Discussion
A particle in a one-dimensional box of length L can be excited from n = 1 to n = 2 state with frequency ν. If the box length is doubled, the frequency needed to achieve this transition is :
Answer: A
The frequency becomes ν/4.
For a particle in a box the energy levels and the 1 → 2 transition energy are
\[ \begin{aligned} E_{n} &= \frac{n^{2}h^{2}}{8mL^{2}} \\ h\nu &= E_{2} - E_{1} = \frac{3h^{2}}{8mL^{2}} \end{aligned} \]
The frequency is proportional to 1/L2. Doubling L divides it by 22 = 4, so the new frequency is ν/4.
ν/2 assumes the energy varies as 1/L, and 2ν and 4ν have the dependence the wrong way round: a longer box lowers the energy levels and brings them closer together.
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