A (probably simple) quantum problem - energy eigenfunctions?

[jeebs]

Hi people,
I have this problem to do, and its only worth one mark which makes me think it must be easy, but our lecturer has not taught us very well at all, never explains anything.

Anyway, there's a particle confined in an infinite potential well within the region -L/2 < x < L/2, where the potential is zero. At a certain time it is described by the wavefunction:
psi(x) = Acos(pi.x/L) + 2Asin(2pi.x/L). [where A = sqrt(2/5L)]

I am supposed to write down psi(x) in terms of the eigenfunctions of energy.

First off, I don't really know what an eigenfunction of energy is, or is supposed to look like, so i don't know what i am supposed to be writing psi(x) in terms of. All i have really done with this question is write down the time-independent Schrödinger equation with the V(x) term missing, since the potential is zero, so i have:

-H/2m.(d^2/dx^2)psi(x) = E(psi(x)) [where H = h/2 pi]

Other than that i have just aimlessly messed around with the equation above.
Can anybody help me out? I am desperate for help with this stuff, its starting to frustrate me now.
Thank you.

 

[malawi_glenn]

It is "just" a second order differential equation which needs to fullfil boundary conditions psi(L/2) = psi(-L/2) = 0

You should have studied second order differential equations before studying quantum mechanics.

 

[turin]

You should have studied second order differential equations before studying quantum mechanics.

... or during.

jeebs,

That there time-independent Schroedinger equation is the eigenvalue problem. "An eigenfunction of the energy" should really say "an eigenfunction of the Hamiltonian". And so, what you have written there is that the operator, H, acts on the wave function, psi, by simply multiplying it by some value, called the energy, E. The eigenvalue problem is especially profound in quantum mechanics, more so than in any other subject, because in quantum mechanics only the normalized projective states matter, so multiplication by a constant doesn't change any physical quantity.

Anyway, the point is, the solution to the time-independent Schroedinger equation is the eigenfunction of the energy. Then, like malawi_glenn said ...

 

[jeebs]

aah thanks for your answers... I'm getting there with this quantum mechanics business, little by little... i understand stuff a bit more every day.