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}$$\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.

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

$\frac{A}{2}$$\hbar$ω(-$\frac{d}{dq}$$\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}$$\hbar$ω=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