I have this question:
An electron with Energy $E = 40 eV$ hits a potential barrier with $E_0 = 30 eV$. What is the wavelength of the electron after hitting the potential barrier?
I worked from the energy $E = p\cdot v \Rightarrow m_e \cdot E = p^2$ and combined it with the DeBroglie Wavelength $\lambda \cdot p = h$ which yields
$$ \lambda = \frac{h}{\sqrt{m_e \cdot (E-E_0)}} $$
However, the sample solution says the wave number is
$$ k = \frac{2\pi}{\lambda} = \frac{\sqrt{2\cdot m_e \cdot (E-E_0)}}{\hbar} $$
Which is exactly what I got, except for the $2$ inside the root.
Where does that factor come from? Why is my lacking of it wrong?
Note: I tagged this homework although it's not really homework, but homework-like head-scratching.
