Louis de Broglie considered Einstien's two ideas about energy and combined them for a photon.

Particle like description of matter

$$ E = m c^2 $$

and wavelike behaviour from

$$ E = hf $$

so

$$ hf = mc^2 $$

$$ h = \frac{mc^2}{f} $$

but

$$ f = \frac{c}{\lambda} $$

so

$$ h = mc\lambda $$

but

$$ p = mc $$

so

$$ h = p\lambda $$

or

$$ p = \frac{h}{\lambda} $$

De Broglie just conjectured that this might apply to other paticles like the electron as well.

Given an electron of energy 150 eV what is its momentum?

Assuming electron is non-relativistic with speed less than 10% speed of light

$$ E = \frac{1}{2}m v^2 $$

so

$$ E = \frac{p^2}{2m} $$

so

$$ p = \sqrt{(2mE)} $$

so

$$ p = \sqrt{(2 \times 9.11 \times 10^{-31} \times 150 \times 1.6 \times 10^{-19})} $$

$$ p = 6.61 \times 10^{-24} \text{ kgm/s} $$

so wavelength is

$$ \lambda = \frac{h}{p} $$

$$ \lambda = 9.9 \times 10^{-11} \text{ m}$$

This is approximately the same as the separation between atoms in a crystal lattice.

If electrons behave as waves we would expect diffraction to occur.

The pattern of rings seen above is just such a diffraction pattern.

The amazing thing that arose from de Broglie's postulate is that all particles have an associated wavelength and will exhibit wave like properties under the right conditions.

Neutron diffraction is now an established technique for studying the arrangement of matter at sub-atomic level.

The electron microscope uses these techniques to see thigns visible light can't detect.