Electron in a 1-D box. Quantum Numbers

[teroenza]

Homework Statement

Imaging an electron (s=1/2) 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

One-D particle in a box energies

E= $\frac{pi*2*hbar^2}{2mL^2}$*n^2

m is the mass
L is the box's length

The Attempt at a Solution

I think my biggest question is, do the l and $m_{l}$ exist for a particle in a box. In atoms the electrons are "orbiting" and have these numbers associated with their angular momentum, but not in a box.

I know spin in intrinsic and have accounted for it. I know the particle's energy depends only on n, and so all sub-levels associated with one n value are at the same energy and thus degenerate.

My original solution has a table listing all possible n, l, m_l, and spin, but I now believe this is wrong. I think it should now perhaps include just n and spin. The exclusion principle will guide how I draw the electrons that are allowed to exist in the energies.

Thank you

 

[mark726]

for your post! You are correct in your understanding that in a one-dimensional rigid box, the energy levels only depend on the principle quantum number (n) and the spin of the particle. This is because in a one-dimensional system, there is no angular momentum (l) or magnetic quantum number (m_l) associated with the particle's motion.

To answer your question about the degeneracy of the energy levels, we can use the energy equation you provided. Since the energy levels only depend on n and spin, we can see that for each n value, there are two possible spin states: spin up and spin down. This means that each energy level is two-fold degenerate.

In the case of six electrons in the same box, we can use the Pauli exclusion principle to determine the occupancy of the lowest energy levels. The first two electrons will occupy the lowest energy level (n=1) with opposite spin states. The next two electrons will occupy the second energy level (n=2) with opposite spin states, and so on. This results in a total of six electrons occupying the lowest three energy levels, with two electrons in each energy level.

I have attached a sketch of the energy levels and their occupancy for six electrons in a one-dimensional rigid box. I hope this helps clarify any confusion and good luck with your studies!