# Why is an electron modelled as a plane wave in the free electron model? [closed]

Why is it a plane wave, is it something that can be visualised?

## closed as unclear what you're asking by John Rennie, Ruslan, ZeroTheHero, Michael Seifert, Jon CusterApr 12 '17 at 21:11

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• It makes the math easier. – Jon Custer Apr 12 '17 at 21:11

## 1 Answer

A free model means that the potential is zero, if you plug in that into Schrodinger's equation you get

$$-\frac{\hbar^2}{2m_e}\nabla^2 \Psi({\bf r},t) = i\hbar \frac{\partial}{\partial t}\Psi({\bf r},t)$$

whose solutions are of the form

$$\Psi({\bf r},t) \sim e^{-i({\bf k}\cdot {\bf r} - \omega t)}$$

which is a plane wave

• Can this be considered to represent anything? – James Apr 12 '17 at 12:34
• @Cameron It represents a electron that has not potential confining it, i.e. a free electron. – taper Apr 12 '17 at 12:59
• @Cameron Perhaps the question should be: why does the free electron model work as well as it does? After all, there are ion cores everywhere, and lots of nearby electrons ... the electrons are hardly free ... – garyp Apr 12 '17 at 13:23