Calculating States in a 3D Cubic Potential Well

[neelakash]

Homework Statement

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

Homework Equations

$$\ E_n=\frac{\hbar^2\pi^2}{\ 2 \ m \ a^2}\ (\ {n_x}^2 + \ {n_y}^2 + \ {n_z}^2)$$

The Attempt at a Solution

We may denote $$\ (\ {n_x}^2 + \ {n_y}^2 + \ {n_z}^2)=\ n^2$$
and express n in terms of E_n

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?

 

[pam]

E and n are discrete. No integral, just add up combinations of nx, ny, nz to give n.

 

[neelakash]

E and n are discrete. No integral, just add up combinations of nx, ny, nz to give n.

I see.But should I sum them?

 

[Avodyne]

Your method is not correct.

The answer is the number of triples (nx,ny,nz) of positive integers such that nx²+ny²+nz² is less than a certain constant times E.

Counting these exactly is difficult. But there is a simple way to do it approximately.

Consider (nx,ny,nz) as coordinates of a point in a three-dimensional space. Any point that is inside a one-eighth sphere of radius (constant)E counts, and any point outside does not count. Now consider that, on average, there is one point per unit volume of this one-eighth sphere ...

 

[neelakash]

Hmmm...
Then it is easier to calculate the volume of the sphere (since there is one state per unit volume) instead of calculating the no. of states.

the squared maximum radius is $$\ n^2=\frac{\ E_n \ 2 \ m \ L^2}{\hbar^2 \pi^2}$$

The co-ordinates $$\ n_x,\ n_y,\ n_z$$ that results in a greater radius,also involves greater energy.Hence,they are excluded.

So, the required number of states is $$\frac{4}{3}\pi\ n^3$$

what was wrong with my approach?

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.