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
u_{1}(q)=A*q*exp((q[itex]^{2}[/itex])/2)
3. The attempt at a solution
Ok, so I know that the Quantum Harmonic Oscillator Hamiltonian (H[itex]_{QHO}[/itex]) is :
(H[itex]_{QHO}[/itex])=[itex]\frac{1}{2}[/itex][itex]\hbar[/itex]ω(((d^2)/(dq^2))+q^2) and I know that:
(H[itex]_{QHO}[/itex])u_{1}(q)=Eu_{1}(q)
but how do I show that it's an eigenfunction? Simply subbing it into the eqn doesn't appear to help.
Many thanks in advance.
