Question

Show that the first few frequencies of light that is emitted when electrons fall to the n^th level from levels higher than n, are approximate harmonics (i.e. in the ratio 1 : 2 : 3...) when n >> 1.

Answer 1

`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, ν_mn are approximately in the order 1, 2, 3...........