Required math: calculus
Required physics: Schrödinger equation
Reference: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 2.9.
The average or expectation value of the energy of a particle in an infinite square well can be worked out either by using the series solution in the form
or directly using an integral, using and :
Since in the general case is a sum over stationary states, as in
and the functions are orthogonal, all the terms of the form
where in the integral evaluate to zero, and the remaining terms where are all independent of time since the complex exponentials cancel out, so is independent of time. Thus if we know the wave function at any given time, we can use it to work out at all times.
For example, if we have a parabolic wave function at
for , we can work out directly from it. Applying the normalization condition we can find .
We get for :
Working out the second derivative and then the integral (use Maple or do by hand) we get .