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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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    $\begingroup$ Not to worry, the homework tag is entirely appropriate here. $\endgroup$ – David Z Aug 26 '12 at 17:36
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Hint to the question(v1): The electron travels at non-relativistic speed. What is the non-relativistic formula for kinetic energy of a point particle?

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  • $\begingroup$ D'oh, silly me. I always neglect those pesky constants. $\endgroup$ – bitmask Aug 26 '12 at 18:00

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