The energy of a photon is equal to the kinetic energy of a proton. The energy of the photon is $E$.

Question

The energy of a photon is equal to the kinetic energy of a proton. The energy of the photon is $E$. Let $\ell_1$ be the de-Broglie wavelength of the proton and $\ell_2$ be the wavelength of the photon. The ratio $\ell_1 / \ell_2$ is proportional to:

Professor PC Thomas Classes · Answer

$\begin{gathered}\ell_2=\frac{h c}{E} \ \ell_1=\frac{h}{\sqrt{2 m E}} \ \frac{\ell_1}{\ell_2}=\frac{h}{\sqrt{2 m E}} 	imes \frac{E}{h c} \ \frac{\ell_1}{\ell_2}=\frac{\sqrt{E}}{c \sqrt{2 m}} \ \frac{\ell_1}{\ell_2} \propto E^{1 / 2}\end{gathered}$