phi^* \left(\frac{\partial}{\partial x} V(x,t)\right)\Phi ~dx \\
&= \left\langle - \frac{\partial}{\partial x} V(x,t)\right\rangle = \langle F \rangle.
\end{align}</math>

As explained in the introduction, this result does ''not'' say that the pair <math>(\langle X\rangle,\langle P\rangle)</math> satisfies [[Newton's second law]], because the right-hand side of the formula is <math>\langle F(x,t)\rangle,</math> rather than <math>F(\langle X\rangle,t)</math>. Nevertheless, as explained in the introduction, for states that are highly localized in space, the expected position and momentum will ''approximately'' follow classical trajectories, which may be understood as an instance of the [[correspondence principle]].

Similarly, we can obtain the instantaneous change in the position expectation value.
<math display="block">\begin{align}
\frac{d}{dt}\langle x\rangle &= \frac{1}{i\hbar}\langle [x,H]\rangle + \left\langle \frac{\partial x}{\partial t}\right\rangle \\[5pt]
&= \frac{1}{i\hbar} \left \langle \left [x,\frac{p^2}{2m} + V(x,t) \right ] \right \rangle + 0 \\[5pt]
&= \frac{1}{i\hbar} \left \langle \left [x,\frac{p^2}{2m} \right] \right \rangle \\[5pt]
&= \frac{1}{i\hbar 2 m} \left \langle [x,p] \frac{d}{dp} p^2 \right\rangle \\[5pt]
&= \frac{1}{i\hbar 2 m}\langle i \hbar 2 p\rangle \\[5pt]
&= \frac{1}{m}\langle p\rangle
\end{align}</math>

This result is actually in exact accord with the classical equation.

== Derivation of the Schrödinger equation from the Ehrenfest theorems ==
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 journal | last1 = Bondar | first1 = D. | last2 = Cabrera | first2 = R. | last3 = Lompay | first3 = R. | last4 = Ivanov | first4 = M. | last5 = Rabitz | first5 = H. | title = Operational Dynamic Modeling Transcending Quantum and Classical Mechanics | doi = 10.1103/PhysRevLett.109.190403 | journal = Physical Review Letters | volume = 109 | issue = 19 | pages = 190403 | year = 2012 | pmid =  23215365|arxiv = 1105.4014 |bibcode = 2012PhRvL.109s0403B | s2cid = 19605000 }}</ref> We begin from
<math display="block">\begin{align}
m\frac{d}{dt} \left \langle \Psi(t) \right | \hat{x} \left | \Psi(t) \right \rangle &= \left \langle \Psi(t) \right | \hat{p} \left | \Psi(t) \right \rangle, \\[5pt]
\frac{d}{dt} \left \langle \Psi(t) \right | \hat{p} \left | \Psi(t) \right \rangle &= \left \langle \Psi(t) \right | -V'(\hat{x}) \left | \Psi(t) \right \rangle.
\end{align}</math>

Application of the [[product rule]] leads to
<math display="block">\begin{align}
\left \langle \frac{d\Psi}{dt} \Big | \hat{x} \Big | \Psi \right \rangle + \left \langle \Psi \Big | \hat{x} \Big | \frac{d\Psi}{dt} \right \rangle &= \left \langle \Psi \Big | \frac{\hat{p}}{m} \Big | \Psi \right \rangle, \\[5pt]
\left \langle \frac{d\Psi}{dt} \Big | \hat{p} \Big | \Psi \right \rangle + \left \langle \Psi \Big | \hat{p} \Big | \frac{d\Psi}{dt} \right \rangle &= \langle \Psi | -V'(\hat{x}) | \Psi \rangle,
\end{align} </math>
Here, apply [[Stone's theorem on one-parameter unitary groups|Stone's theorem]], using {{mvar|Ĥ}} to denote the quantum generator of time translation.  The next step is to show that this is the same as the Hamiltonian operator used in quantum mechanics.  Stone's theorem implies
<math display="block">i\hbar \left | \frac{d\Psi}{dt} \right \rangle = \hat{H} | \Psi(t) \rangle ~,</math>
where {{mvar|ħ}} 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 {{mvar|Ĥ}} are derived:
<math display="block">im [\hat{H}, \hat{x}] = \hbar \hat{p}, \qquad i [\hat{H}, \hat{p}] = -\hbar V'(\hat{x}).</math>

Assuming that observables of the coordinate and momentum obey the [[canonical commutation relation]]  {{math|1=[''x̂'', ''p̂''] = ''iħ''}}. Setting <math>\hat{H} = H(\hat{x}, \hat{p})</math>, the commutator equations can be converted into the differential equations<ref name=Bondar2012 /><ref name=Transtrum2005>{{Cite journal | last1 = Transtrum | first1 = M. K. | last2 = Van Huele | first2 = J. F. O. S. | doi = 10.1063/1.1924703 | title = Commutation relations for functions of operators | journal = Journal of Mathematical Physics | volume = 46 | issue = 6 | pages = 063510 | year = 2005 |bibcode = 2005JMP....46f3510T | url = http://scholarsarchive.byu.edu/cgi/viewcontent.cgi?article=1371&context=facpub }}</ref>
<math display="block">m \frac{\partial H (x,p)}{\partial p} = p, \qquad \frac{\partial H(x,p)}{\partial x} = V'(x),</math>
whose solution is the familiar [[Hamiltonian (quantum mechanics)|quantum Hamiltonian]]
<math display="block">\hat{H} = \frac{\hat{p}^2}{2m} + V(\hat{x}).</math>

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|[''x̂'', ''p̂'']}}.

The implications of the Ehrenfest theorem for systems with classically chaotic dynamics are discussed at Scholarpedia article [http://www.scholarpedia.org/article/Ehrenfest_time_and_chaos Ehrenfest time and chaos]. Due to exponential instability of classical trajectories the Ehrenfest time, on which there is a complete correspondence between quantum and classical evolution, is shown to be logarithmically short being proportional to a logarithm of typical quantum number. For the case of integrable dynamics this time scale is much larger being proportional to a certain power of quantum number.

== Notes ==
{{Reflist}}

==References==
{{Commons category}}
*{{citation|first=Brian C.|last=Hall|title=Quantum Theory for Mathematicians|series=Graduate Texts in Mathematics|volume=267 |publisher=Springer|year=2013|bibcode=2013qtm..book.....H | isbn=978-1461471158}}

[[Category:Theorems in quantum mechanics]]
[[Category: Mathematical physics]]