Question

An electron from various excited states of hydrogen atom emit radiation to come to the ground state. Let ${\lambda }_n,{\lambda }_g$ be the de Broglie wavelength of the electron in the $n^{th}$ state and the ground state respectively. Let ${\Lambda }_n$ 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)

• A. ${\Lambda }_n\approx A+\frac{B}{{\lambda }_n^2}$
• B. ${\Lambda }_n\approx A+B{\lambda }_n$
• C. ${\Lambda }_n^2\approx A+B{\lambda }_n^2$
• D. ${\Lambda }_n^2\approx \lambda$

Answer 1

Answer: (A)

$P_n=\frac{h}{{\lambda }_n},P_g=\frac{h}{{\lambda }_g}$
$k=\frac{P^2}{2m}=\frac{h^2}{2m{\lambda }^2},E=-k=-\frac{h^2}{2m{\lambda }^2}$
$E_n=-\frac{h^2}{2m{\lambda }_n^2},E_g=-\frac{h^2}{2m{\lambda }_g^2}$
$E_n-E_g=\frac{h^2}{2m}\left(\frac{1}{{\lambda }_g^2}-\frac{1}{{\lambda }_n^2}\right)=\frac{hc}{{\Lambda }_n}$
$\frac{h^2}{2m}\left(\frac{{\lambda }_n^2-{\lambda }_g^2}{{\lambda }_g^2{\lambda }_n^2}\right)=\frac{hc}{{\Lambda }_n}$
${\Lambda }_n=\frac{2mc}{h}\left(\frac{{\lambda }_g^2{\lambda }_n^2}{{\lambda }_n^2-{\lambda }_g^2}\right)$
${\Lambda }_n=\frac{2mc{\lambda }_g^2}{h}\frac{{\lambda }_n^2}{{\lambda }_n^2\left(1-\frac{{\lambda }_g^2}{{\lambda }_n^2}\right)}$
$=\frac{2mc{\lambda }_g^2}{h}{\left[1-\frac{{\lambda }_g^2}{{\lambda }_n^2}\right]}^{-1}$
$=\frac{2mc{\lambda }_g^2}{h}\left[1+\frac{{\lambda }_g^2}{{\lambda }_n^2}\right]$
$=\frac{2mc{\lambda }_g^2}{h}+\left(\frac{2mc{\lambda }_g^4}{h}\right)\frac{1}{{\lambda }_n^2}$
$=A+\frac{B}{{\lambda }_n^2}$
$A=\frac{2mc{\lambda }_g^2}{h},B=\frac{2mc{\lambda }_g^4}{h}$