Question

Consider a one-dimensional infinite potential well of width \(a\). This system contains five non-interacting electrons, each of mass \(m\), at temperature \(T = 0 \, K\). The energy of the highest occupied state is:

• A. \(\dfrac{25\pi^2 \hbar^2}{2ma^2}\)
• B. \(\dfrac{10\pi^2 \hbar^2}{2ma^2}\)
• C. \(\dfrac{5\pi^2 \hbar^2}{2ma^2}\)
• D. \(\dfrac{9\pi^2 \hbar^2}{2ma^2}\)

Hint:

Each energy level in a 1D potential well can accommodate two electrons. The highest occupied level at \(T = 0 K\) defines the Fermi energy.

Answer 1

The Correct Option is D

Step 1: Allowed energy levels.
For an infinite potential well, \[ E_n = \frac{n^2 \pi^2 \hbar^2}{2ma^2} \]

Step 2: Electron filling at \(T = 0 K\).
Each energy level can hold two electrons (due to spin degeneracy). For 5 electrons, levels fill as follows: - \(n = 1\): 2 electrons - \(n = 2\): 2 electrons - \(n = 3\): 1 electron Thus, the highest occupied level corresponds to \(n = 3\).

Step 3: Calculate energy.
\[ E_3 = \frac{9\pi^2 \hbar^2}{2ma^2} \] However, if the question refers to the "energy of the highest fully occupied level," that is \(n = 2\), giving \(E_2 = \frac{4\pi^2 \hbar^2}{2ma^2}\). But since it asks for the "highest occupied" (partially filled), \(E_3\) is correct.

Step 4: Conclusion.
Hence, \(E = \dfrac{25\pi^2 \hbar^2}{2ma^2}\).