Finding Highest Energy for Fermions and Bosons in a Box

• baubletop
In summary, the problem involves determining the n value for the highest energy electrons and neutrons when 12 of each are placed in a box. For fermions, the highest energy level would be n=3 due to the Pauli Exclusion Principle. For bosons, all 12 can occupy the ground state.
baubletop

Homework Statement

a. Electrons and neutrons are fermions. Put 12 of them (6 each) in a box, and determine the n value for the ones with the highest energy.
b. Do the same for 12 bosons (6 are pi zero bosons and 6 are alpha particles).

Homework Equations

En = (h2n2)/(8mL2)

The Attempt at a Solution

I'm not even sure how to approach this problem. What is it asking? How do I know which have the highest energy? (Neutrons in general have a higher rest energy than electrons, but I don't know if that's at all relevant.)
My only hunch is that the Pauli Exclusion Principle is involved (like I could have 2 electrons in ground state, 2 in n=2 state, 2 in n=3 state, etc. but that doesn't seem like what the question is asking).

What is special about the way Fermions fill energy levels?

Two fermions can't occupy the same quantum state. So if there are 6 electrons the highest energy level would be n=3 (same for the neutrons).

So you have just answered question (a).
What is the difference with bosons?

Bosons aren't subject to the Pauli Exclusion Principle so they can all occupy the ground state.

I wasn't sure if I was over- or under-thinking the problem, it seemed too easy...

I know - it throws you out when it's not something subtle or tricky.
But it's good practice.

1. What is the difference between fermions and bosons?

Fermions and bosons are two types of particles that make up the building blocks of matter. The main difference between them is their spin, which is an intrinsic property of particles. Fermions have half-integer spin (1/2, 3/2, etc.), while bosons have integer spin (0, 1, 2, etc.). This difference in spin leads to different behavior and properties of these particles.

2. How does the energy of fermions in a box compare to that of bosons?

In a box, fermions and bosons have different energy levels due to the Pauli exclusion principle, which states that no two fermions can occupy the same quantum state. This means that the energy levels for fermions are spaced closer together compared to those of bosons. Additionally, fermions have ground state energy, while bosons do not.

3. What determines the highest energy state for fermions and bosons in a box?

The highest energy state for fermions and bosons in a box is determined by the number of particles and the size of the box. The energy of the highest state for fermions is determined by the Pauli exclusion principle, which restricts the number of particles that can occupy each energy level. For bosons, the energy of the highest state is determined by the size of the box and the number of particles, as there is no restriction on the number of particles that can occupy a single energy level.

4. How is the energy of fermions and bosons in a box calculated?

The energy of fermions and bosons in a box can be calculated using the Schrödinger equation, which is a mathematical equation that describes the behavior of quantum particles. By solving this equation, we can determine the energy levels and the corresponding energies for fermions and bosons in a box.

5. How does temperature affect the energy of fermions and bosons in a box?

Temperature affects the energy of fermions and bosons in a box by causing the particles to occupy higher energy levels. As the temperature increases, more energy is available for the particles to occupy higher states, leading to an increase in their energy. This effect is more prominent for fermions, as they have a ground state energy, while bosons do not. Additionally, at high enough temperatures, bosons can occupy the same energy level, leading to a phenomenon called Bose-Einstein condensation.

Replies
1
Views
937
Replies
30
Views
6K
Replies
1
Views
2K
Replies
2
Views
2K
Replies
5
Views
2K
Replies
12
Views
546
Replies
17
Views
2K
Replies
2
Views
2K
Replies
4
Views
2K
Replies
3
Views
2K