= Two-electron states in the first two shells of an isotropic oscillator
The spatial ground orbital of a three-dimensional <quantum harmonic oscillator> has $l=0$ and energy $3\hbar\omega_0/2$; its first excited shell has $l=1$ and energy $5\hbar\omega_0/2$. Two identical <Electrons> both in the ground orbital must form the <singlet state>, giving one state at total energy $3\hbar\omega_0$.
At energy $4\hbar\omega_0$, one electron occupies each shell. There are two ground-shell spin orbitals and six first-excited-shell spin orbitals, hence twelve <Slater determinants>. Equivalently, $0\otimes1$ gives $L=1$; symmetric spatial combinations with the <singlet state> give three states, while antisymmetric spatial combinations with the <triplet states> give nine. Distinct occupied orbitals allow both spatial exchange symmetries, so both spin sectors must be kept.
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