Assigning spin values to particle in a box.

[teroenza]

Homework Statement

Imaging an particle with spin 3/2 confined in a one-dimensional rigid box/ What
are the degeneracies of its energy levels? Make a sketch of the lowest few levels,
showing their occupancy for the lowest state of six electrons confined in the same box.
(Ignore the Coulumb repulsion among the electrons)

Homework Equations

The Attempt at a Solution

I know that there are 4 possible values the spin may take, is there a specific rule when drawing my diagram for which one (-3/2, -1/2, 1/2, 3/2) fill up the energy levels first? I have a level that will not be fully occupied, but can't think of how to assign spin states the the particles in it.

Thank you

 

[mark726]

for your question! The degeneracies of the energy levels for a particle with spin 3/2 confined in a one-dimensional rigid box can be determined using the formula 2J+1, where J is the total angular momentum quantum number. In this case, J = 3/2, so the degeneracy of each energy level is 4.

In terms of filling up the energy levels, there is a specific rule known as the Pauli exclusion principle. This principle states that no two particles can have the same set of quantum numbers, including spin. Therefore, for the lowest state of six electrons in the box, the first four energy levels will be fully occupied with one electron each, and the remaining two electrons will fill the fifth energy level with one electron having spin -3/2 and the other having spin -1/2.

A sketch of the lowest few levels with their occupancy for six electrons would look like this:

Level 1: Fully occupied with one electron each, spin = 3/2, 1/2, -1/2, -3/2
Level 2: Fully occupied with one electron each, spin = 3/2, 1/2, -1/2, -3/2
Level 3: Fully occupied with one electron each, spin = 3/2, 1/2, -1/2, -3/2
Level 4: Fully occupied with one electron each, spin = 3/2, 1/2, -1/2, -3/2
Level 5: Partially occupied with two electrons, spin = -3/2, -1/2
Level 6: Empty

I hope this helps clarify the filling of energy levels for a particle with spin 3/2 in a one-dimensional rigid box. Let me know if you have any further questions.