Consider a neutron (mass m ) of kinetic energy E and a photon of the same energy. Let $\displaystyle \lambda_{\mathrm{n}}$ and $\displaystyle \lambda_{\mathrm{p}}$ be the de Broglie wavelength of neutron and the wavelength of photon respectively. Obtain an expression for $\displaystyle \frac{\lambda_{\mathrm{n}}}{\lambda_{\mathrm{p}}}$.

Obtaining an expression for \(\displaystyle \lambda_{\mathrm{n}} / \lambda_{\mathrm{p}}\) \[\begin{aligned} & \mathrm{E}=\frac{\mathrm{hc}}{\lambda \mathrm{p}} \Rightarrow \lambda \mathrm{p}=\frac{\mathrm{hc}}{\mathrm{E}} \\ & \lambda \mathrm{n}=\frac{\mathrm{h}}{\mathrm{p}}=\frac{\mathrm{h}}{\sqrt{(2 \mathrm{mE})}} \\ & \frac{\lambda \mathrm{n}}{\lambda \mathrm{p}}=\frac{\mathrm{h}}{\sqrt{ }(2 \mathrm{mE})} \times \frac{\mathrm{E}}{\mathrm{hc}} \\ & \frac{\lambda \mathrm{n}}{\lambda \mathrm{p}}=\sqrt{ }\left(\frac{\mathrm{E}}{2 \mathrm{~m} c^{2}}\right) \end{aligned} \]