Required math: calculus
Required physics: Schrödinger equation
Reference: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 2.37.
As an example of the infinite square well potential suppose the particle starts off with the wave function
for (and zero elsewhere).
First, we can find by normalizing (using software for the integral):
The general solution as a function of time is the series
where are the stationary states of the square well. We find the by considering the sum at :
Because the stationary states are orthogonal functions, we have
A quick glance at this result might make you think that for all because of the sine term. However, we need to be careful, since the denominator factors to
and thus has zeroes at . We can redo these integrals for these specific values of and we get
The full solution is therefore
The particle will be found with energy with probability of and with energy with probability 0.1. Thus
The mean position is found from
Working out the integrand, we get
We can now do the integral using software with the result
The mean position is the midpoint of the well.