Understanding Eigenfunctions and Eigenvalues in Quantum Mechanics

[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?

 

[olgranpappy]

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?

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?

 

[varunag]

In case you are not aware of eigenfunctions:
http://mathworld.wolfram.com/Eigenvalue.html"

 

[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.

 

[gabbagabbahey]

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