Electrons and Matter Waves — Question 2

Question 2

Equal electron and photon wavelengths imply equal momenta but different energies

A photon and a nonrelativistic electron each have wavelength $\lambda=.100\,\mathrm{nm}$. Use $hc=1240\,\mathrm{eV\,nm}$ and $m_ec^2=511000\,\mathrm{eV}$.

1. Find their common momentum times c, and compare photon energy with electron kinetic energy.

2. Derive their energy ratio and check the nonrelativistic assumption.

Question 2 - Solution

Equal electron and photon wavelengths imply equal momenta but different energies

(a) Method and calculation

$$\boxed{p_\gamma c=p_ec=hc/\lambda=12400\,\mathrm{eV}},\quad E_\gamma=pc.$$$$\boxed{K_e=\frac{(pc)^2}{2m_ec^2}=150.450\,\mathrm{eV}}.$$

(b) Interpretation and further calculation

$$\boxed{\frac{E_\gamma}{K_e}=\frac{2m_ec^2}{pc}=82.4194}.$$$$\boxed{\frac{K_e}{m_ec^2}=2.94423\times10^{-4}\ll1},\quad v/c\simeq pc/(m_ec^2)=.0242661.$$

Physical check

The de Broglie relation determines momentum, not a universal energy law. Massive particles and photons have different dispersion relations.