Wavelength of an electron of energy E is

${\lambda _e} = {h \over {\sqrt {2{m_e}E} }}$

$\Rightarrow$ $\lambda _e^2 = {{{h^2}} \over {2{m_e}E}}$ .....(1)

Wavelength of a photon of same energy E is

${\lambda _p} = {{hc} \over E}$

$\Rightarrow$ $E = {{hc} \over {{\lambda _p}}}$ .....(2)

Equating (1) and (2), we get

${{hc} \over {{\lambda _p}}} = {{{h^2}} \over {2{m_e}\lambda _e^2}}$

$\Rightarrow$ ${\lambda _p} = {{2{m_e}c} \over h}\lambda _e^2$

$\Rightarrow$ ${\lambda _p} \propto {\lambda _e}^2$