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2-3: Relation to the Classical Theory

[Newtonian Equation of Motion]
Newtonian equation of motion in a one-dimensional space is written

where F is a force exerted on a particle of mass m, and V ( x ) is the potential energy. The momentum of the particle, p, is given by

So that the equation of motion is rewritten

As discussed previously, the fundamental equation of quantum mechanics, the Schroedinger equation, was promulgated as the wave equation for the de Broglie wave accompanying a substance particle motion. Hence the Schroedinger equation must be closely related to Newtonian equation of motion. What is the relation? In order to study this, let us start with the concept of mean value in quantum mechanics.

[The Mean Value of Position and Its Dispersion]
In quantum mechanics, the position of a particle is determined only probabilistically (according to the probability). Therefore, the result would varies whenever it is measured, even if the measurement is done under the same conditions. The mean value of these results, , is obtained by carrying out the integration

where x is the operator denoting the position coordinate. Of course, x is as well a usual variable representing the position of the particle, but it is also considered as an operator in quantum mechanics. And in Eq. (4) is the probability that the particle will be found at x. The wave function should, of course, be normalized.
Accordingly, in the state represented by the wave function , the mean value of the position of a particle, , is given by Eq. (4).
There are many kinds of physical quantities like a position coordinate x, or the momentum of a particle p, or the energy of a particle E, etc. Let us represent these physical quantities by an operator O. Thus, the mean value of a physical quantity O can be obtained by integrating over the whole space as

If we measure the position of a particle many times under the same condition, the data would be distributed around the mean value. The dispersion of the distribution, , must be given by

[The Operator of Momentum]
In quantum mechanics, the position operator is the usual variable x . As explained on the page, 1-3: The Schroedinger Equation the momentum operator p is defined by the differential operator, i.e.,

In general, a physical quantity in quantum mechanics is expressed by an operator. (We should think that the usual variable for the position of a particle, x , is an operator as well.) Hence, a product of two different physical quantities is not equal to the result obtained by interchanging the order of product of these two quantities. Thereby the usual common sense in classical theory is not valid in quantum mechanics.
Accordingly, the mean value of momentum, , in a state , is written

[Ehrenfest's Theorem]
Let us study the time-variation of the mean value of momentum, . The derivative of with time t can be written

Namely, we can say that "the time-variation of the mean value of the momentum of a particle is equal to the mean value of the force exerted on the particle". This is Ehrenfest's theorem.
You should notice that this theorem is of the same form as Newtonian equation of motion in the classical mechanics. In this sense, Newtonian equation of motion is involved in quantum mechanics.
It is not so difficult to prove Ehrenfest's theorem Eq. (9) as shown on the page,
2-3-A: Proof of Ehrenfest's Theorem.
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