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## Hamiltonian of the Quantum Harmonic Oscillator-Eigenfunction & Eigenvalue

1. The problem statement, all variables and given/known data
Show that the equation below is an eigenfunction for the Quantum Harmonic Oscillator Hamiltonian and find its corresponding eigenvalue.

2. Relevant equations
u1(q)=A*q*exp((-q$^{2}$)/2)

3. The attempt at a solution
Ok, so I know that the Quantum Harmonic Oscillator Hamiltonian (H$_{QHO}$) is :
(H$_{QHO}$)=$\frac{1}{2}$$\hbar$ω(((-d^2)/(dq^2))+q^2) and I know that:
(H$_{QHO}$)u1(q)=Eu1(q)

but how do I show that it's an eigenfunction? Simply subbing it into the eqn doesn't appear to help.