The energy of a photon is related to its wavelength by the equation:
\[
E = \frac{hc}{\lambda_p}
\]
Where:
- \( h \) represents Planck's constant.
- \( c \) is the speed of light.
- \( \lambda_p \) denotes the photon's wavelength.
For an electron, the de Broglie wavelength is expressed as:
\[
\lambda_e = \frac{h}{\sqrt{2mE}}
\]
Where:
- \( m \) signifies the mass of the electron.
- \( E \) is the electron's energy.
The ratio of these wavelengths is calculated as follows:
\[
\frac{\lambda_p}{\lambda_e} = \frac{\frac{hc}{E}}{\frac{h}{\sqrt{2mE}}} = \frac{c}{\sqrt{2mE}}
\]
Consequently, the wavelength ratio is:
\[
\frac{\lambda_p}{\lambda_e} = \frac{\sqrt{2mE}}{c}
\]