This one-dimensional expression corresponds to the definition of force under quantum interpretation
Proof of Ehrenfest's Theorem
I even read a post on this platform about explaining Ehrenfest’s theorem. That publication was interesting, as it also presented details of the language of quantum physics. Well, so I decided to delve deeper into the subject by publishing the algebraic development of Ehrenfest’s theorem.
We will make the proof corresponding to the one-dimensional case.
The quantum interpretation of the momentum of a particle is based on the average value of the momentum operator:
that is,
On both members of this expression, the operation of function differentiation with respect to the time variable will be applied
Let us now develop the partial derivative that is present in the second term of the expression
some factors of this expression will be replaced by algebraic segments of the Schrödinger equation, shown below
and, with respect to the complex conjugate term,
also, given that the Schrödinger wave function is an analytic function, then
So that now we can consolidate the expression in development
Expanding the partial derivatives with respect to the variable “x” and at the same time simplifying algebraically
grouping common factors
and then,
We can transport this result to the expression of the integrand
and as a strategy, divide the right side into two expressions of mathematical integrals:
and,
On the first of them we will apply the integration by parts, this written next already having an algebraic modification
the second term there can be rewritten expanded as
and then the rule of integration by parts is applied to the term under integration
so, with these results, we can write the following equality,
We rewrite here that expression which we are processing as part of the resolution strategy:
and in it we will replace the results achieved, that is,
After the simplifications are carried out, the result will be as follows
which can be synthetically rewritten as,
Note that, in theory, for the Schrödinger wave function we have
In that case its derivative (for the Schrödinger wave function) would similarly tend to zero, so that
therefore, the expression of the derivative in relation to the time variable “t” of the expected value of the moment, will be restricted to the expression of the second integral that was waiting to be developed:
Note now that the second member of the expression above corresponds to the negative of the expected value of a potential gradient, with this we can rewrite the expression above in the synthetic form:
and if we call the second the expected value of a force “F”, then
