Harmonics Oscillator Homework: Solving Schrodinger Equation

[Winzer]

Homework Statement

Show:
\psi_1=N x e^{-\frac{x^2}{\sigma}} is an eigenfunction of the total energy operator(H).

Homework Equations

psi=N x exp[-x^2/K]

The Attempt at a Solution

I plugged in the above to the Schrodinger Equation-time indep. for Harmonic oscillator but I keep getting an x^2 dependence inside the calculation. Do I not plug this in?

 

[Winzer]

oh, I should be putting psi into d^2(psi)/dy^2 + (e-y^2) psi =0 where y=sqrt(mw/h) x. the asymtotic solution: e-->0. This is what is I should plug it into right?