Normalization of harmonic-oscillator wave functions.
Verify that the wave functions for the n = 0 and n = 1 states of the SHO are correctly normalized as given in Table 4-1. Use the following outline or some other method.
(a) To evaluate -x∫xe-x2dx, note that -x∫xdx -x∫xdy e-(x2-y2) is equal to o∫2Π dθ o∫xdr re-r2
(b) Adapt your result from (a) to verify the normalization of the ground state of SHO
(c) Examine the normalization condition as applied to the second-state wave function. Can you see a way to apply the results of (a) and (b) to evaluate the integral?
[Hint: What happens if you regard the SHO parameter a as a variable and differentiate the ground-state normalization integral with respect to it?]