Required math: calculus
Required physics: Schrödinger equation
References: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 2.48.
Earlier, we looked at the case of a particle in the infinite square well with an initial wave function that was triangular:
We find from normalization:
Because the first derivative of is discontinuous at , we might encounter problems in calculating the second derivative, which we need to find the mean value of the energy if we use integration, since this is
We can express the first derivative of wave function as a step function :
We’ve seen that the derivative of the step function can be taken as the delta function, so
Using the delta function directly, we get
using from above. The final result is the same as that from summing the series as we did earlier.