GATE 2025 PH – Question 63
Five identical non-interacting particles of mass $m$ and spin 3/2 are confined to a one-dimensional well of length $L$. If the lowest total energy is $N\pi^2\hbar^2/(2mL^2)$, the value of $N$ (in integer) is _____
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Correct answer: 8
Explanation
**Spin-3/2 particles** have $2s+1=4$ spin states. The particles are identical fermions (half-integer spin), so by the Pauli principle no two can have the same set of quantum numbers (spatial level and spin state).
**Energy levels in the 1D well:** $E_n=\dfrac{n^2\pi^2\hbar^2}{2mL^2}$.
**Filling five particles at the lowest total energy:**
- The level $n=1$ can hold 4 particles (one for each of the 4 spin states).
- The fifth particle goes into $n=2$.
**Total energy:**
$$E=\frac{\pi^2\hbar^2}{2mL^2}\left[4(1)^2+1(2)^2\right]=8\,\frac{\pi^2\hbar^2}{2mL^2}.$$
So $N=\mathbf{8}$.