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
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]
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.
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.
were to satisfy Newton's second law, the right-hand side of the second equation would have to be
This means, in the case of Newton's second law, the right side would be in the form of
An exception occurs in case when the classical equations of motion are linear, that is, when
For general systems, if the wave function is highly concentrated around a point
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.
[5] 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
The Heisenberg picture moves the time dependence of the system to operators instead of state vectors.
Ehrenfest's theorem follows simply upon projecting the Heisenberg equation onto
One may pull the d/dt out of the first term, since the state vectors are no longer time dependent in the Heisenberg Picture.
For the very general example of a massive particle moving in a potential, the Hamiltonian is simply
Suppose we wanted to know the instantaneous change in the expectation of the momentum p. Using Ehrenfest's theorem, we have
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.
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.
Here, apply Stone's theorem, using 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.
Since these identities must be valid for any initial state, the averaging can be dropped and the system of commutator equations for Ĥ are derived:
Assuming that observables of the coordinate and momentum obey the canonical commutation relation [x̂, p̂] = iħ.
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.
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.