Discussion
For the particle in 1-D box, the value of quantum number n = 0 is ruled out because :
Answer: D
n = 0 is ruled out because it implies \(\Psi = 0\) everywhere in the box.
The wave functions of a particle in a box of length L are \(\Psi_{n} = \sqrt{2/L}\,\sin(n\pi x/L)\). Putting n = 0 gives \(\sin 0 = 0\) at every x, so \(\Psi = 0\) throughout. Then \(|\Psi|^{2}\) is zero everywhere and the probability of finding the particle anywhere is zero, which means there is no particle at all.
The energy \(E_{n} = n^{2}h^{2}/8mL^{2}\) would simply be zero, not divergent, and the wave function would vanish rather than diverge. Nothing about n = 0 makes the particle classical; the lowest allowed state is n = 1, with non-zero zero-point energy.
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