In quantum mechanics, Ehrenfest’s theorem gives a general expression for the
time evolution of an observable’s expectation value .
The time-dependent Schrödinger equation is as follows,
where prime denotes differentiation with respect to time :
Given an observable operator and a state ,
the time-derivative of the expectation value is as follows
(due to the product rule of differentiation):
The first two terms on the right can be rewritten using a commutator,
yielding the general form of Ehrenfest’s theorem:
In practice, since most operators are time-independent,
the last term often vanishes.
As a interesting side note, in the Heisenberg picture,
this relation proves itself,
when one simply wraps all terms in and .
Two observables of particular interest are the position and momentum .
Applying the above theorem to yields the following,
which we reduce using the fact that commutes
with the potential ,
because one is a function of the other:
This is the first part of the “original” form of Ehrenfest’s theorem,
which is reminiscent of classical Newtonian mechanics:
Next, applying the general formula to the expected momentum
To find the commutator, we go to the -basis and use a test
By inserting this result back into the previous equation, we find the following:
This is the second part of Ehrenfest’s theorem,
which is also similar to Newtonian mechanics:
There is an important consequence of Ehrenfest’s original theorems
for the symbolic derivatives of the Hamiltonian
with respect to and :
These are easy to prove yourself,
and are analogous to Hamilton’s canonical equations.