# Eigenfunction concept

1. Sep 28, 2008

### samdiah

I am a seond year Quantum Chemistry student. I am having a hard time understanding these concepts. I was wondering I can get help in this concept.

How can it be demonstrate mathematically in the Hamiltonian operator that the function
φ(x) = A sin(2x) + B cos(2x)

is an eigenfunction of the Hamiltonian operator:
H=-h^2 d^2
2m dx^2

What is the eigenvalue equal to?

2. Sep 28, 2008

### olgranpappy

Yikes. That is very difficult to read. You should try and learn some TeX. It is good for the soul, and for the typesetting. for example:

$$H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}\;.$$

I believe that If you click on the above equation it will show you the TeX source that I used to write the equation in such a pretty manner.

Anyways. In order to do what you want you need to take the derivative of
$$A\sin(2 x)+B\cos(2 x)$$
twice.

what do you get?

Then multiply by
$$-\frac{\hbar^2}{2m}\;.$$

That is what the symbols on the right hand side of the equation for $H$ are instructing you to do.

What is the end result?

Is it proportional to
$$A\sin(2 x)+B\cos(2 x)$$
?

What is the proportionality constant?

3. Sep 29, 2008

### varunag

In case you are not aware of eigenfunctions:
Eigenvalue -- Wolfram Mathworld

4. Sep 30, 2008

### samdiah

I found the two derivatives and I found that the function φ(x) = A sin(2x) + B cos(2x)
is an eigenfunction of the Hamiltonian operator:

H=-h2 d2
2m dx2

and the proportionality constant is
2h2
m

Can someone confirm with me if this is right or what did I do wrong?

Thanks so much for all the help.

5. Oct 1, 2008

### gabbagabbahey

Yes, you have the correct eigenvalue (proportionality constant).