# What's the wavelength of an electron after hitting a potential barrier?

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.

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Not to worry, the homework tag is entirely appropriate here. – David Z Aug 26 '12 at 17:36