At what kinetic energy $K$ will the de Broglie wavelength of an electron be exactly equal to its Com

Question

At what kinetic energy $K$ will the de Broglie wavelength of an electron be exactly equal to its Compton wavelength? (Let $E_0 = m_0 c^2$ be the rest mass energy of the electron).

Professor PC Thomas Classes · Answer

The Compton wavelength is $\lambda_C = \frac{h}{m_0 c}$.
The de Broglie wavelength is $\lambda_d = \frac{h}{p}$, where $p$ is the relativistic momentum.
Equating the two: $\frac{h}{p} = \frac{h}{m_0 c} \implies p = m_0 c$.
The total relativistic energy $E$ is given by $E^2 = (m_0 c^2)^2 + (pc)^2$.
Substituting $p = m_0 c$:
$E^2 = (m_0 c^2)^2 + (m_0 c^2)^2 = 2(m_0 c^2)^2 = 2E_0^2 \implies E = \sqrt{2}E_0$.
The kinetic energy $K$ is the total energy minus the rest energy:
$K = E - E_0 = \sqrt{2}E_0 - E_0 = (\sqrt{2} - 1)E_0$.