Electron waves, electron momentum and atomic size

Electric potential energy

Electric potential energy of an electron at distance r from a hydrogen nucleus = -e²/4πε_or = ke²/r This is NEGATIVE. You have to put energy into the atom to remove the electron from it. The electron is held in an electrostatic potential well in just the same way as a stone on the Earth’s surface is held in a gravitational potential well of the Earth.

Electron wavelength and momentum

Electron wavelength = λ = h/mv
Electron momentum = mv
So for a large wavelength we must have a small momentum

deBroglie standing waves

deBroglie standing waves fitting into the ‘box’ (atom).
The smaller the atom the smaller the wavelength of the standing waves that could fit into it.

It follows that if the diameter of the atom is smaller the wavelength of the electron wave must also be smaller and since mv = h/l the momentum of the electron must be larger.

Electron kinetic energy

The kinetic energy of the electron E_K = ½ mv² = ½(mv)²/m = ½ [(electron momentum)²/m]
A larger momentum will therefore also mean a larger kinetic energy.

Total electron energy

Electron total energy = electron kinetic energy + electron potential energy

Minimum size for an atom

At some critical radius (r_C ) the kinetic energy is equal to the potential energy and so the total energy of the electron is zero. This means that it is just about enough energy to escape from the potential well of the atom.

At this value: ½(mv)²/m = e²/4peπελ)²/m = e²/4πε_or_C
But r_C = λ/4

And so ½(h/4r_C)2/m = e²/4πε_or_C



Using h= 6.63x10^-34 Js, ε_o as 8.84x10^-12 Fm^-1 , m = 9x10^-31 kg and e = 1.6x10^-19 C we have:
r_C = (6.63x10^-34)2xπx8.84x10^-12/8x9x10^-31x(1.6x10^-19)² = 6.62x10^-11 m = 0.0662 nm.

This is then the smallest possible radius for a hydrogen atom. If we try to make the atom smaller we have to ‘squeeze’ the electron wave. A shorter electron wave means a greater kinetic energy and so the electron will break free from the atom.