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?]