Homework Help: Ladder operators and a 1-D QHO

1. The problem statement, all variables and given/known data



2. Relevant equations



3. The attempt at a solution
I solved part a) correctly, I believe, giving me

ψ = e[itex]^{-(√(km)/\hbar)x^{2}}[/itex]

and a normalization constant A = ((π[itex]\hbar[/itex])/(km))[itex]^{-1/4}[/itex]

I'm having difficulty with part b. I'm not exactly sure how I create a linear combination. I think I'm supposed to write the linear combination like so:

a + a[itex]^{+}[/itex] = 2(k/2)[itex]^{1/2}[/itex]x

(a + a[itex]^{+}[/itex])(2k[itex]^{-1/2}[/itex]) = x

However, I am not sure how I use the operator methods to show that xψ0 is the first excited state.

If anyone could nudge me in the right direction, I'd be eternally grateful! Thank you.