Question

Consider a neutron (mass $m$) of kinetic energy $E$ and a photon of the same energy. Let $\lambda_n$ and $\lambda_p$ be the de Broglie wavelength of the neutron and the wavelength of the photon, respectively. Obtain an expression for $\frac{\lambda_n{\lambda_p}$.}

Hint:

For de Broglie wavelength problems, use $\lambda = \frac{h}{p}$ and carefully relate the momentum or energy for each particle or photon.

Answer 1

We are given the equation for energy \(E\): \[ E = \frac{hc}{\lambda_p} \quad \Rightarrow \quad \lambda_p = \frac{hc}{E} \] Now, using the relation \(\lambda_n = \frac{h}{p}\), where \(p\) is the momentum, we can substitute the expression for \(p\): \[ \lambda_n = \frac{h}{p} = \frac{h}{\sqrt{2mE}} \] Now, substituting the expression for \(p\): \[ \frac{\lambda_n}{\lambda_p} = \frac{\frac{h}{\sqrt{2mE}}}{\frac{h}{Ehc}} = \frac{E}{\sqrt{2mc^2}} \] Thus, we get the expression: \[ \frac{\lambda_n}{\lambda_p} = \sqrt{\frac{E}{2mc^2}} \] This gives us the desired expression for \(\frac{\lambda_n}{\lambda_p}\).