Hamiltonian of the Quantum Harmonic Oscillator-Eigenfunction & Eigenvalue

[Calcifur]

Homework Statement

Show that the equation below is an eigenfunction for the Quantum Harmonic Oscillator Hamiltonian and find its corresponding eigenvalue.

Homework Equations

u₁(q)=A*q*exp((-q^{2})/2)

The Attempt at a Solution

Ok, so I know that the Quantum Harmonic Oscillator Hamiltonian (H_{QHO}) is :
(H_{QHO})=\frac{1}{2}\hbarω(((-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.

Many thanks in advance.

 

[vela]

Simply subbing it into the eqn doesn't appear to help.

It should. Either that or you need to solve the differential equation, which is a much harder task.

Show us what you got when you plugged u₁ into the equation.

 

[Calcifur]

Show us what you got when you plugged u₁ into the equation.

Ok, so here goes:

H_QHO*U₁(q)=E*U₁(q)

\frac{1}{2}\hbarω(\frac{-d^{2}}{dq^{2}}+q^{2}).A.q exp(\frac{-q^{2}}{2})=E*U₁(q)

\frac{A}{2}\hbarω(-\frac{d}{dq}\frac{d}{dq}(q.exp(\frac{-q^{2}}{2}))+(q^{3})exp(\frac{-q^{2}}{2}))=E*U₁(q)

which eventually comes to:

(q^{3}+q^{2}-2q-1)exp(\frac{-q^{2}}{2})\frac{A}{2}\hbarω=E*U₁(q)

So does this mean that : (q^{3})+q^{2}-2q-1) is the corresponding eigenvalue?

Is my method correct?

Many thanks.

 

[vela]

You must have calculated the second derivative incorrectly. You should get
$$u_1''(q) = (q^3-3q)e^{-q^2/q}.$$

 

[Calcifur]

You must have calculated the second derivative incorrectly.

Thanks, my mistake. So I now have:

\frac{\hbarω}{2}A.exp(\frac{-q^{2}}{2})(-q^{3}-q^{2}+3q)=E.U_{1}(q).

So is -q^{3}-q^{2}+3q the corresponding eigenvalue? Or can I simplify even further?

Many thanks.

 

[vela]

That's still not correct. The eigenvalue is a constant. It can't depend on q.

 

[Calcifur]

I think I've figured it out now. Many thanks Vela