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 .
Reference: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 1.4.
The square modulus of the wave function which is the solution to the Schrödinger equation is interpreted as a probability density. As an example consider the wave function given by
We can normalize by requiring
Plugging in the formula and doing the integral gives
where we’ve taken the positive real root for . Note that could also be multiplied by a phase factor for any real without affecting normalization. This can be important in some applications where we need to add together wave functions.
Given this value for , we can plot 1. Here, we’ve taken and :
Since has its maximum at , that is where the particle is most likely to be found. The probability of the particle being found to the left of is
If , then drops to zero at so . If , then is an isosceles triangle symmetric about so .
The expectation value of is
where we used Maple to simplify the integration. If , then as expected.