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# A hydrogen atom initially at rest in the n = 3 state decays to the ground state with the emission of a photon. (a) Calculate the wavelength of the emitted photon. (b) Estimate the recoil momentum of the atom and the kinetic energy of the recoiling atom. Where does this energy come from?

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### 1 Concepts and Principles

$\textbf{Bohr Model}$: Bohr postulated that:

1. Electrons occupy only certain nonradiating, stable, circular orbits selected by quantization of the angular momentum $L$.

$\begin{gather*} L=mvr=\dfrac{nh}{2\pi}=n\hbar,\quad \text{for integer }n \end{gather*}$

1. Radiation of frequency $f$ (wavelength $\lambda$) occurs when the electron jumps from an allowed orbit $E_i$ to one of lower energy $E_f$. $f$ (or $\lambda$) is given by the frequency condition

$\begin{gather*} E=hf=h\dfrac{c}{\lambda}=E_i-E_f\tag{1} \end{gather*}$

The quantization of the orbit radii lead to energy quantization. The allowed energy levels are

$\begin{gather*} E_n=-\dfrac{(13.6\;\mathrm{eV})Z^2}{n^2} \tag{2} \end{gather*}$

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