## Infinite square well wavefunction.

Posted by peeterjoot on May 31, 2010

# Motivation.

Work problem 4.1 from [1], calculation of the eigensolution for an infinite square well, with boundaries . It’s actually a bit tidier seeming to generalize this slightly to boundaries , which also implicitly solves the problem. This is surely a problem that is done in 700 other QM texts, but I liked the way I did it this time so am writing it down.

# Guts

Our equation to solve is . Separation of variables gives us

With , we have

and the usual boundary conditions give us

We must have a zero determinant, which gives us the constraints on immediately

So our constraint on in terms of integers , and the corresponding integration constant

One of the constants can be eliminated directly by picking any one of the two zeros from 2.4

So we have

Or,

Because probability densities, currents and the expectations of any operators will always have paired and factors, any constant phase factors like above can be dropped, or absorbed into the constant , and we can write

The only thing left is to fix by integrating , for which we have

This last trig term vanishes over the integration region and we are left with

which essentially completes the problem. A final substitution back into 2.8 allows for a final tidy up

# References

[1] R. Liboff. *Introductory quantum mechanics*. Cambridge: Addison-Wesley Press, Inc, 2003.

## Leave a Reply