Consider a model system of five non interacting fermions in a single 3-dimensional harmonic oscillator. The Hamiltonian of a single particle is $\hat{\mathrm{H}}=\frac{1}{2 \mathrm{m}}\left\{\left(\mathrm{p}_{\mathrm{x}}^{2}\right)+\left(\mathrm{p}_{\mathrm{y}}^{2}\right)+\left(\mathrm{p}_{\mathrm{z}}^{2}\right)\right\}+\frac{1}{2}\left(\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{z}^{2}\right)$ where m is the mass of the particle, on is the angular frequency, $\hat{\mathrm{P}}_{\mathrm{x}}, \hat{\mathrm{P}}_{\mathrm{y}}$ and $\hat{\mathrm{P}}_{\mathrm{z}}$ are the momentum operators. The ground state energy of the system of non-interacting fermions is
(a)$\frac{21}{2} \mathrm{h}\omega$
(b)$\frac{15}{2} \mathrm{h}\omega$
(c)$\frac{5}{2} \mathrm{h}\omega$
(d)$\frac{25}{2} \mathrm{h}\omega$
Answer
Answer: A ✓ checked by 4AB · confidence high
The source book printed no answer; this one was worked out and checked — see the explanation.
Explanation
3-D oscillator levels (n + 3/2)ħω with degeneracy (n+1)(n+2)/2. Two spin-½ fermions fill n = 0 (2 × 3/2 = 3), the next three go into the triply degenerate n = 1 level (3 × 5/2 = 15/2): total 21/2 ħω.