Even in the light of the Compton Effect, critics of the early single-photon interference experiments dismissed the importance of the observation by noting that a photon doesn’t have mass. Through some fancy hand-waving, they argued that the low-light interference could be caused through splitting and recombining the light quanta’s wavefront. Decisive proof would come when particles with mass would show interference.

In 1924, French physicist Prince Louis-Victor de Broglie (pronounced “de Broy,” and by the way, he did belong to the French royalty) proposed that maybe it’s not just light that has this dual personality, maybe it’s everything! He reasoned that if the quanta of light could be both a wave and a particle, then maybe the same could be true of electrons.

Remember that although a photon doesn’t have mass, it does have momentum p given by:

De Broglie’s hypothesis was that matter—which is commonly described by our perception as “solid”—can also behave as a wave. He took the concept used to find the momentum of a photon, and applied it to particles, proposing that the wavelength associated with a particle is:

where the momentum p of a particle equals the product of its mass m, and velocity v. In the same way, he used Planck’s relationship to propose that the frequency f of the matter wave is related to the energy E of the particle by:

MATTER WAVES AND THE BOHR ATOM

As you may remember from chapter 4, Bohr was able to explain the discrete spectral lines emitted by the hydrogen atom by forcing the electrons into a limited number of permitted orbits (Figure 94a). However, like Planck before him, he did this without having a physical justification.

Figure 94 De Broglie used his proposed matter waves to explain why Bohr’s atomic orbits would be quantized. (a) Bohr explained the discrete spectral lines of hydrogen by enforcing a limited number of orbits. However, he did this without having a physical justification. (b) De Broglie proposed that, like in a guitar string fastened at the ends to rigid supports, allowed orbits would be those where a complete number of electron waves would fit on the orbit. (c) A higher-energy orbit would thus fit a larger number of complete vibrations than a lower-energy orbit.

The first theoretical success of de Broglie’s matter waves came when he used them to explain why certain electron orbits are allowed, and why others are not. He proposed that the allowed orbits for an electron are those that support a standing wave of specific wavelength, energy, and frequency (i.e., Bohr’s energy levels); much like a guitar string sets up a standing wave when plucked.

As shown in Figure 94b, de Broglie proposed that if one were to straighten an orbit into a string and fix the ends to rigid supports, it could be set to vibrate. However, since the ends are fixed, the only vibrations that it can support are those where full wavelengths fit between the ends. De Broglie proposed that these vibrations would correspond to his matter waves, so a higher frequency vibration would correspond to an electron at a higher energy state. As shown in Figure 94c, de Broglie’s view of Bohr’s atom consisted of electron waves of different wavelengths. An electron wave would transition between these wavelengths (each at one of Bohr’s energy levels), giving off or absorbing photons with an energy hf equal to the difference in energy ΔE between the electron waves.