Question

Find the kinetic energy of an electron whose de broglie wavelength is 34.0 nm.

Answer 1

The De Broglie wavelength of the electron is
$\lambda=34.0 nm=34 \cdot 10^{-9} m$
And we can use De Broglie's relationship to find its momentum:
$p= \frac{h}{\lambda}= \frac{6.6 \cdot 10^{-34} Js}{34 \cdot 10^{-9} m}=1.94 \cdot 10^{-26} kg m/s$

Given $p=mv$, with m being the electron mass and v its velocity, we can find the electron's velocity:
$v= \frac{p}{m}= \frac{1.94 \cdot 10^{-26} kgm/s}{9.1 \cdot 10^{-31} kg}= 2.13 \cdot 10^4 m/s$

This velocity is quite small compared to the speed of light, so the electron is non-relativistic and we can find its kinetic energy by using the non-relativistic formula:
$K= \frac{1}{2}mv^2= \frac{1}{2}(9.1 \cdot 10^{-31} kg)(2.13 \cdot 10^4 m/s)^2=2.06 \cdot 10^{-22} J$