a localized wave pulse runs down the rope. It transfers an impulse to your friend’s hand, just as if you had thrown a ball to her.
The marriage of wave and particle came with Einstein’s corpuscle of light (the photon). It was, for a long time, a union blessed by very few sci-entists. From 1905 until the mid- 1920s, the idea that light could be both wave and particle was too much for most scientists to swallow. Compton’s 1923 experiments on the scattering of X-rays by electrons in atoms con-vinced some. Bose’s derivation of Planck’s radiation law concon-vinced oth-ers. Proof positive came in 1927 when Clinton Davisson and Lester Germer, working at Bell Labs in the United States, and, in de pen dently, George Thomson (son of J. J., the electron’s discoverer), working at Ab-erdeen University in Scotland, found that electrons striking the surface of a solid crystalline material exhibited diffraction and interference ef-fects (see Figure 42). From the known spacing of atoms in the crystal and from the mea sured angle at which electrons preferentially bounced from the crystalline surface, they could even mea sure the wavelength of the electrons.
In these early experiments, the wavelength of the electrons was com-parable to the spacing between atoms in a solid. Later, experimenters learned how to slow down neutrons so much that their wavelength greatly exceeded the spacing of atoms in a solid. (I will explain this inverse con-nection between speed and wavelength in answer to the next question.) As a result, a neutron drifting lazily through a material “reaches out” via its wavelength to interact simultaneously with many atoms, behavior hardly expected of a particle smaller than a single atomic nucleus.
65. What is the de Broglie equation? What is its signifi cance? By chance, the young French Prince Louis- Victor de Broglie (pronounced,
FIGURE 41 A wave pulse, like a particle, carries energy and momentum from one place to another.
roughly, “Broy”) received his bachelor’s degree in physics in 1913, the same year as Niels Bohr’s groundbreaking work on the quantum the-ory of the hydrogen atom. I don’t know if quantum physics was yet on de Broglie’s mind at that time, but later, in his Nobel Prize address of 1929, he did speak of his attraction to “the strange concept of the quantum, [which] continued to encroach on the whole domain of physics.”
Following ser vice in World War I, de Broglie took up graduate study in physics and, in 1924, as part of his doctoral dissertation at the Univer-sity of Paris, he offered the deceptively simple but powerful equation that now bears his name. The de Broglie equation is written
λ = h/p
On the left, the symbol λ (lambda) stands for wavelength— evidently a wave property. The p in the denominator on the right stands for momentum— clearly a particle property. Linking the two is Planck’s constant h, which appears in every equation of quantum physics. This equation was known to be true for photons— if you believed in photons.
De Broglie asserted it to be true for electrons and all particles. When Davisson, Germer, and Thomson mea sured the wavelengths of electrons a few years later, they found that indeed their mea sure ments conformed to de Broglie’s equation. The de Broglie equation has stood the test of
FIGURE 42
Experimental results of Davisson and Germer showing that electrons of 54 eV, after striking a nickel crystal, emerge mostly in a certain direction because of diffraction and interference of the electron waves. Image adapted from Nobel Lectures, Physics
time and remains a pillar of quantum physics. It is as simple in appear-ance and, in its way, as powerful as Einstein’s E = mc2.*
De Broglie said later that two things led him to his equation, which expresses what we now call the wave- particle duality. One was the evi-dence supplied by Compton’s then- recent 1923 work that X-rays exhibit particle as well as wave properties. The other was de Broglie’s observa-tion that in the classical world, waves are often quantized but particles, apparently, are not. He was thinking of the fact that the strings on a vio-lin and the air column in a fl ute vibrate with only certain selected fre-quencies, not arbitrary frequencies. He wondered whether the quantiza-tion then known to exist in atoms might be explained as the result of a vibrating wave— whether, in effect, the atom is a musical instrument.
The structures of Einstein’s and de Broglie’s equations differ in a simple yet signifi cant way. In E = mc2, the E and the m are both “upstairs.” E is directly proportional to m. More mass means more energy. In λ = h/p by contrast, λ is “upstairs” and p is “downstairs” (in the denominator).
Wavelength and momentum are inversely proportional. This means that increasing a particle’s momentum decreases its wavelength. So protons in more powerful accelerators have shorter wavelength than protons in less powerful accelerators, making them better able to probe phenomena at subnuclear distances. Neutrons slowed to very small speed— and small momentum— acquire relatively long wavelengths that enable them to reach out to interact with many atoms at a time. There are implications on the human scale, too. When you walk down the street, you have mo-mentum. If de Broglie is right, you must have a wavelength as well. Where is it? Why don’t you experience it? It is there, but because your momentum is so enormous on an atomic scale, your wavelength is too small by many orders of magnitude to detect. A 150- pound person strolling at two miles per hour has a wavelength of 4 × 10−34 inch— not encouraging for mea sure-ment. “But,” you might say, “I can move more slowly, in order to decrease
* You may wonder why Einstein’s equation does not contain Planck’s constant h. It is because E = mc2 is basically a classical equation, which happens to be valid in the quantum world as well.
my momentum and increase my wavelength.” Good thought, but it’s hope-less. If you creep at 1 inch per century, your wavelength will be 5 × 10 −23 inch, far less than the diameter of a single proton. We humans, like it or not, are denizens of the classical world.
But when you shrink the mass enough to enter the particle world, wavelength becomes very signifi cant indeed. Because of its wave nature, an electron within an atom spreads out to encompass the whole atom.
Similarly, neutrons and protons within the nucleus spread themselves over the nuclear volume. Only when a particle is accelerated to great energy does its wavelength shrink to less than the size of a nucleus or even the size of a single neutron or proton. Then the high- energy particle, with its shrunken wavelength, becomes a good probe of the smallest distances.
66. How are waves related to quantum lumps? When de Broglie offered his idea that all particles have wave properties, the full theory of quantum mechanics was yet to appear (it was developed in the following two years, stimulated in part by de Broglie’s work). In 1924, scientists still imagined electrons in atoms to be tracing out “Bohr orbits.” De Broglie’s ingenious idea was to suppose that the reason only certain orbits were allowed was the self- reinforcement, or constructive interference, of the wave that accompanied the particle in its orbit. As indicated in Figure 43, a wave could— and usually would— interfere destructively with itself after completing one trip around the orbit. Then, as de Broglie imagined it, the wave would simply wipe itself out, averaging to zero after multiple trips around. Such an orbit would simply not exist. But for certain se-lected orbits, the circumference would be an exactly integral number of
Wave destructively
interferes with itself Wave contructively reinforces itself p
r
p r
FIGURE 43
De Broglie’s idea of a wave interfering with itself.
wavelengths, and the wave, after completing one or any number of trips around, would constructively reinforce itself. That, suggested de Broglie, would explain why only certain orbits were permitted within the atom.
The orbiting electron would be like the stretched string on a guitar, ca-pable of vibrating with only certain selected wavelengths. Quantization in the atom then becomes no more mysterious than the “quantization”
of vibrational frequencies in a musical instrument.
When de Broglie applied this reasoning to circular orbits in the hy-drogen atom, he found exact agreement with the experimentally known energies in that atom. The lowest- energy state, or “ground state,” was the one in which one wavelength stretched around the circumference, the fi rst excited state had two wavelengths stretched around the circumfer-ence, the next state three wavelengths, and so on. The soon- developed quantum mechanics upended this picture of the atom, but not totally.
De Broglie’s idea of self- reinforcing waves within atoms survived the revolution. Yet there were big changes. The biggest, perhaps, was that the electron is not to be looked at as a particle accompanied by a wave. Rather, it is a wave— one spread over the whole three- dimensional space within an atom, not just stretched out along an orbit. Consequently, the oscilla-tion of the electron wave is not just circumferentially around the nucleus, it is also radially in and out toward and away from the nucleus, and can be a combination of both the roundabout and in- and- out vibrations.
Figure 44, for example, shows two possible distributions of electron intensity (really of electron probability) for the fourth state (the third ex-cited state) in a hydrogen atom. The shaded regions show where the elec-tron is likely to be found if something— such as a high- energy X-ray—
probes the atom in search of a particle amid the wave. In the upper diagram, the high- intensity regions are concentric circles (with an inten-sity peak also right at the nucleus). This state of motion is one with zero angular momentum, meaning that the electron, instead of circling around the nucleus, can be visualized as running back and forth radially, toward and away from the nucleus. It is analogous, is a way, to the waves on the surface of still water that spread out radially when a pebble is dropped into the water. The direction of propagation of such a wave is away from the center, not around the center.
By contrast, the lower diagram shows the wave pattern for a state of motion with the most angular momentum that is possible at this en-ergy (actually, three units). This pattern corre-sponds to circular motion around the nucleus and thus matches de Broglie’s original idea. It is an example with four full wavelengths around the circumference. The reason you see eight dark regions in the diagram rather than four is that a peak of intensity occurs at every minimum as well as every maximum of the wave, so there are two peaks per wavelength. Notice in the diagram that the dark regions, although concentrated at a certain radius, do have some in- and- out spread.
You can never completely corral a wave.
Just a reminder of how these diagrams relate to energy quantization. There can be two or three or four wavelengths in and out or round-about, but never two- and- a-third wavelengths or four- and- a-half wavelengths. Because of the re-striction to a whole number of wavelengths, there is a restriction to only certain energy val-ues, just as de Broglie originally hypothesized.
Back at Question 3 I discussed the
corre-spondence principle, the idea introduced by Niels Bohr that when a quan-tum property changes by a small fraction from one state to the next, quantum behavior closely mimics classical behavior. This is illustrated in Figure 45, which shows the electron intensity pattern in the twentieth state of the hydrogen atom with the greatest possible angular momentum for that energy. You see, very clearly, a wave following a circular path far from the atomic nucleus. There are forty intensity peaks, corresponding to the twenty full wavelengths around the circumference, and there is a lot of empty space (actually, almost empty space) between the orbit and the nu-cleus. The intensity is bunched up radially and the pattern begins to re-semble a classical circular orbit— although, to be sure, still showing a clear
FIGURE 44 radially in and out, the other with the electron circling the nucleus.
wave pattern. A little closer to the nucleus there is an allowed state of motion with nine-teen wavelengths around its orbit, and a little farther out there is one with twenty- one wave-lengths around its orbit. These permitted or-bits, although quite distinct, each with its own energy, are, percentagewise, not far apart, again illustrating the correspondence principle, since classically an orbit could have any radius at all.
67. How do waves relate to the size of atoms? Imagine that you have a proton in one hand and an electron in the other. You release them at some distance apart with only empty space between them. What happens? They are drawn together by an electric force (since unlike charges attract). Classically, the electron would either dive straight into the proton or spiral around it in ever smaller loops as it radiates away energy. That classical expectation is not unlike what happens when you release a marble within a bowl. It will either roll straight down, eventually settling at the center as friction dissipates its energy, or it will spiral downward, still ending up at the center. It will have found the point of lowest energy. The electron would “like” to do the same, but its wave nature gets in the way. As the electron orbits closer and closer to the nucleus, its wave “pulls in” to an ever shorter wavelength. Enter de Broglie.
Shorter wavelength means larger momentum. As the space over which the electron ranges gets smaller, its wavelength gets shorter, and it moves ever faster as its momentum grows. You see the momentous signifi cance of the inverse relationship between wavelength and momentum.
And as momentum grows, kinetic energy grows. As the electron gets confi ned to less and less space, its energy of motion gets larger and larger.
It is almost as if there is a repulsive force countering the attractive elec-tric force. In energy terms, there are two competing effects. Elecelec-tric at-traction lowers the energy— the potential energy— as the electron gets closer to the nucleus, while, at the same time, the electron’s wave nature causes an increase in energy— its kinetic energy— as it gets closer to the
FIGURE 45
nucleus. The two effects fi nd a point of balance where the total energy is a minimum— a point where moving in toward the nucleus would cause the total energy to increase as the kinetic energy effect dominates and moving away from the nucleus would also cause the total energy to in-crease as the potential energy effect dominates. That point of balance for a single electron and proton is at a distance of about 10−10 m (a tenth of a nanometer), very large by usual particle standards— in fact, about one hundred thousand times larger than the size of the proton itself.
Thanks to the wave nature of matter, atoms are “huge”— huge relative to the size of any composite particle such as a proton, neutron, or pion.
The small mass of the electron also plays a role in fi xing the relatively large size of the atom. Momentum is the product of mass and speed, so it is not only small speed but also small mass that contributes to large wavelength. Because the electron is far less massive than any other par-ticle (except the photon and neutrinos), it has, typically, less momentum and greater wavelength than other particles and so occupies more space.
(The neutron provides an exception to this rule. It can be slowed so much that despite its large mass it can have a large wavelength and, as I mentioned before, spread itself over a space as large as that occupied by several atoms.)
There are two other questions related to the wave nature of the elec-tron and the size of atoms. One is why heavy atoms, despite having more powerful electric attractive forces, do not collapse to a size much smaller than the size of a hydrogen atom. The other question is why an atom in an excited state can be quite a bit larger than in its ground state. Here I amplify answers to these questions given at Question 31.
If you considered the fate of a single electron near, say, a uranium nucleus, it would indeed fi nd its point of least energy, its point of bal-ance, much closer to the nucleus than in a hydrogen atom. But as you add more and more electrons, they feel a less and less powerful force, on average. The ninety- second electron in the uranium nucleus is pulled inward by ninety- two protons and pushed outward by ninety- one other electrons. The net pull on it is about the same as in a hydrogen atom, so that last electron settles into a state of motion not very different in size from the size of a hydrogen atom.
As to why excited atoms are larger than ground- state atoms: The wave for an electron in its lowest state has just one cycle of oscillation. It rises and falls just once over a dimension that defi nes the size of that state of motion. In an excited state, the electron wave undergoes two or more cycles of oscillation. So the excited- state wave needs more “elbow room” to complete its multiple cycles of oscillation. That is a bit of an oversimplifi cation but it gives the general idea. If the excited- state wave, with its multiple oscillations, were squeezed down to a size as small as the ground- state wave, it would have a shorter wavelength, a greater momen-tum, and a greater kinetic energy. It could then lower its total energy and fi nd a point of balance at a larger size.
68. What is diffraction? What is interference? There is something basically “fuzzy” about waves. They don’t follow narrowly defi ned paths like the paths of baseballs or spacecraft. They occupy a region of space, have no well- defi ned boundaries, spread as they propagate, and can over-lap one another.
When a wave passes through an opening or by an edge, it bends (Fig-ure 46). That is called diffraction. It can be seen in water waves that pass a ship at anchor, or it can be experienced indirectly by the fact that your wireless phone usually works even if there is a building between you and the cellular antenna. The diffraction effect is more pronounced for larger wavelength, which explains why longer- wavelength am radio signals bend around obstacles more readily than shorter- wavelength fm signals do.
Driving in the canyons of a big city, you are likely to fi nd am stations to be a bit more reliable than FM stations.
Driving in the canyons of a big city, you are likely to fi nd am stations to be a bit more reliable than FM stations.