What is proof of the ehrenfest's thereom?

Answer 1

Statement:The ensemble average behavior of a wave function resembles that of a classical particle.

Proof:

We have followed the mathematical formulation governed by Schrodinger's picture

(a)(1)

Last term in above equation vanishes, since does not depend upon explicitly. From (a) and its complex conjugate, (1) can be written as (2)

Let's evaluate the second integral in the square brackets of (2). (3)

According to divergence theorem, LHS of (3) is nothing but surface integral of the quantity in parentheses. (4)

Since the volume integral is being evaluated over entire space, the wave function is presumed to vanish at large distances and so does LHS of (4). Thus, (5)

This makes (2) to look like (6)

Once again, the quantity in square brackets of (6) can be simplified as (7)

Using end result of (7) in (6), (8)

Let's now calculate the time derivative of expectation x - component of momentum: (9)

Invoking Schrodinger's equation (a) and its complex conjugate again, (9) proceeds as follows: (10)

First integral has two terms. It turns out that second term of the integral cancels first. This can be seen below - (11)

or (12)

LHS of (12) vanishes due to the same argument as given for (4). Thus, (13)

Hence first step of (10) and (13) suggest that the first integral in (10) vanishes. Moreover we obtain a rather simplified form of (10) that proves Ehrenfest's theorem: