Can only two electrons be in ground state? Are the energy levels the same thing as the energy shells? I can't find a straight answer

Question

So for a model like this one, can there be two electrons in one energy level?
And I don't understand the Pauli principle that two electrons can't be on the same energy level when there are 2 electrons in the first shell of most atoms???
Does this then mean that shells and energy levels aren't the same thing?

Gert · Accepted Answer

Wikipedia says it well:

The Pauli exclusion principle is the quantum mechanical principle
which states that two or more identical fermions (particles with
half-integer spin) cannot occupy the same quantum state within a
quantum system simultaneously.

The 'same quantum state' here can be understood as a set of quantum numbers  $n$, $l$, $m_l$ and $m_s$ to be the same.
For example, if $2$ electrons are in the $text{1s}$ orbital, that is $1,0,0$, then they must have a different spin quantum number $m_s$, one with $m_s=+1/2$ and the other with $m_s=-1/2$. Both will have the same energy (ground state in this case).

trula · Answer

The Pauli principle does not say, they can not have the same energy, the can not have the exact same state, So with the same energy and angular momentum they have to have different spin.

Yejus · Answer

Of course two electrons can be in the same energy level. Pauli's principle only states that two fermions (in this case, electrons) cannot occupy the same state. This means two electrons can still have different spins and occupy the same energy level, while having different states.

Answered by Yejus on July 23, 2021

Can only two electrons be in ground state? Are the energy levels the same thing as the energy shells? I can't find a straight answer

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