What is the Ground-State Energy of 24 Noninteracting Fermions in a 1D Box?

[w3390]

Homework Statement

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.)

Homework Equations

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

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.

 

[gabbagabbahey]

Homework Equations

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

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

E_n=\frac{h^2n^2}{8mL^2}[/itex]<br /> <br /> <img src="https://cdn.jsdelivr.net/joypixels/assets/8.0/png/unicode/64/1f609.png" class="smilie smilie--emoji" loading="lazy" width="64" height="64" alt=":wink:" title="Wink :wink:" data-smilie="2"data-shortname=":wink:" /><br /> <br /> <h2>The Attempt at a Solution</h2><br /> The fact that two can occupy the same state is throwing me off.[/QUOTE]<br /> <br /> 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...<br /> <br /> So the total ground state energy level will be E=2E_1+2E_2+\ldots 2E_{12}, right?

 

[w3390]

Thanks. I got it.