**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.

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