Required math: calculus
Required physics: none
Reference: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 2.23.
Here are a few simple examples of integrals involving the Dirac delta function. The delta function is defined by the two conditions:
Since it is zero everywhere except at it follows that
for any ‘ordinary’ function . A simple extension of this is
This follows by making the substitution . Then and we get
This integral is provided the limits of integration include , that is, , or .
And a final example:
since the limits of integration don’t include .