Reference: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 1.5.

The delta function well gives rise to a wave function that decays exponentially either side of the delta function:

We can normalize in the usual way:

By symmetry, and

Therefore

A plot of is shown, with vertical yellow lines indicating , for the case :

The probability that the particle lies outside is

In this case, the probability of being greater than one standard deviation is a constant, independent of .