The purpose of this project is to help undergraduate students better visualize the behavior of identical particles, due to their spin-statistics. So we present here the following, very simple, physical model:
The energy levels in the chosen potential, will be defined by the known expression for energy levels of bound states of some specific particle (i.e: an electron). They will not change due to characteristics (such as mass) of the different particles that populate them.
So, for example, in the case of the 2D infinite square potential well, the energy levels will be given by the expression:
) = (ℏ2π2)/(2mL2)(nx2
+ ny2) for nx
1, 2, ...}
is some constant.
From the above equation we can see that the “ground state”, in this case, has the energy E0 ≡ E(1, 1) = 2C
and we can express any other energy level in terms of this “ground state”. For instance: E(3, 2) = (13)/(2)E0
Also, all interactions between particles in the potential well, will not be considered at all.
Only the spin-statistics of a particle will be considered when populating the energy levels, so that identical fermions
that have the same spatial wave function will be in different spin states (anti-symmetrical), and identical bosons
with the same spatial wave function will be in the same spin states (symmetrical).
Usually, in reality, when a magnetic field is present, the spin degeneracy in the energy levels is removed in such a way, that the state with the spin quantum number ms = s
has the lowest energy and ms = − s
the highest energy. But in this model there is no magnetic field, so there is a spin degeneracy. Nevertheless, the population of the energy levels in the chosen potential will be done as if there is a magnetic field present, in the following manner:
For each energy level (with quantum number(s) n⃗ = (nx, ny)
in the 2D case and n⃗ = (n)
in the 1D case), the “preferred” spin state will be the “least energetic” ψ(n⃗, ms = s)
state, and the energy level will be populated in an ascending manner (spin-wise), up to the “most energetic” ψ(n⃗, ms = − s)
So, particles that are fermions
will populate each energy level in the manner described above. But for bosons
, since they “prefer” to be in the same “least energetic” state, they will all be in the “ground state” ψ(n⃗, ms = s)
, where E(n⃗) = E0
Additionally, if the energy level is degenerate, its population will be done in the above manner, but for each spin quantum number, an amount of particles that match the level’s degeneracy will be populated. For example, in the case of eq. (1↑
), E(2, 1) = E(1, 2)
, so for a fermion with spin s = (1)/(2)
, this level’s population will be done in the following order: ψ((2, 1), ms = (1)/(2))
then ψ((1, 2), ms = (1)/(2))
then ψ((2, 1), ms = − (1)/(2))
and then ψ((1, 2), ms = − (1)/(2))