(a) Show by direct substitution in the Schrodinger equation for the one-dimensional harmonic oscillator that the wave function ψ1 (x) = A1xe –a2x2/2, where a2 = mw/h, is a solution with energy corresponding to n = 1 in Eq. (40.22).
(b) Find the normalization constant A,.
(c) Show that the probability density has a minimum at x = 0 and maxima at x = ± 1/a, corresponding to the classical turning points for the ground state n = 0.