Explain the de Broglie hypothesis of mater waves.

de Broglie hypothesis (1924): Radiation has a dual nature: waves (interference) and particles (photoelectric effect). Nature loves symmetry, so de Broglie suggested that every moving material particle also has a wave associated with it, called a matter wave or de Broglie wave.

Derivation: For a photon, \(E=h\nu\) (Planck) and \(E=mc^2\) (Einstein). So \(h\nu=mc^2\), giving \(\dfrac{hc}{\lambda}=mc^2\) and \(\lambda=\dfrac{h}{mc}=\dfrac{h}{p}\). Applying the same relation to a particle of mass m moving with velocity v:

\[ \boxed{\lambda=\frac{h}{mv}=\frac{h}{p}=\frac{h}{\sqrt{2mE}}} \]

For a charge q accelerated through a potential V: \(\lambda=h/\sqrt{2mqV}\); for an electron, \(\lambda=12.27/\sqrt V\) Å.

Points: the wavelength is significant only for microscopic particles such as electrons and neutrons. The hypothesis was confirmed by the Davisson–Germer electron diffraction experiment, and it explains Bohr's quantisation (\(2\pi r=n\lambda\Rightarrow mvr=nh/2\pi\)).