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An electron from various excited states of a hydrogen atom emit radiation to come to the ground state. Let λn,λ9 be the de Broglie wavelength of the electron in the nth state and the ground state respectively. Let Λn be the wavelength of the emitted photon in transition from the nth state to the ground state. For large n,(A,B are constants)

A
2nA+Bλ2n
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B
2nλ
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C
nA+Bλ2n
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D
nA+Bλn
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Solution

The correct option is D nA+Bλ2n
vn=(2πKe2h)1nEn=12mv2nλn=hmvn=(h2m2πKe2)nλn>>>λghcλn=EnE1=12m(v21v2n) hcn=12m((hmλg)2(hmλn)2)hcn=h22m(1λ2g1λ2n)=h22mλ2g(1λ2gλ2n)nhc=2mλ2gh21(1λ2gλ2n)
Using binomial expansion,
Λn=(2mcλ29h)(1+λg2λ2n)Λn=A+Bλ2n
(Since λgλn<<1)

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