Waves show wave properties such as wavelength, and experience the wave phenomena.
Objects show particle properties such as momentum and kinetic energy and can collide with each other.
de Broglie’s hypothesis states that particles such as electrons can also show wave properties.
Momentum of a particle, \(p\) and its wavelength, \(\lambda\) is related by the equation:
\[p=\frac{h}{\lambda}\]
where \(h=\) Planck’s constant
Since momentum of a particle, \(p\) can be related to its mass, \(m\) and velocity, \(v\) through the formula \(p=m v\), de Broglie’s wavelength can also be expressed as:
\[ \lambda=\frac{h}{m v} \]
where
\(h =\) Planck’s constant
\(m=\) mass of particle
\(v=\) velocity of particle
From the above equation, it is seen that wavelengths for large masses are hardly detectable (unobservable).

Example

Given that the mass of an electron is \(9.11 \times 10^{-31} kg\). Determine the de Broglie’s wavelength of an electron moving with a velocity of \(3 \times 10^7 m s ^{-1}\).

\[
\begin{aligned}
\lambda & =\frac{h}{m v} \\
& =\frac{6.63 \times 10^{-34}}{\left(9.11 \times 10^{-31}\right)\left(3 \times 10^7\right)} \\
& =2.43 \times 10^{-11} m
\end{aligned}
\]

de Broglie’s hypothesis was confirmed through electron diffraction experiment which showed the presence of wave properties of electron (Diagram 7.4 and Diagram 7.5).

Diagram 7.4 Diffraction pattem of electrons (NorbertMitzel, CC BY-SA 4.0, via Wikimedia Commons)
Diagram 7.5 Diffraction pattem of red laser light (Wisky, CC BY-SA 3.0, via Wikimedia Commons)
Light and electrons show waveparticle duality because they have both wave and particle properties.

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