Consider hypothetical situation. We have a one dimensional box of length L with perfectly reflecting walls. Photons as well as electrons can be present in the box if they form standing waves with nodes at the end. The photon can be only completely absorbed by electron if it can go to a higher excited state. An electron can go from higher excited state to lower excited state only if it emits photons which can form standing wave in the box. Select correct option(s). Take $\frac{h}{m_{e} L c} = 1$
(c: speed of the light, $m_{e}$ : mass of the electron):

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An electron in state n = 1 can go to n = 3

A photon corresponding to 5th harmonic can be emitted by an electron in 5th excited state.

A photon corresponding to 4th harmonic can be emitted by an electron in 5th excited state.

A photon corresponding to 2nd harmonic can be emitted by an electron in 2nd excited state.

answer is A, C, D.

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Detailed Solution

$\begin{array}{l} \frac{n λ}{2} = L \\ E_{p h o t o n} = \frac{h c}{λ} = \frac{h n c}{2 L} = \frac{m_{e} c^{2} n}{2} \\ E_{e l e c t r o n} = \frac{p^{2}}{2 m_{e}} = \frac{{(\frac{h}{λ})}^{2}}{2 m_{e}} = \frac{m_{e} c^{2}}{8} n^{2} \end{array}$

$\begin{array}{l} O p t i o n 1 : I f e l e c t r o n j u m p s f r o m n = 1 t o n = 3, i t w i l l r e q u i r e e n e r g y = \frac{m_{e} c^{2}}{8} (3^{2} - 1^{2}) = m_{e} c^{2} \\ T h i s e n e r g y c a n b e g i v e n b y a p h o t o n i n i t s 2 n d h a r m o n i c . \end{array}$

$\begin{array}{l} O p t i o n 3 : A n e l e c t r o n i n 2 n d e x c i t e d s t a t e c a n g o f r o m n = 3 t o n = 1 \\ E n e r g y r e l e a s e d b y e l e c t r o n = \frac{m_{e} c^{2}}{8} (3^{2} - 1^{2}) = m_{e} c^{2} \\ w h i c h m a t c h e s w i t h e n e r g y o f p h o t o n i n 2 n d h a r m o n i c . \end{array}$

$\begin{array}{l} O p t i o n 4 : A n e l e c t r o n i n 5 t h e x c i t e d s t a t e c a n g o f r o m n = 6 t o n = 4 \\ E n e r g y r e l e a s e d b y e l e c t r o n = \frac{m_{e} c^{2}}{8} (6^{2} - 4^{2}) = \frac{5}{2} m_{e} c^{2} \\ w h i c h m a t c h e s w i t h e n e r g y o f p h o t o n i n 5 t h h a r m o n i c . \end{array}$

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