Required math: calculus
Required physics: Schrödinger equation
Reference: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 2.6.
As another example of an explicit case of a particle in the infinite square well, suppose we modify the example in the last post so that the second stationary state has a constant phase relative to the first. That is:
we can first normalize by noting that is the same as in the last example since and are orthogonal. So
The time-dependent form is then:
where is given by the infinite square well formula.
The modulus is calculated in the usual way
The mean position is calculated by integrating this expression multiplied by :
where we have used the infinite square well formula to substitute for the functions, and done the integral with Maple. This reduces to the answer from before when .
For we have
and for we have
Thus the phase has no effect on the amplitude of the oscillation; but it shifts the oscillation in time.