Quantum statistics: density of states problem

Suske

When you consider a electron L×L×L box, I think I understand how to derive the DOS-spectrum.
Unfortunately, when a small change is made to the problem, I really don't understand what to do, so I probably don't understand the theory at all..

This is the question:

1. The problem statement, all variables and given/known data
Consider a particle with mass M in three dimensions, confined to an L × L square in the x,y-direction, and held in a parabolic potential (1/2)kz^2 in the z-direction.

Q1: Determine the one-particle spectrum for the system
Q2: Calculate the density of one-particle states

2. Relevant equations

D_n(E) = (dΩ_n(E))/d(E) (1)
U = 2 Ʃ_nxƩ_nyƩ_nzε(n) (2)
λ_n = 2L/n (3)
p_n = h/λ (4)

3. The attempt at a solution
I take formula (2) and change it to an integral (I don't know why I can do this, but I think it is for large n, but then it's also strange because in Q1 I have to determine the one-particle spectrum! (or isn't that the spectrum for one-particle, and does it mean something else?)

then for nx and ny I can use (3) and (4) to show that te allowed energies are something like

ε = ((h^2)/(8mL^2))*((nx)^2 + (ny)^2)
then how do I plug in the parabolic potential? I have absolutely no clue at all. Catastrophe in my brain!
I would be very grateful if you would help me with this problem!

Suske

TSny

The Schrodinger equation for this system is separable in the coordinates x, y, z. So, the energy of a single particle state will be a sum of an energy for each variable: E = E_x + E_y + E_z. You've already got the first two terms taken care of. Considering that the motion in the z direction occurs under the potential (1/2)kz², what would be the appropriate expression for E_z?

Suske

Thank you very much!

So then I put V(z,t) = is (1/2)kz^2,

Which gives me Ez_n = (h/2pi)&sqrt(k/m)*(n + 0.5)?
And can I then transform that into (h^2)(k^2)*(n+0.5)/(2m)? <-- this is probably wrong?

Usually with just a particle in a LxLxL box I can 'count' the number of states with a specific energy with a sphere, but how does this apply here?

Thanks!

Suske

TSny

Good. E_z = [itex]\frac{h}{2π}[/itex]ω(n_z + ½) where ω = [itex]\sqrt{k/m}[/itex]. (What are the allowed values of n_z?)

Write out the total energy E = E_x + E_y + E_z in terms of the quantum numbers n_x, n_y, and n_z. That gives the energy spectrum of the one-particle states. You can express it as E = a(n_x² + n_y²) + b(n_z + ½) where a and b are just constants.

When finding the density of one-particle states, I believe you treat the quantum numbers as large compared to 1. So, I think it’s safe to drop the ½ compared to n_z. Then, E = a(n_x ²+ n_y²) + bn_z

As you noted, if you were dealing with a free particle in a cube, you can count the number of states by considering a sphere. This is because in that case E = a(n_x ²+ n_y² + n_z²). So, if you construct a 3D Cartesian coordinate system with axes denoting n_x, n_y, n_z then a specific value of E corresponds to the surface of a sphere in this coordinate system.

But now you have E = a(n_x ²+ n_y²) + bn_z . For a certain value of E, you’ll need to think about the shape of the surface defined by this relation. Also, the allowed values of n_x, n_y, and n_z will restrict you to only a portion of that surface. Then you need to think about how to use the surface to find the total number of states with energy less than or equal to E.

Suske

Thanks again!

Oke. Only positive integers are allowed for n_x, n_y, n_z
Therefor, the surface of the paraboloid (right?) is four times 'too much'

so for a certain E, the amount of states (n_x, n_y, n_z)

equals 1/4 [itex]\int\int[/itex] sqrt (1+(4(a/b)^2)*r^2) r d \Theta dr

I put it in polar coordinates.. but now I don't know what the limits of the integral are? And whether this is correct at all..

I used [itex]\int\int[/itex] sqrt{1 + (derivative to x)^2 + (derivative to y)^2} dA = surface;

Could you please help me find the integral-limits? and when you integrate this, i'm afraid it won't leave anything like (n_x, n_y, n_z), so you can't just convert this back to a 'normal' n..!

After that, you just rewrite your equation so you have n in terms of e, and then you differentiate to e to have your DOS(e)?

Almost there?

Thanks again!

TSny

Right, a paraboloid that may be expressed as E = a(x² + y²) + bz where, for convenience, we let x, y, z denote n_x, n_y, n_z.

To find the volume enclosed by the paraboloid, slice it in circular disks perpendicularly to the z axis. For a certain value of z, show that the radius of the disk is r = ([itex]\frac{E-bz}{a}[/itex])^(1/2).

Integrate to sum up the volumes of all of the disks from z = 0 to z = E/b (right?).

The resultant volume is the total number of quantum states N with energies less than or equal to E except, as you noted, you need to take 1/4 of this volume because we only have quantum states in the region where x, y, z are positive. Also, for electrons we have to allow for two states of spin. So, this introduces a factor of 2.

I got (might be wrong!) N = [itex]\frac{πE^{2}}{4ab}[/itex].

From this you can get the density of states.

TSny

I see that the problem just says you have a particle of mass M. So, we don't know if it has spin. If we ignore spin, then of course N will be half of what I stated.

Suske

Thank you very much, I got the answer completely right! this is great