Discussion
The wavelength associated with a particle in one-dimensional box of length L is (n refers to the quantum number)
Answer: A
The wavelength is 2L/n.
The wavefunctions of a particle in a box, \(\psi_{n} = \sqrt{2/L}\,\sin(n\pi x/L)\), must vanish at both walls, so they are standing waves with a whole number of half-wavelengths fitting into the box:
\[ L = n\frac{\lambda}{2} \Rightarrow \lambda = \frac{2L}{n} \]
For the ground state (n = 1) the box holds half a wavelength, so \(\lambda = 2L\). L/n counts whole wavelengths instead of half-wavelengths, L/2n halves the answer again, and nL would make the wavelength grow with the quantum number, when higher states have more nodes and shorter wavelengths.
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