Discussion
A one-dimensional box contains three identical particles in the ground state of the system. Find the ratio of the total energies of these particles if they were spin-1/2 fermions, to that if they were bosons.
Answer: C
The ratio is 2.
The levels of a one-dimensional box are \(E_{n} = n^{2}E_{1}\). In the ground state, bosons can all occupy the lowest level, while spin-1/2 fermions obey the Pauli principle: at most two (spin up and spin down) per level.
\[ \begin{aligned} E_{\text{bosons}} &= 3E_{1} \\ E_{\text{fermions}} &= 2E_{1} + 4E_{1} = 6E_{1} \\ \text{ratio} &= 6/3 = 2 \end{aligned} \]
A ratio of 1 ignores the Pauli principle. 14/3 is what one gets by putting the three fermions in three different levels, \(1 + 4 + 9 = 14\), which forgets the spin degeneracy. 1/3 inverts the comparison in a different way and has no basis.
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