N spin 1/2 particles in 3D harmonic oscillator potential

[darthmonkey]

Homework Statement

The 3-dimensional harmonic oscillator potential holds N identical non-reacting spin 1/2 particles

a)How many particles are needed to fill the low lying states through E=(3+3/2)$\bar{h}$ω

b)What is the total energy of the system

c)what is the fermi energy

Homework Equations

E=(n+3/2)$\bar{h}$ω
n=nx+ny+nz=3

The Attempt at a Solution

a)(1,0,0)*3,(1,1,0)*3,(2,0,0)*3,(1,1,1),(2,1,0)*6,(3,0,0)*3
19 states for 38 total particles
b)6E1+12E2+20E3=147$\bar{h}$ω
c)Ef=(3+3/2)$\bar{h}$ω
-------------------------------------------------------------------
So I'm pretty sure I did part a correct. If so part b should just be the sum of the number of particles at each energy right?

For part c isn't the given energy the fermi energy since I'm finding all the states below it?

I'm not really sure about my reasoning on the part b and c

 

[mark726]

. Please let me know if I'm on the right track.

Hello,

First of all, your solution for part a is correct. You have correctly counted the number of particles needed to fill the low lying states through the given energy.

For part b, you are correct that the total energy of the system should be the sum of the energies of all the particles in the system. However, you have to be careful with the notation. The total energy of the system should be given as the sum of the individual energies of each particle, not the number of particles at each energy. So the correct expression for the total energy would be:

Etotal = 6E1 + 12E2 + 20E3

Additionally, you should also include the energy of the particles at the (1,0,0)*3 state, which is E0 = 3/2 * hω. So the final expression for the total energy would be:

Etotal = E0 + 6E1 + 12E2 + 20E3 = (3+3/2)hω + 6(1+3/2)hω + 12(2+3/2)hω + 20(3+3/2)hω = 147hω

For part c, you are correct that the given energy is the Fermi energy. However, the expression for the Fermi energy is given as:

Ef = (nx+ny+nz+3/2)hω

So the correct expression for the Fermi energy would be:

Ef = (3+3/2)hω = 9/2hω

I hope this helps clarify your reasoning for parts b and c. Good job on part a!