If electrons also behaved like waves, then their waves would only fit on their orbits around the nucleus in certain ways, just like standing waves on vibrating strings. When the electron wavelength fitted exactly on circular orbits of 1 λ, 2 λ, 3 λ, etc. this would result in standing electron waves.
De Broglie knew that a moving a particle, such as an electron, has a certain momentum p = m.v, where m.v denotes mass x speed. Combining this momentum with the Planck formula for the energy of a light particle and Einstein's energy-mass equivalence, he proposed that this determined the associated wavelength of a moving electron. A photon has an associated wavelength and also an associated momentum, something that was already confirmed by the photoelectric effect [21]. Combining E = h.f, E = m.c2 and p = m.v he arrived at a very simple formula for the wavelength of an electron: λ = h/(mv) or λ = h/p. This means that the greater the speed v is, the shorter the associated electron wavelength will be. When jumping to a higher orbit the speed of the electron will be slowed down and its wavelength will therefore change accordingly.
Figure 4.10: The first four De Broglie harmonic electron orbits in the hydrogen atom.
Figure 4.10 shows the De Broglie standing electron waves, from left to right, for the lowest orbit, the "tonic" (n = 1), up to and including the 4th orbit (n = 4), the 3rd "harmonic". Incidentally, upon further study of those De Broglie orbits I hope you will notice that, unlike with the vibrating string with fixed ends, one complete wavelength fits on the lowest orbit, on the next orbit fit two complete wavelengths, etc. Standing wave orbits are not possible with an odd number of half wavelengths. This has to do with the fact that after having traveled a full orbit, the moving wave front has to arrive exactly in phase with its 'tail' to constructively interfere. The electron wave just travels around and is not reflected at any boundary as happens in a vibrating string. Reflecting on fixed boundaries invokes 180o phase shifts which corresponds to halved wavelengths. In short, only complete wavelengths are suitable for circular standing waves [22].
Only four years later De Broglie's hypothesis became confirmed experimentally, albeit in a rather circuitous way. Coincidence - serendipity - played a major role. The wave character of electrons was demonstrated in 1927 by Clinton Davisson (1881-1958) and Lester Germer (1896-1971) in experiments where electrons were reflected on a crystal lattice of nickel. The results of their experiment, preferred directions in which the scattered electrons reflected, could only be explained with interference - which implied wave behavior. This is in fact the same line of thought that Young followed with his double-slit experiment in 1805. Interference presumes waves.
Two years after the confirmation of electron waves by Davisson and Germer, Louis De Broglie received the physics Nobel Prize in 1929. Which was a well-earned recognition for his bold out-of-the-box thinking. For a good understanding of what the Davisson-Germer experiment entailed and what it confirmed, we should first pay some attention to diffraction grids.
Diffraction grids, a whole series of parallel slits
Diffraction gratings or diffraction grids follow exactly the same principle as Young's double-slit. The practical difference is that diffraction grid result shows considerably sharper lines of a higher intensity. A diffraction grid consists of a great number of parallel slits with very small and precisely equal mutual distances. The earliest form of a diffraction grid was a smoke blacked glass plate where a large number of parallel scratches was carefully applied to the soot. When lit with parallel traveling monochrome light, coming from a distant source, each slit will function as a Huygens wave source vibrating synchronously with the other slits, all slits producing then the same circular extending waves. A positive lens is applied then to focus these waves on a photosensitive screen. In a number of very specific locations on that screen, all arriving waves will just differ exactly a whole number of wavelengths, thus creating constructive interference.
When hitting the projection screen, sharply focused by the lens, these waves will all reinforce each other precisely and only in very definite angles producing sharp defined light lines. Destructive interference will occur in just slightly different directions because of the contribution to the interference from multiple slits. That way much sharper interference lines can be obtained than with a simple double-slit. With such diffraction grids the wavelengths of all kinds of EM-waves, not restricted here to the visible spectrum, can be measured with great precision. The geometric principle, where the angle that the light rays make with the slit holder determines the path length difference, is completely identical to the double-slit principle. See figure 4.11 for a geometric construction of the path length differences for a diffraction grid.
Figure 4.11: A diffraction grid with slits. Source: Wikimedia Commons.
In figure 4.11, g is the distance between the slits, φ is the angle at which the light measurements are made, and d is the path length difference between the waves originating from adjacent slits. The relationship between g, d and φ can be used to calculate the wavelength. Just for completeness, the formula is: sin (φ) = n.λ/g where n is the sequence number of the intensity maximum counted from the primary middle maximum.
But to understand quantum physics it isn't necessary to memorize or even understand this formula. Forget it if you will. It's not really relevant in understanding the subject of this book, quantum physics. Just remember that every diffraction grid or grating will produce interference effects in very distinct directions by creating synchronous wave sources departing from the slits.
Figure 4.12: Reflection on a diffraction grid with grooves such a DVD.
Source: Wikimedia Commons.
Reflection of light on parallel grooves, such as with light falling on the tracks of a DVD, will also produce diffraction effects. See figure 4.12. The principle as depicted in figure 4.11 remains the same for reflection, only the beam I0 arrives in this case from the right and synchronous wave sources will now also depart to the right. This explains the rainbow-like effects that you will observe when white light falls on a DVD. White light contains different wavelengths (colors) which need different angles for constructive interference and thus projects maxima at different locations. In this case the lenses in our eyes act as focusing lenses and our retina act as the photosensitive screen.
Diffraction experiment [23]: You can try this for yourself with a DVD or CD. Use light from different sources, an incandescent lamp, a fluorescent lamp and an LED lamp. What is the difference? Can you explain the difference? Can you roughly estimate the groove distance from what you observe if you know that yellow light has a wavelength of around 575 nanometers?
Serendipity with electrons
Serendipity often plays a role in important discoveries in physics. Think of Wilhelm Röntgen's discovery of X-rays because a screen with barium platinocyanide, that he happened to have standing somewhere in his lab, lit up. How wave behavior of electrons was confirmed experimentally is another excellent example of serendipity. From 1921