Showing f is a solution to quantum oscillator SWE

[infinitylord]

Homework Statement

For a 1-dimensional simple harmonic oscillator, the Hamiltonian operator is of the form:
H = -ħ²/2m ∂_xx + 1/2 mω²x²
and
Hψ_n = E_nψ_n = (n+1/2)ħωψ_n

where ψ_n is the wave function of the nth state.

defining a new function f_n to be:

f_n = xψ_n + ħ/mω ∂_xψ_n

show that f_n is a solution to the low energy SWE. I.e. that:

Hf_n = (E_n - ħω)f_n

The Attempt at a Solution

I know that:
ψ_n = C_ne^-mωx2/2H_n(x)
where H_n(x) is an nth order Hermite Polynomial.
I was wondering what the procedure for determining this would be. I'm assuming I could plug this expression for ψ_n into the expression for f_n, and then apply the Hamiltonian operator. But I'm not sure how that would simplify, and it also seems very complicated to plug in. Any help would be appreciated

 

[TSny]

I believe you are meant to let the operator H act on the expression for f_n and show that Hf_n reduces to (E_n - ħω)f_n.

It will require a fair amount of manipulation and use of the fact that Hψ_n = E_nψ_n. But nothing else is needed.