= Exclusion and the energy of two electrons in three spatial levels
Consider an original noninteracting model with three <orthonormal> <spatial orbitals> $A,B,C$ at <energies> $0,\Delta,3\Delta$, where $\Delta>0$. Both <spin> projections are available at each <energy>, so there are six <spin orbitals>. The <Quantum Hamiltonian> is $H=h\otimes I+I\otimes h$, with no interaction between particles and no dependence on <spin>. Comparing its allowed <ground states> isolates the effect of <fermionic> exchange symmetry.
For two distinguishable spin-one-half particles, the spatial <ground state> is $A\otimes A$ and any of four independent two-<spin> states is possible: $E_0=0$ and <ground-state degeneracy> four. For two identical <electrons>, both can still occupy $A$, using $A\uparrow$ and $A\downarrow$, but there is just one allowed <ground state>: the symmetric spatial factor times the <spin-one-half singlet state>,
$$
\Psi_0=A(\mathbf r_1)A(\mathbf r_2)\frac{|\uparrow\downarrow\rangle-|\downarrow\uparrow\rangle}{\sqrt2},\qquad E_0=0.
$$
Exclusion has reduced the <ground-state degeneracy> without raising the <ground state> <energy>.
Now restrict both particles to the same <spin> projection $\uparrow$. Distinguishable particles can still both occupy $A\uparrow$, so $E_0=0$. Identical <electrons> must occupy different <spatial orbitals>; the lowest choice is $A,B$, with
$$
\Psi_{\uparrow\uparrow}=\frac{A(\mathbf r_1)B(\mathbf r_2)-B(\mathbf r_1)A(\mathbf r_2)}{\sqrt2}|\uparrow\uparrow\rangle,\qquad E_0=\Delta.
$$
The <spin> factor is a <spin-one-half triplet state>, and the spatial factor is antisymmetric. These are <ground states> within the restricted parallel-<spin> sector, not the unrestricted two-<electron> <ground state>. Both restricted models have <ground-state degeneracy> one.
\Image[/pauli-three-level-ground-energy.png]
{title=Ground energies in a three-spatial-level model depend on which spin states are available}
{description=Original model with chosen <spatial orbital> <energies> $0,\Delta,3\Delta$, no interactions and no <spin> splitting. The first pair of panels allows both <spin> projections; the second pair restricts both to $\uparrow$. Labels $g_0$ count independent <ground states>; particle subscripts appear only in the distinguishable cases.}
The first excited two-<electron> <energy> in the unrestricted model is $\Delta$, with four independent <quantum states>: one occupant in $A$, one in $B$, and two <spin> choices for each. A basis adapted to <particle exchange operators> separates these into one <spin-one-half singlet state> with symmetric spatial factor and three <spin-one-half triplet states> with antisymmetric spatial factor. All four have equal <energy> in this model. Singlet versus triplet does not by itself determine an <energy> gap; an interaction or a <spin>-dependent term is needed to split them.
As a further count, the full two-<electron> space has <dimension> $\binom62=15$. Its <energies> in units of $\Delta$ and corresponding multiplicities are $(0,1,2,3,4,6)$ and $(1,4,1,4,4,1)$. For distinguishable spin-one-half particles the ordered <tensor product> instead has <dimension> $6^2=36$, with multiplicities $(4,8,4,8,8,4)$ at those same <energies>. These are exact model counts, not atomic measurements. The higher restricted <ground state> <energy> follows from removing forbidden <quantum states>; the <Quantum Hamiltonian> contains no additional “exclusion force”.
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