Deriving Electron’s Kinetic Energy from Uncertainty Principle

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Say, electron is confined within a region of size ‘r’.

The uncertainty principle tells us:

\Delta x \Delta p \geq \frac{\hbar}{2}

using $\Delta x \sim r$

\Delta p \geq \frac{\hbar}{2r}

If the electron is confined to a region of size r, its momentum can not be smaller than its uncertainty in its momentum

\therefore p \sim \Delta p

or,

$p \geq \frac{\hbar}{2r}$

Using an order-of-magnitude approximation

p \sim \frac{\hbar}{r}

substituting this estimate into classical expression for kinetic energy we get,

$KE=\frac{p^2}{2m} \sim \frac{\hbar^2}{2mr^2}$

KE=\frac{\hbar^2}{2mr^2}

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