Wave–Particle Duality and de Broglie

The Core Idea: Matter as Waves

Imagine a guitar string. It can vibrate, but only at specific frequencies that create standing waves—patterns where the ends remain fixed. If you try to create a wave that does not fit an integer number of half-wavelengths between the fixed ends, the wave interferes destructively with itself and disappears. De Broglie suggested that electrons orbiting a nucleus behave like such a string. For an electron orbit to be stable, the circumference of the orbit must contain a whole number of electron wavelengths. If the wavelength does not fit perfectly, the electron wave cancels itself out, and that orbit is forbidden. This simple geometric requirement naturally leads to the quantisation of energy levels, providing the physical basis for the Bohr model’s arbitrary postulate.

The de Broglie Wavelength

De Broglie proposed that any particle with momentum \( p \) has an associated wavelength \( \lambda \), given by the relation:

where \( h \) is Planck’s constant (\( 6.626 \times 10^{-34}\ \mathrm{J\,s} \)) and \( p \) is the momentum of the particle. For a non-relativistic particle of mass \( m \) moving at velocity \( v \), the momentum is \( p = mv \), so the wavelength can be written as:

This equation reveals a crucial inverse relationship: the faster and heavier a particle is, the shorter its wavelength. For macroscopic objects, such as a baseball, the mass is so large that the wavelength is infinitesimally small, far smaller than any atomic scale, which is why we do not observe wave-like behaviour in everyday life. However, for electrons, which have a very small mass (\( 9.109 \times 10^{-31}\ \mathrm{kg} \)), the wavelength can be comparable to the size of atoms, making quantum effects observable.

Standing Waves and Quantisation

To see how this leads to quantisation, consider an electron in a circular orbit of radius \( r \). The condition for a stable standing wave is that the circumference of the orbit must be an integer multiple of the wavelength:

where \( n \) is a positive integer (\( n = 1, 2, 3, \dots \)). Substituting the de Broglie expression \( \lambda = h/p \) into this condition gives:

Rearranging this equation yields:

Since \( p = mv \), the product \( pr \) is the angular momentum \( L \) of the electron. Thus, we obtain:

or, using \( \hbar = h/2\pi \):

This is precisely the quantisation condition postulated by Bohr. The de Broglie hypothesis therefore provides a physical justification for Bohr’s first postulate: angular momentum is quantised because only orbits that support standing electron waves are stable.

Experimental Verification

The wave nature of electrons was confirmed experimentally by Clinton Davisson and Lester Germer in 1927. They fired a beam of electrons at a nickel crystal and observed an interference pattern, analogous to X-ray diffraction. The angles at which the intensity of the reflected electrons peaked matched the predictions of Bragg’s law, which is used for X-rays. This demonstrated that electrons behave as waves with wavelengths given by \( \lambda = h/p \). Later, the double-slit experiment with electrons showed that even single electrons, sent one at a time, build up an interference pattern over time, confirming that each individual electron exhibits wave-like behaviour.