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
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 .