Discussion
For a quantum mechanical system of N identical spin − 1/2 particles in one-dimensional box of length L, the Fermi wave number is :
Answer: B
The Fermi wave number is \( k_{F} = \frac{\pi N}{2L} \).
In a box of length \( L \) the allowed wave numbers are \( k = n\pi/L \), \( n = 1, 2, 3, \ldots \). Each level holds two spin-1/2 particles (spin up and spin down), so N particles fill the levels up to \( n_{F} = N/2 \):
\[ k_{F} = \frac{n_{F}\pi}{L} = \frac{\pi N}{2L} \]
\( \frac{\pi N}{L} \) is what results from ignoring spin and putting one particle per level. The options without \( \pi \) drop the \( \pi/L \) spacing of the allowed \( k \) values.
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