# Identical particles and degenrate energy levels

## Homework Statement

Five electrons (with mass m) whose interaction can be neglected, are in the same 3-dim harmonic oscillatorpotential
$$V(x,y,z) = \frac k2 (x^2 + y^2 + z^2)$$

What is the ground state energy?

## The Attempt at a Solution

I have the energy for the potential. It is:
$$E_{n_x,n_y,n_z} = (n_x+n_y+n_z + \frac 32)\frac{\hbar\omega}2$$

My question is about the degeneracy.Since it's electrons, at most 2 of them can be in the same state, but can more than 2 electrons have the same energy?

Relating to this question: Should the ground state energy for this system be

$$E=2E_{111}+2E_{211}+E_{121}$$

or are the two states (211) and (121) not allowed to have more than 2 electrons totally, ie

$$E=2E_{111}+2E_{211}+E_{221}$$

nrqed
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## Homework Statement

Five electrons (with mass m) whose interaction can be neglected, are in the same 3-dim harmonic oscillatorpotential
$$V(x,y,z) = \frac k2 (x^2 + y^2 + z^2)$$

What is the ground state energy?

## The Attempt at a Solution

I have the energy for the potential. It is:
$$E_{n_x,n_y,n_z} = (n_x+n_y+n_z + \frac 32)\frac{\hbar\omega}2$$

My question is about the degeneracy.Since it's electrons, at most 2 of them can be in the same state, but can more than 2 electrons have the same energy?
of course. As long as they don't all have the same quantum numbers.
Relating to this question: Should the ground state energy for this system be

$$E=2E_{111}+2E_{211}+E_{121}$$

or are the two states (211) and (121) not allowed to have more than 2 electrons totally, ie

$$E=2E_{111}+2E_{211}+E_{221}$$

Wait. Why aren't you starting with the n=0 states??

Avodyne
$$2E_{000}+2E_{100}+E_{010}$$?