\(\lambda_{e}=\frac{h}{\sqrt{2 m_{e} E}}\)   .... \((i)\)

Wavelength of a photon of same energy \(E\) is

\(\lambda_{p}=\frac{h c}{E}\)  or  \(E=\frac{h c}{\lambda_{p}}\)   ..... \((ii)\)

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

\(\lambda_{e}^{2}=\frac{h^{2}}{2 m_{e} E}\)  or  \(E=\frac{h^{2}}{2 m_{e} \lambda_{e}^{2}}\)   ..... \((iii)\)

Equating \((ii)\) and \((iii)\), we get

\(\frac{{hc}}{{{\lambda _p}}} = \frac{{{h^2}}}{{2{m_e}\lambda _e^2}}{\text{ }}\)  or  \({\lambda _p} = \frac{{2{m_e}c}}{h}\lambda _e^2\)

\({\lambda_{p} \propto \lambda_{e}^{2}}\)