Question about Eigenfunctions and as solutions to differential equations.

Question

This is a pretty general question but I am struggling to understand why this reasoning is wrong. I recently had a question on proving that the simple harmonic oscillator $$ψ = Ax^ne^{-mwx^2/2ħ}$$ is a solution to the schrodinger equation for very large |x|. I proceeded to solve the eigenvalue problem and ended up with $$E=(n+\frac{1}{2})ħω$$ I got the problem wrong because of this even though I thought the way to prove this would be to show that $ψ(x)$ is an eigenfunction of the Energy operator. I know there is another way to prove it by negating all of the portion of the schrodinger equation for large $|x|$besides $$\frac{1}{2}mw^2x^2$$ I contacted the professor but haven't heard back. I appreciate your time. I think all of the equations above are correct but don't have book in front of me.

Answer 1

Schroedingers equation for the harmonic oscillator in time-independent form is:

$$-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2}+\frac{1}{2}m\omega^2x^2\psi=E\psi,$$

and its solutions can be compactly written as:

$$\psi= ae^{-bx^2}H_n(cx).$$

In general $H_n$ is a nontrivial polynomial of degree $n$, a fact that can be easily derived from it's representation $H_n(x):=(-1)^ne^{x^2}\frac{d^n}{dx^n}e^{-x^2}$. All you need to do is to work out the leading order term infront of $x^n$ for $\psi$, which from the last formula is seen to be $(2c)^n$. This implies that at large $|x|$, the $x^n$ term will dominate the wave function.