I am to calculate the number of states in a 3Dcubic potential well with impenetrable walls that have energy less than or equal to E

We know (E_n)=[(hbar^2*pi^2)/(2mL^2)] [(n_x)^2+(n_y)^2+(n_z)^2]
=[(hbar^2*pi^2)/(2mL^2)] n^2


Then, we can evaluate the integral n(E')dE' for E'=0 to E'=E

I am not sure if this would give the correct answer.Can anyone please help?

Hi kolahal_b,

Yes, this is the right idea. Usually you are dealing with systems with fairly large energies, so there are very many states, and you can approximate counting the states by measuring volume. In the 3 dimensional space with coordinates n_x, n_y, and n_z, the every point with integer coordinates represents a state, and each such point can be thought of as sitting at the center of a unit cube. The total number of states in a given region is then (to a very good approximation, when the region is large) the same as the number of unit cubes, which is the same as the volume of the region.

Therefore, the number of states is simply the volume of a sphere (n_x)^2 + n_y^2 + n_z^2 < N^2, which is 4/3*pi*N^2. Since N^2 can be calculated from the energy, this should answer the question.

Hope that helps!

--Stuart Anderson

kolahal_b, I feel you lean on forums too much. You are constantly asking questions which should be straight forward to you. You ask on these forums, SciForums and NRich. And those are just 3 I post on. You don't show you make much of an attempt on any of the questions. You've been asking what amount to entire courses worth of homework for 1st and 2nd year levels.

Asking for help is fine, provided it's used as a last resort. You seem to be using it as a first resort.

Correction: I meant 4/3*pi*N^3, cubed, not squared.

Dear mr homm,

What I did previously may have meaning only for a large system...where E and N can be thought of as continuous.

∫N(E) dE= (some constant) (∫√E dE)

Should this integral not do?Of course we can represent the energy in terms of volume V...but that would involve N

The energy states can be represented as (each bracketted term represent a state)

E(1)-(1 1 1)

E(2)-(2 1 1)(1 2 1)(1 1 2)

E(3)-(2 2 1)(2 1 2)(1 2 2)

E(4)-(3 1 1)(1 3 1)(1 1 3)

E(5)-(2 2 2)

E(6)-(3 2 1)(1 3 2)(2 1 3)(3 1 2)(2 3 1)(1 2 3)

So if we are asked how many states are there with energy less than or equal to E(3),the answer should be 1+3+3=7

Given all the values of h,m,L, this should conform with the integral apprach.The upper limit must be E(3) and the lower limit is E(1)

Sitting at the centre of the cube, we would see the axes of n_x,n_y and n_z projected in perpendicular direction...It appears that this is also correct...as we look around, we see the states are distributed within a sphere...

But do the answers match?I used the formula...it does not match!!!

Am I doing something wrong somewhere?

AlphaNeumeric,I do not wish to give response here---this is not the appropriate place.Check you email and PM inbox.

OK,now I see mr homm's idea.We are excluding those states whose co-ordinates combine to form a radius greater than that permissible by the energy.

My approach was to calculate directly the number of states. The direct value of N gives the number of states with the specified energy E...but it does not include the states with lower values of energy.Therefore I tried to put the problem into an integral...

The integration should not be valid here because n(E) is not a continuous variable.