# Homework Help: Ground-state energy of fermions

1. Oct 11, 2009

### w3390

1. The problem statement, all variables and given/known data

What is the ground-state energy of 24 identical noninteracting fermions in a one-dimensional box of length L? (Because the quantum number associated with spin can have two values, each spatial state can be occupied by two fermions.) (Use h for Planck's constant, m for the mass, and L as necessary.)

2. Relevant equations

E=h^2/(8mL^2)[n1+n2+n3+....n24]

3. The attempt at a solution

Since the question states that each spatial state can be occupied by two fermions, I thought it would be 48h^2/8mL^2, simplifying to 6h^2/mL^2. However, this is incorrect. Any help would be much appreciated. The fact that two can occupy the same state is throwing me off.

2. Oct 11, 2009

### gabbagabbahey

I don't understand this equation; I thought the energy levels of a particle in a box were proportional to $n^2$:

[tex]E_n=\frac{h^2n^2}{8mL^2}[/itex]

3. The attempt at a solution
The fact that two can occupy the same state is throwing me off.[/QUOTE]

Well, the first two fermions can occupy the $n=1$ state, but the next two will have to go in a higher energy level, $n=2$, and the next two will have to go in the $n=3$ level, and so on...

So the total ground state energy level will be $E=2E_1+2E_2+\ldots 2E_{12}$, right?

3. Oct 11, 2009

### w3390

Thanks. I got it.