Reference: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 1.4.
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.