Time-Independent Schrödinger Equation — David J. Griffiths
Physics — Quantum Mechanics · expectation_energy
Problem
In Problem 2.7(d) you got the expectation value of the energy by summing the series in Equation 2.21, but I warned you (in footnote 21) not to try it the "old fashioned way," ⟨H⟩=∫Ψ(x,0)∗H^Ψ(x,0)dx, because the discontinuous first derivative of Ψ(x,0) renders the second derivative problematic. Actually, you could have done it using integration by parts, but the Dirac delta function affords a much cleaner way to handle such anomalies.
(a) Calculate the first derivative of Ψ(x,0) (in Problem 2.7), and express the answer in terms of the step function, θ(x−a/2), defined in Equation 2.146.
(b) Exploit the result of Problem 2.23(b) to write the second derivative of Ψ(x,0) in terms of the delta function.
(c) Evaluate the integral ∫Ψ(x,0)∗H^Ψ(x,0)dx, and check that you get the same answer as before.