The wavelength ${\lambda }_e$of an electron and ${\lambda }_p$of a photon of same energy E are related by

Wavelength of an electron of energy E is

${\lambda }_e=\frac{h}{\sqrt{2m_{e}E}}---\left(i\right)$

Wavelength of a photon of same energy E is

${\lambda }_p=\frac{hc}{E}orE=\frac{hc}{{\lambda }_p}---\left(ii\right)$

Squaring both sides of Eq. (i), we get

${\lambda }_e^2=\frac{h^2}{2m_{e}E}orE=\frac{h^2}{2m_e{\lambda }_e^2}---\left(iii\right)$

Equating (ii) and (iii), we get

$$\begin{gathered} \frac{hc}{{\lambda }_p}=\frac{h^2}{2m_e{\lambda }_e^2} \\ {\lambda }_p=\frac{2m_{e}c}{h}{\lambda }_e^2 \\ {\lambda }_p\propto {\lambda }_e^2 \end{gathered}$$