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:

Homework Statement

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

Homework 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)

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 = $\frac{h}{2π}$ω(n_z + ½) where ω = $\sqrt{k/m}$. (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 $\int\int$ 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 $\int\int$ 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 = ($\frac{E-bz}{a}$)^(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 = $\frac{πE^{2}}{4ab}$.

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