The Bohr Model of the Hydrogen Atom — Question 9

Question 9

Atomic recoil makes an emitted photon slightly less energetic than the internal gap

An initially stationary atom has an internal transition gap $\Delta=10.2\,\mathrm{eV}$. Use a final atomic rest-energy scale $Mc^2=9.38\times10^8\,\mathrm{eV}$ and nonrelativistic recoil. Photon momentum is $p=E_\gamma/c$.

1. Write energy conservation and derive the leading recoil correction for $\Delta\ll Mc^2$.

2. Evaluate the energy correction and fractional photon-energy decrease.

Question 9 - Solution

Atomic recoil makes an emitted photon slightly less energetic than the internal gap

(a) Method and calculation

$$\boxed{\Delta=E_\gamma+\frac{E_\gamma^2}{2Mc^2}}.$$$$E_\gamma=\Delta-\epsilon\Rightarrow\boxed{\epsilon\simeq\Delta^2/(2Mc^2)}.$$

(b) Interpretation and further calculation

$$\boxed{\epsilon=5.54584\times10^{-8}\,\mathrm{eV}}.$$$$\boxed{\epsilon/\Delta=\Delta/(2Mc^2)=5.43710\times10^{-9}}.$$

Physical check

Photon and atom recoil momenta are equal and opposite. This is a kinematic correction within the supplied model, separate from fine structure or other changes in the internal gap.