Harmonic oscillator verify both solutions to schrodinger equation

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Homework Statement

a particle of mass m moving in one dimension has potential energy
V(x)=0.5m [[omega(subscript0)]^2] x^2

verify that
psi0 (proportional to) exp [(-m omega0 x^2)/2 h bar]
and
psi1 (proportional to) exp [(-m omega0 x^2)/2 h bar]

are both solutions of the time independent schrodinger equation, and find the corresponding eigenvalues

Homework Equations

SE:
(-hbar^2)/2m d2psi/dx^2 +0.5m [[omega(subscript0)]^2] (x^2)psi=E psi

The Attempt at a Solution

(-hbar^2)/2m * (constant^2)([(-m^2 omega0^2)/(4 (h bar)^2)]exp [(-m omega0 x^2)/2 h bar]+0.5m [[omega(subscript0)]^2] (x^2)*(constant)exp [(-m omega0 x^2)/2 h bar]=E exp [(-m omega0 x^2)/2 h bar]

 

[jbsteele]

E=(-hbar^2)/(2m*constant^2)The same equation can be used to calculate E for psi1. Therefore, both psi0 and psi1 are solutions to the time independent schrodinger equation with the same corresponding eigenvalue E