## Related questions with answers

Question

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?

Solution

VerifiedStep 1

1 of 7### 1 Concepts and Principles

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

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

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

- 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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