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Question
Show that the first few frequencies of light that is emitted when electrons fall to the nth level from levels higher than n, are approximate harmonics (i.e. in the ratio 1 : 2 : 3...) when n >> 1.
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Solution
`v_(mn) = cRZ^2 [1/(n + p)^2 - 1/n^2]`, where m = n + p, (p = 1, 2, 3, ...) and R is Rydberg constant.
For p << n.
`v_(mn) = cRZ^2 [1/n^2 (1 + p/n)^-2 - 1/n^2]`
`v_(mn) = cRZ^2 [1/n^2 - (2p)/n^3 - 1/n^2]`
`v_(mn) = cRZ^2 (2p)/n^3 ≃ ((2cRZ^2)/n^3)p`
Thus, νmn are approximately in the order 1, 2, 3...........
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