Prince Louis-Victor de Broglie was a French graduate student in 1924. It had been 19 years since Einstein had blurred the distinction between a particle and a wave. As de Broglie thought about these issues, it seemed that nature should have some kind of symmetry. If light waves could have a particle-like nature, why shouldn’t material particles have some kind of wave-like nature? In other words, could matter waves exist?

With no experimental evidence to go on, de Broglie reasoned by analogy with Einstein’s equation $E=hf$ for the photon and with some of the ideas of his theory of relativity. The details need not concern us, but they led de Broglie to postulate that if a material particle of momentum $p=mv$ has a wave-like nature, then its wavelength must be given by

where $h$ is Planck’s constant. This is called the de Broglie wavelength.

For example, an electron with $1.0\text{eV}=1.6\times {10}^{-19}\text{J}$ of kinetic energy has a speed of $5.9\times {10}^5\text{m/s}$. Although fast by macroscopic standards, this is a slow electron because it gains this speed by accelerating through a potential difference of a mere 1 V. At this speed, its de Broglie wavelength is found to be $\lambda =h/mv=1.2\text{nm}$. This is a small wavelength, but comparable to the wavelengths of x rays and a factor of 10 larger than the spacing of atoms in a crystal.

What does it mean for matter—an electron or a proton or a baseball—to have a wavelength? Would it obey the principle of superposition? Would it exhibit interference and diffraction? The classic test of “waviness” is Young’s double-slit experiment. FIGURE 38.12 shows the intensity pattern recorded after 50 keV electrons passed through two slits separated by $1.0\mu \text{m}$. The pattern is clearly a double-slit interference pattern, and the spacing of the fringes is exactly as predicted for a wavelength given by de Broglie’s formula. And because the electron beam was weak, with one electron at a time passing through the apparatus, it would appear that each electron—like photons—somehow went through both slits, then interfered with itself before striking the detector!

FIGURE 38.12 A double-slit interference pattern created with electrons.

Surprisingly, electrons—also neutrons—exhibit all the behavior we associate with waves. But electrons and neutrons are subatomic particles. What about entire atoms, aggregates of many fundamental particles? Amazing as it seems, research during the 1980s demonstrated that whole atoms, and even molecules, can produce interference patterns.

FIGURE 38.13 shows an atom interferometer. You learned in Chapter 33 that an interferometer, such as the Michelson interferometer, works by dividing a wave front into two waves, sending the two waves along separate paths, then recombining them. For light waves, wave division can be accomplished by sending light through the periodic slits in a diffraction grating. In an atom interferometer, the atom’s matter wave is divided by sending atoms through the periodic intensity of a standing light wave.

FIGURE 38.13 An atom interferometer.

Full Alternative Text

You can see in the figure that a laser creates three parallel standing waves of light, each with nodes spaced a distance $\lambda \text{/}2$ apart. The wavelength is chosen so that the light waves exert small forces on an atom in the laser beam. Because the intensity along a standing wave alternates between maximum at the antinodes and zero intensity at the nodes, an atom crossing the laser beam experiences a periodic force field. A particle-like atom would be deflected by this periodic force, but a wave is diffracted. After being diffracted by the first standing wave at A, an atom is, in some sense, traveling toward both point B and point C.

The second standing wave diffracts the atom waves again at points B and C, directing some of them toward D where, with a third diffraction, they are recombined after having traveled along different paths. The detector image shows interference fringes, exactly as would be expected for a wave but completely at odds with the expectation for particles.

The atom interferometer is fascinating because it completely inverts everything we previously learned about interference and diffraction. The scientists who studied the wave nature of light during the 19th century aimed light (a wave) at a diffraction grating (a periodic structure of matter) and found that it diffracted. Now we aim atoms (matter) at a standing wave (a periodic structure of light) and find that the atoms diffract. The roles of light and matter have been reversed!