An electron from various excited states of hydrogen atom emits radiation to come to the ground state. Let λ_n, λ_g, be the de Broglie wavelength of the electron in the n^th state and the ground state respectively. Let Λ_n be the wavelength of the emitted photon in the transition from the n^th state to the ground state. For large n, (A, B are constants)

$Λ_{n}^{2} \approx λ$

$Λ_{n} \approx A + \frac{B}{λ_{n}^{2}}$

$Λ_{n} \approx A + B λ_{n}$

$Λ_{n}^{2} \approx A + B λ_{n}^{2}$

Solution

$Λ_{n} \approx A + \frac{B}{λ_{n}^{2}}$

The wavelength of emitted photon from n^th state to the ground state,

$\frac{1}{Λ_{n}} = R Z^{2} (\frac{1}{1^{2}} - \frac{1}{n^{2}}) Λ_{n} = \frac{1}{R Z^{2}} {(1 - \frac{1}{n^{2}})}^{- 1}$

Since n is very large, using the binomial theorem

$Λ_{n} = \frac{1}{R Z^{2}} (1 + \frac{1}{n^{2}}) Λ_{n} = \frac{1}{R Z^{2}} + \frac{1}{R Z^{2}} (\frac{1}{n^{2}}) A s w e k n o w, λ_{n} = \frac{2 π r}{n} = 2 π (\frac{n^{2} h^{2}}{4 π^{2} m Z e^{2}}) \frac{1}{n} \propto n Λ_{n} \approx A + \frac{B}{λ_{n}^{2}}$

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