Use Gaussian multiplication on the Hermite polynomials in the attached document. These give the un-normalized wave functions for the levels of the harmonic oscillator.
The 3rd and 4th Hermite polynomials are, respectively:
H_2(x) = 4x^2 - 2
H_3(x) = 8x^3 - 12x
When multiplies by the Gaussian, e^-ax^2 /2 (please see the attached file), these give the un-normalized wave functions for the v = 2 and 3 levels of the quantum harmonic oscillator.
a. Show that the resulting wave functions are orthogonal to each other.
b. Find the normalization constants for the two wave functions.