Discussion
Sixteen non-interacting electrons are confined in an infinite box of side a. The energies of the least energetic and the most energetic electrons in the ground state of the system are :
Answer: A
The energies are \(\frac{\hbar^{2}\pi^{2}}{2ma^{2}}\) and \(\frac{32\hbar^{2}\pi^{2}}{ma^{2}}\).
For a box of side a treated in one dimension, the levels are \(E_{n} = n^{2}\pi^{2}\hbar^{2}/2ma^{2}\). By the Pauli principle each level holds two electrons (spin up and down), so 16 electrons fill n = 1 to 8 in the ground state.
\[ \begin{aligned} E_{1} &= \frac{\pi^{2}\hbar^{2}}{2ma^{2}} \\ E_{8} &= \frac{64\pi^{2}\hbar^{2}}{2ma^{2}} = \frac{32\pi^{2}\hbar^{2}}{ma^{2}} \end{aligned} \]
Zero is not an allowed energy in an infinite well, equal energies would require all 16 electrons in the lowest level, which Pauli forbids, and \(128\hbar^{2}\pi^{2}/ma^{2}\) is the n = 16 level, reached only if each level held one electron.
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