**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.

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Chris KranenbergEq. 17 and Eq. 18 should have the h-bar term squared.

gwrowePost authorFixed now. Thanks.

KeithA nicer way to do this would be to use the identity for sin^3(x), it spits out the coefficients directly but it would only work for an example like this.

UzzielHi Keith, I was trying to do a similar thing but I got stuck (and that’s why I looked this up! )

Which identity worked for you?

MateuszTry to expand sin^3(x) into exponential form and simplify it 🙂