[illu5tri0us]

when tryign to find the first 10 energy states of a particle in a 2d box, is it safe to assume that the difference between the energy levels in the x and the y direction can only b 1?! or was that stated somewhere andi missed it?

[Maz]

difference between the energy level in the x and y direction can b 1? 

I am not sure what you mean by that.  It sounds like you haven't solved the schrod. eq. for a 2D PIB yet. 

I will assume that you have solved, or have seen the derivation of the energy eigen state for the inf. sq. well (a.k.a. 1D PIB). 

Set up your schrod. equation again in the same fashion, but instead of only using (d/dx)², rewrite it as del² operator.  Note, however, that your del operator only spans the x and y directions. 

So your sch. eq. will look like hbar^2/2m*(d²/dx² + d²/dy²)*psi(x,y) + V(x,y)*psi(x,y) = E*psi. 

But you know (PIB) that V(x,y) = 0 for x,y inside the area of interest.

Now simply solve your second order differential equation.  Assume that psi is separable i.e., that it can be written as the product of two separate functions A(x)*B(y). 

Solve using separation of variables.  It really turns out quite nicely.  In fact, you could probably have guessed the soln from the 1D PIB, and you will be able to guess the soln to the 3D PIB after this, i'm sure. 

[illu5tri0us]

yah that was part of the process that i've already done. thanks. i end up getting something for E, which are the eigen values (correct me if i'm wrong):

 E total = E1 + E2 =- h^2/8m (n^2/ 9L^2 + s^2/L^2)

where n and s are the energy states of x and y
L is the height of the box and 3L is the width of the box.

i was asked to find the energies and degeneracies of the first ten states. then i started doing combos of when n=1 & s=1, n=2 & s=1, n=1 & s=2... etc...

however, my problem is... am i allowed to say n=1 & s=3, where the "difference" between the states is 2 as opposed to n=2 & s=3 where the difference between them is 2. i guess i can better illustrate:

n       s        or       n  s
1      1                       
1      2
2      1
1      3                   2    3             ?

[Maz]

I don't believe there is any reason you can't hold n at say 1 and push s all the way up to 10.  However, my guess is that when the question asked for the 'first 10', the teacher wanted the lowest 10.  Obviously 1² + 10² will NOT be lower then 2² + 2². (Extreme example, but you get the idea)

Also, you're energy doesn't quite look right.  I think you dropped a pi².  Also, the /8m is surprising to me.  You may want to double check that, although it could be that you are taking the limits at areas I'm not used to. 

Perhaps you're going -a/2 to a/2? or something like that?  But that isn't a really big deal, I was expecting it to be /2m.

[illu5tri0us]

yah i think your response makes sense to me about the lowest energies.

the reason taht i got what E was is that h is not h bar.
E = -h^2/8m ... = h bar^2 pi ^2 / 2m

because hbar = h/2pi

[Maz]

lol.  And here i thought you were just being lazy about typing hbar.  that makes sense then.  Turns out I was the one being lazy.  I should have checked that.  heh.

[illu5tri0us]

haha... i fi were u i wouldn't have been thinking about that either... but

it is the lowest energy that i'm trying to find. so it can b 1 to 3 and what not. just got that clearified today. that would mean, i would have to do a lot more calculations...