Degenerate states of 2 particles in a 1D harmonic oscillator potential

• captainjack2000
In summary, the two particles have the same number of degeneracies, but for fermions there is no pauli exclusion principle.
captainjack2000

Homework Statement

"Two non-interacting particles are placed in a one-dimensional harmonic oscillator potential. What are the degeneracies of the two lowest energy states of the system if the particles are
a)identical spinless bosons
b)identical spin-1/2 fermions?

The Attempt at a Solution

I think degeneracy is equal to n^2
for b) two fermions can be in the same spatial state if they are in different spin states. one would be in spin up and spin down so this would affect the degeracy. Apart from that..I'm not sure

exactly so there is no degeneracy for fermions (at least for the wavefunction of the entire state)

and for bosons there is no such pauli exclusion principle.

i would like someone to confirm however, if we are asked for the degeneracy are we meant to write a number i.e. 2 for the boson case or are we expected to write a spiel about pauli exclusion etc.

also isn't degeneracy given by something like $g_n =\sum_{l=0}^n l(l+1)$ - can someone tell me whether this is true or not?

Why is there no degeneracy for fermions - is that just because there are only two particles? What would the degeneracy be if there were three particles? What do you mean at least for the wavefunction of the entire state?

didn't read your question properly. my above argument is tru for the ground state only. but in the first excited state it's possible to have two fermions with different wavefunctions and still have the same degenaracy. i.e. if you use the symmetric spatial wavefrunction then you need to put it in a spin singlet state - can you figure out the other possibility?

actually it looks like the degeneracy relation is $g_n=\sum_{l=0}^{n-1} (2l+1)=n^2$

The number of degeneracies is just the number of configurations you can have to get a specific energy. How many different ways can you arrange 2 fermions/bosons to get them to have the lowest total energy?

@Latentcorpse: It looks like your formula is for the degeneracies of the hydrogen atom (or a hydrogenic atom), I don't think that applies to this problem...? (Correct me if I'm wrong)

1. What are degenerate states in a 1D harmonic oscillator potential?

Degenerate states refer to the situation where two or more energy levels have the same energy value. In the case of a 1D harmonic oscillator potential, this means that two or more states have the same energy value, but different quantum numbers.

2. How do degenerate states arise in a 1D harmonic oscillator potential?

Degenerate states in a 1D harmonic oscillator potential arise when the energy spacing between adjacent energy levels becomes smaller and smaller, eventually becoming zero. This can happen when the potential well becomes very deep, or when the quantum number n (representing the energy level) becomes very large.

3. What is the significance of degenerate states in a 1D harmonic oscillator potential?

Degenerate states have important implications for the behavior of particles in a 1D harmonic oscillator potential. For example, particles in degenerate states have the same energy, but different wavefunctions, leading to different probabilities of finding the particle at different positions.

4. How are degenerate states of two particles related in a 1D harmonic oscillator potential?

In a 1D harmonic oscillator potential, the degenerate states of two particles are related by the principle of exchange symmetry. This means that if the quantum numbers of the two particles are swapped, the energy and wavefunction of the system remains unchanged.

5. Can degenerate states of two particles in a 1D harmonic oscillator potential be distinguished?

No, degenerate states of two particles in a 1D harmonic oscillator potential cannot be distinguished from each other based on energy or wavefunction alone. This is because they have the same energy and different wavefunctions, or the same wavefunction and different energies.

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