A rapidly occurring string of developments, starting in 1922, would reveal a much different physical world than Bohr, Sommerfeld, Pauli, Einstein, or anyone else had ever imagined. In this key Chapter 3, I describe those developments and provide a first glimpse into that world, our world.
COMPTON SCATTERING (PROOF OF THE PARTICLE NATURE OF THE QUANTUM OF LIGHT)
The outstanding twenty-seven-year-old American experimental physicist Arthur Holly Compton (shown directly behind Einstein in the second row of Fig. 1.1) had been appointed professor and head of physics at Washington University in St. Louis, Missouri, in 1920. Later, at the University of Wisconsin over the winter of 1922–1923, he found an extraordinary result while working with x-rays (high-energy electromagnetic radiation with wavelengths approximately the sizes of atoms). Monochromatic x-rays shining on graphite reflected back with changed and lengthened wavelengths. It was as if violet light reflecting off of a substance would change its color to red in the process. What was observed is now called the Compton effect.
Nothing like that had ever been seen to happen with waves of any kind. In fact, what was seen couldn't be explained by waves at all. The only viable explanation was that electromagnetic radiation, as Einstein had shown theoretically in 1916, had momentum in the same sense that a particle has momentum. The electrons in the atoms of graphite (carbon) would recoil from impact and thereby absorb some of the momentum of an incoming x-ray quantum, so that the latter would reflect away in another direction with reduced energy and momentum and a correspondingly increased wavelength.
But Bohr and two colleagues questioned Compton's conclusions. Bohr simply would not accept that light itself would be quantized and have momentum. Compton had assumed conservation of energy in the collision, and Bohr argued (defying one of the universal fundamentals of physics) that conservation of energy had never been proved on an atomic level.
Compton then found a way to confirm the existence of the recoiling electrons and prove that energy and momentum were indeed conserved. And Hans Geiger and Walther Bothe in Germany obtained the same results. The collection of results from the experiments was irrefutable evidence of the particlelike nature of the electromagnetic quantum, by inference also including the light quantum. With these experiments and their interpretation, the quantum of light was finally accepted by the physics community as behaving as both a particle and a wave (as Einstein first proposed it with his analysis of the photoelectric effect in 1905). The “particle” part of wave-particle duality had been confirmed. Einstein had finally been vindicated in the view that he had advocated nearly alone for over twenty years. Compton would in 1927 receive the Nobel Prize in Physics “for his discovery of the effect named after him.”
In a paper presented in 1928, Compton used the Greek word for “light”—photon—to refer to single particles of light or x-rays, and this has become accepted use for particles of electromagnetic radiation in general. (But Gilbert Lewis is described as naming the photon in 1926.1) I use the term “light” in this broader context in the rest of this book, though it is more commonly thought of as applying to light in the visible range of the electromagnetic spectrum (refer to Appendix A).
PARTICLES AS WAVES (TURN-AROUND IS FAIR PLAY?)
Louis de Broglie (pronounced “dee-BROY”; third from the right in the second row of Fig. 1.1 and, more closely, in Fig. 3.1) was a graduate student at the Sorbonne in Paris in 1923. He reasoned that if light could have particlelike properties, perhaps particles, like the electron, could have wavelike properties. He backed his thoughts with calculations. It was a radical idea, but when the outside thesis examiner, Paul Langevin, wrote to ask a separate opinion, Einstein examined the work and replied: “He has lifted a corner of the great veil.”2
Fig. 3.1. Louis de Broglie. (Image from Deutscher Verlag, courtesy of AIP Emilio Segre Visual Archives, Brittle Books Collection.)
Louis Victor Pierre Raymond de Broglie was born in 1892 into an aristocratic French family with the titles of both a French duke and a German prince. His brother, Maurice, seventeen years older, had become head of the family after their father died. Louis was just fourteen at the time. Maurice had started a career in the military, as was the family tradition, but there he got involved with wireless communications, became interested in science, and eventually left the service to obtain a doctorate under Langevin at the College de France. He would go on to build a laboratory in his mansion in Paris and become a recognized researcher on x-rays.
Louis was tutored at home, and guided by Maurice into general studies. But then he opted into the sciences through exposure to Maurice and the lab. One year of compulsory military service was extended a further five years with the outbreak of the war, after which he worked with his brother and wrote several papers.
Louis pictured the electrons in Bohr's orbits as being related to standing waves, like the harmonic vibrations in a guitar string shown in Figure 3.2(a). (These are “standing” because they are trapped and don't travel anywhere.) The half-wavelength string shown in the bottom figure would vibrate up and down at the string's “fundamental” frequency, determining the string's basic pitch. The extent of the “down” swing in this case is shown as a solid line, and the extent of the “up” swing is shown as a dashed line. These extremes mark the “amplitude” of the vibration. The three “harmonics,” shown above the fundamental, contribute to the “timbre” of the string's sound. These harmonics vibrate at successive integral multiples of the fundamental frequency.
De Broglie surmised that the full wavelength of a wave, like that shown second from the bottom in Figure 3.2(a), must fit an exact number of times around the circumference of an orbit. Five such wavelengths are shown to fit around the n = 5 Bohr orbit, as shown in Figure 3.2(b). This condition would result in the same quantized energies and orbits that had been determined originally by Bohr. But because de Broglie's electron was viewed as a standing wave and not a particle in orbit, there would be no acceleration of charge and no associated radiative slowdown and collapse. So, with de Broglie's standing wave the conflict with classical physics and the continuing fundamental objection to the Bohr model would be avoided.
Fig. 3.2. (a) Note the “fundamental” (bottom) and three lowest harmonic vibrations of a single guitar string. Each vibrational mode is called a “standing wave” because the string just vibrates in each case between the extremes of the solid line and the dashed line and the waves shown don't travel anywhere. (b) De Broglie envisioned the states of the electron in the hydrogen atom as a set of standing waves that circle within the Bohr orbits and close on themselves. The n = 5 standing wave is drawn over an n = 5 Bohr orbit.
De Broglie also suggested that electrons by themselves as free particles should undergo a wavelike diffraction and interference, much as had been observed for particles of light, as illustrated in Figure 2.2 by analogy to what happens with waves in water.
This wavelike nature of the electron was glimpsed in 1926 by a team of two American physicists, Clinton Davisson and Lester Germer, when they saw preliminary evidence of diffraction and interference as they directed a beam of electrons at a nickel crystal. (The regular lattice of atoms in the crystal would act to diffract the electrons to interfere in somewhat the same way that the two openings in the breakwater acted to diffract water waves to interfere, as illustrated in Fig. 2.2.) The two researchers were unaware at the time of de Broglie's ideas. Diffraction (when waves spread out past openings) and the resulting interference (when they come together) were confirmed the following year by these men and independently by the British physicist George Thomson (son of J. J. Thomson), who in a somewhat different experiment got similar results as he passed beams of electrons through thin metal films.
Diffraction and interference do not occur unless something is wavelike! The classical concept of particles always behaving as pointlike specifically located objects had been successfully challenged. Their (sometimes, at least) wavelike nature had been demonstrated as fact!
(De Broglie would in 1929 receive the Nobel Prize in Physics “for his discovery of the wave nature of electrons.” Davisson and Thomson would share the prize in 1937 “for their experimental discovery of the diffraction of electrons by crystals.”)
I interrupt my chronological narrative here to describe more recent experiments that further elucidate the wave nature of electrons and light quanta.
Fig. 3.3. Electrons are shot one at a time at a barrier with two slits in it. Each electron produces a dark dot on a white screen. With many electrons this results is an interference pattern with a dark bar of dots at the center, fading to two lighter bars of dots on either side and still lighter bars (not shown) beyond those. Counting the density of the dots would produce a graph of intensity resembling the wave heights shown in Figure 2.2(c). (Image from Wikipedia Creative Commons; file: Double-slit.png; assumed author: NekoJaNekoJa~commonswiki. Licensed under CC BY-SA 3.0.)
The dual particle/wave nature of electrons has been beautifully demonstrated in a two-slit diffraction and interference experiment. The setup for the experiment is shown schematically in Figure 3.3. Everything shown is enclosed in an evacuated chamber. Electrons are shot one at a time from an “electron gun” at the midpoint between two slits in an otherwise-blocking sheet of material, and their impacts are recorded as a tiny spots on an observing screen. The accumulation of tens of thousands of spots, each from one of tens of thousands of electrons, produces an interference pattern (shown schematically on the screen) that resembles the interference pattern of a wave.
Fig. 3.4. Here we have the same two-slit experiment as in Figure 3.3, except that there is a dark screen that produces a white dot when the electron strikes, and we see the dots of the individual electrons. We see the growth of the interference pattern: (a) with 11 electrons having been shot, (b) with 200, (c) with 6,000, (d) with 40,000, and (e) with 140,000. Note in (a), but it applies to (a) through (e): each electron, shot as a particle by itself, interferes with itself to impact the screen as if it has traveled through both slits as a wave. (Image from Wikipedia Creative Commons; file: Double slit experiment results Tanamura 2.jpg; user: Belsazar, with permission of Dr. Akira Tanamura. Licensed under CC BY-SA 3.0.)
Figure 3.4 shows the actual gradual accumulation of spots for the experiment where the screen this time is black and the spots where the electrons impact the screen are white: (a) shows the accumulation of spots from just 11 electrons, (b) for 200 electrons, (c) for 6,000 electrons, (d) for 40,000 electrons, and (e) for 140,000 electrons. The particle nature of the electrons is in evidence by the fact that they are shot from the gun one particle at a time and hit the screen one particle at a time, in each case creating only one spot. But somehow that single electron has a wave property that allows it, if not traveling through both slits, to at least sense the locations of both slits, so that its single spot of impact is located in such a way as to contribute, with the spots of other separately shot electrons, to the formation of an interference pattern.
For the electron, which we think of as a particle, we thus have wavelike behavior. Could the electron actually have split somehow and traveled through two slits to interfere with itself?
Now, remember the two-slit experiment conducted by Thomas Young to show that light was wavelike, as described following the discussion of Figure 2.2. In a recent experiment by Lymon Page of Princeton University, monochromatic light (light all of a single, distinct wavelength) was reduced in intensity so that single quanta, essentially single photons, were shot one at a time through the two slits in a setup like that shown in Figure 3.3. A similar accumulation of single spots occurred, producing a wavelike interference pattern. We pose the similar question: How does a single, indivisible photon travel through both slits and interfere with itself?
Classical ideas, classical physics, can in no way explain the wave nature of individual particles. The double-slit experiment, either for single electrons or for single light quanta, remains one of the best demonstrations that we live in a quantum world. And, as you can see, our quantum world is strange. Quantum mechanics provides an explanation for this strangeness. Stay tuned.
(I would say that we have shed some light on this subject but have not as yet fully illuminated it [puns intended].)
There is a practical side to the wavelike behavior of particles.
It had been determined some considerable time before 1926 that the magnification that could be achieved in optical microscopes would be limited by the wavelength of light. Wavelength needed to be short compared to the size of the object being viewed. With light one could see blood cells, about ten microns (10–5 meters) in size, but not much that was smaller.
In 1931, realizing that electrons could be produced with wavelengths 100,000 times shorter than for visible light, Max Knoll and Ernst Rusk invented the electron microscope. Now one could “see” almost to the very size of the atom. The manufacture of commercial units was begun in England in 1935.
(I resume now my chronological narrative of early discoveries in the development of quantum mechanics.) Though Bohr and Sommerfeld in 1916 had invoked a quantized set of orbits in their model of the atom, their model still used classical physics, point particles, and orbits otherwise resembling those of the planets—still “solid ground” for most physicists at the time. However, with the still persistent issue of expected collapse of orbits due to radiation, with the suggestion of the de Broglie atom, and with the experimental demonstrations of the Compton effect in 1923 and electron diffraction and interference in nickel crystals and metal films in 1926, that ground began to shake and crack apart.
QUANTUM MECHANICS (FINALLY, A SOUND THEORY)
Even before the above-mentioned diffraction experiments, three scientists were exploring separate theoretical approaches toward better explaining the nature and the workings of the atom: Werner Heisenberg in Germany (standing third from the right in the back row in Fig. 1.1), Erwin Schrödinger from Austria (pronounced in English as “SCHROW-dinger”; standing sixth from the right in the center of the back row), and the reclusive Paul Dirac from England (pronounced “deer-ACK”; in front of Schrödinger and behind Einstein). Remarkably, all three approaches produced the same results. However, of the three, Schrödinger's “wave mechanics” version lends itself best to visualization, and we concentrate on his approach to describe the fascinating key concepts of the atom and eventually the broader workings of our universe (or universes, as you will see). First, a quick glimpse of the three approaches and the personalities involved in their development.
Matrix Mechanics
Working in Göttingen in June 1925, Werner Heisenberg struggled to make sense of the spectral lines from hydrogen, a problem that his professor, Max Born, had assigned him two years earlier. Heisenberg came up with the idea that the atom could be treated as a bunch of quantum oscillators (an approach somewhat similar to that used by Planck on light radiation twenty-five years earlier and by Einstein on the specific heat of solids nine years after that). Heisenberg developed an array describing the possible transitions within the atom, a theory of sorts. With encouragement from Wolfgang Pauli (who had come to be respected and feared as a no-holds-barred judge and critic of work in the field), Heisenberg published his theory, the first paper on quantum mechanics. Einstein and Bohr were skeptical but hopeful that it would open up something new. But the theory, though stimulated by consideration of the atom, was rather obscure and had yet to be successfully applied.
Fig. 3.5. Werner Heisenberg in 1927. (Image from AIP Emilio Segre Visual Archives, Segre Collection.)
Werner Karl Heisenberg (shown more closely in Fig. 3.5) was born in Würzburg, Germany on December 5, 1901, the younger of two boys.
His father became professor of Byzantine philology at Munich University and moved the family there when Werner was eight. The aftermath of the war was particularly difficult in Bavaria, where radical socialists sought to declare a Soviet Republic. Werner and his friends formed one of the many militarylike groups that were organized to oppose the movement.
Through it all, Heisenberg excelled in school and won a prestigious scholarship to the university. His coming of age in physics involved study and work at the three major centers of thinking on quantum physics. He began in 1920 as an undergraduate alongside Pauli in Sommerfeld's institute in Munich. Both recognized Heisenberg's potential, and Pauli and Heisenberg were afterward to maintain a close professional communication. When Sommerfeld left for a stint in America, he arranged for Heisenberg to study with Born in Göttingen. Heisenberg subsequently returned to Munich, completed his doctorate, and then responded to Bohr's invitation that he spend a year in Copenhagen, this following Heisenberg's asking penetrating questions of Bohr after one of his Göttingen lectures in 1922.
Pauli had written to Bohr of Heisenberg, and the young physicist was received warmly and personally. The men developed a professional and personal relationship. (This has been portrayed as somewhat of a father/son relationship in the play and movie Copenhagen. It describes Heisenberg's visit with Bohr after Heisenberg had taken over leadership of the Nazi atomic weapons program during World War II.) Bohr, as Heisenberg would soon learn, even more than others, was concerned with the deficiencies being revealed of his own model for the atom.
Born recognized that working with Heisenberg's array of oscillators would require the mathematics of matrices, a set of mathematical tools not generally known to physicists at the time. Born enlisted his student, twenty-two-year-old Pascual Jordan, who had transferred into physics from mathematics, to devise a good theoretical framework. They submitted their findings in collaboration with Heisenberg for publication in October 1925. Their approach would be labeled more specifically matrix mechanics.
Operating with matrix mechanics was cumbersome, but Pauli, working in parallel, mastered it and applied it to successfully derive the hydrogen spectrum. Matrix mechanics got results! But what did the atom look like? The theory wouldn't say.
Dirac's Quantum Mechanics
Paul Dirac's paper recognizing the fundamental physics underlying Heisenberg's original theory was received by the Proceedings of the Royal Society in London before the “three-man paper” on matrix mechanics described above had been submitted for publication.
Fig. 3.6. Paul Dirac at middle age. (Photograph by A. Bortzells Tryckeri, courtesy of AIP Emilio Segre Visual Archives, E. Scott Barr Collection, Weber Collection.)
Paul Adrien Maurice Dirac (shown more closely at middle age in Fig. 3.6) was born in 1902 as the second of three children of an English mother and a Swiss Frenchspeaking father.
His overbearing father, who taught French, insisted that all communication with his son be in French. But Paul found that he could not express himself well in French, and so he chose not to speak very much at all. This carried over as a general characteristic. He tended to be withdrawn. But he was also brilliant.
Dirac was interested in science, but his father steered him into engineering. He graduated in three years from the University of Bristol as an electrical engineer, was unable to find a job in the aftermath of the war, and continued there with the offer of free tuition to earn a degree in mathematics. Then, with the help of a government grant, he was admitted to Cambridge, which earlier he had simply been unable to afford. Even while an engineering student he had read and understood Einstein's relativity, and he'd listened to Bohr lecture at Cambridge; he was impressed by the man, but not by his theory.
Working alone at Cambridge, but exposed to Heisenberg's earlier work by his advisor Ralph Fowler (Rutherford's son-in-law, second from the right in the back row of Fig. 1.1), Dirac by the end of November 1925 had turned out four papers that together constituted his PhD thesis and earned him his doctorate. It was a more complete and careful work than the three-man paper, and it used a different formalism. (Dirac would contribute in an even more fundamental way to quantum mechanics, as will be described in Part Two. He would combine relativity and Schrödinger's equation in a way that naturally produced the spin of the electron. Later he would be elected Lucasian Professor of Mathematics at Cambridge, a post held earlier by Isaac Newton and to be held later by Stephen Hawking.)
Wave Mechanics
Schrödinger, a professor at the ETH in Zurich, was already a published, solid, recognized physicist when in October 1925 he began to address the problems of the Bohr atom.
Fig. 3.7. Erwin Schrödinger in 1933. (Image from Wikipedia Creative Commons; file: Erwin Schrödinger (1933).jpg; author: Nobel Foundation.)
Erwin Schrödinger, shown in Figure 3.7, was born on August 12, 1887, the only child in an upper-middle-class family in Vienna, at a time when the waning Hapsburg Empire and first Austrian republic were centers of contemporary culture. After excelling in the equivalent of American high school and college, Schrödinger went on to graduate school and was in 1914 awarded a doctorate in physics from the University of Vienna.
That year marked the beginning of World War I, and Schrödinger was soon called up as an artillery officer on the Italian front, where he served with distinction. By the war's conclusion in 1918, his father's business had closed. Both of his parents died soon thereafter, and prospects were poor for the young and brilliant physicist in postwar occupied Austria.
Eventually, in 1920, at thirty-two he married his longtime sweetheart, Annemarie Bertel, a working twenty-three-year-old country girl from a good family. But this required that he take a series of positions throughout Europe where he could be paid well enough to support her. His work on wave mechanics was at the height of his career in and around Zurich in 1926.
In 1933, Schrödinger left a distinguished professorship as Planck's successor in Berlin for a position at Oxford because he disliked Germany's sanctioned persecution of the Jews. Hitler had just become chancellor. Moore notes: “Schrödinger was exceptional—very few non-Jewish professors refused to knuckle under to the Nazis,” punctuating this with the further comment, “In the autumn of 1933, 960 professors published a vow in support of Hitler.”3 Schrödinger was not one of them.
Schrödinger returned to Austria in 1936. But after the annexation of Austria by Germany in 1939, Schrödinger found it difficult to remain there, having earlier been entered into the Nazi records as “politically unreliable.” He moved with his wife to take visiting positions at Oxford and at Ghent before being invited by the prime minister of Ireland, Eamon de Valera, to help set up the Institute for Advanced Studies in Dublin. He became director of the School for Theoretical Physics and remained there for seventeen years, during which time he became a naturalized Irish citizen and wrote another fifty publications on various topics, including efforts to develop a unified field theory.
James Watson refers to Schrödinger's book What Is Life, written in 1944, as having inspired Watson to discover (with Francis Crick, Maurice Wilkins, and crystallographer Rosalind Franklin) the double-helix structure of DNA, which, as you may know, allows replication of the genetic code during the process of cell division. (Watson, Crick, and Wilkins received the Nobel Prize, but Franklin died before the prize was awarded, and therefore she was not a candidate. Her work was critical to the discovery, however.)
Schrödinger found Heisenberg's 1925 oscillator approach to quantum mechanics to be obscure and lacking any comfortable physical description. He turned to examining de Broglie's standing-wave theory, following a footnote in one of Einstein's papers. He set to work to try to solve for the electron in the atom as standing waves in three dimensions. His resulting description of the atom as a set of standing waves seemed more tangible than matrix mechanics, involved a more widely known and easily used mathematics, and seemed at the time to offer some possibility of a comfortable connection to the classical physics that many (especially Schrödinger himself) were reluctant to abandon.
A Conflict Over Theories?
Conceived over Christmas 1925 and published in March 1926, Schrödinger's wave mechanics was welcomed by Einstein and Bohr and eventually most in the field (but not without questions); it was challenged especially by Heisenberg, who was increasingly becoming frustrated that the physics community was endorsing Schrödinger's concepts rather than his own. During Schrödinger's lecture in Munich in July, Heisenberg asked questions that Schrödinger had difficulty answering. For example, Schrödinger could not explain the photoelectric effect: How could the electron emerge as a “popped loose” wave? And how would a wave carry the charge of the electron? Schrödinger at first thought this could be done with the charge smeared-out in some fashion. But this would seem to deny the electron's observed particle nature and possession always of a single, indivisible localized basic unit of charge.
Following the meeting, Bohr invited Schrödinger for a visit to Copenhagen. In early October, the two of them, together with Heisenberg, who had returned there from Munich, would try over several days to sort out conflicts and questions in both theories. It was more Bohr than Heisenberg who was relentless in questioning the weakness of wave mechanics. But Heisenberg and Bohr had no answer to Schrödinger's call for a mechanism describing “quantum jumps,” the transformation of the electron from one energy state to another. Schrödinger's wave mechanics would seem to allow for such a mechanism, but Heisenberg's matrix mechanics would not. A broader, all-encompassing formalism seemed to be needed to answer the questions in both theories.
Transformation Theory
The answer was provided in part independently in late 1926 by Pascual Jordan and Paul Dirac. While working at Bohr's invitation during a six-month stay in Copenhagen, Dirac showed that matrix and wave mechanics were just special mathematical cases of his own broader “transformation theory.” This sorted the mathematics, but a good physical interpretation of the theories and resolution of their apparently conflicting physical aspects was still lacking.
(Heisenberg in 1932 would receive the Nobel Prize in Physics “for the creation of quantum mechanics, the application of which has, inter alia, led to the discovery of allotropic forms of hydrogen.” He felt that Jordan and Born should have shared the prize. In 1933, Schrödinger and Dirac would together be awarded the prize “for the discovery of new productive forms of atomic theory.”)
WAVE MECHANICS AND THE HYDROGEN ATOM (HOW IT WAS DONE)
Schrödinger's “wave mechanics” approach to quantum mechanics gives us the best picture of what is going on. I describe it as follows, starting with a simple analogy, and conclude with another analogy that shows his results in comparison with something more familiar to us.
What Schrödinger did in 1925 is a little bit like what many of us learned back in high-school algebra. Remember, we would use the letter x to represent some unknown sought answer, and then we would set up an equation to describe the circumstances related to x that offered clues to its value. We would then solve the equation for x. Sometimes the solution would produce two values of x. These were the only values for x that would fit the circumstances, the only correct and allowed answers.
Schrödinger defined a complex mathematical function represented by the Greek letter Ψ (pronounced “psi”). Ψ would contain all of the information about a particle's position and movement through space and time. Every one of the physical properties of the particle could be extracted from Ψ by performing certain separate mathematical operations on it. One operation would provide the electron's position, another its velocity, a third its energy, and so on.
Instead of solving for numbers for the unknown number x as we did in high school, Schrödinger would solve for the mathematical form of the unknown function Ψ. He set up an equation making the total energy of the atomic system (given by an appropriate mathematical operation on Ψ) equal to the sum of the energy of motion of hydrogen's single electron (given by a different operation on Ψ) and the energy associated with the attraction of the electron to the single-proton nucleus (given by yet another operation on Ψ).
It sounds like a lot of mathematical mumbo-jumbo, but when Schrödinger solved his equation the results were a first-order triumph. With the added inclusion only of the assumption of spin, later to be derived by Dirac as an intrinsic property of the electron (and so no longer an assumption), Schrödinger's model almost exactly predicted all of the observed properties of the hydrogen atom, and did so without the ad hoc postulates required for the Bohr-Sommerfeld model!
When Schrödinger solved his equation, he got not one or two but an infinite series of separate standing-wave solutions for Ψ at discrete, separate, energy levels. These corresponded exactly with the energy levels of the Bohr model shown in Figure 2.7(b), and, with spin included, further split those levels into the closely spaced energies derived by Sommerfeld. These solutions for Ψ represent the only states that mathematically satisfy the wavelike physics of the electron in the hydrogen atom. They are thus the only allowed states for the electron in hydrogen. And the differences in the energies of these states give exactly the hydrogen spectrum that had been characterized (as described in Chapter 2) by Balmer, along with the further fine structure and magnetic- and electric-field splitting of the lines that were observed later on. But, most importantly, unlike the Bohr atom, there aren't any orbits, and so there is no acceleration of the electron and no reason for radiation and collapse!
PROBABILITY (A WAY TO VISUALIZE THE ATOM)
Just a little later, in August 1926, Max Born was beginning to suggest that particles are not physical waves, rather, that wavelike characteristics are only in the mathematics of wave mechanics (which determines among other things where a particle may be found). According to Born, the magnitude (think of wave amplitude) of one of Schrödinger's mathematical standing-wave solutions at any point in space should be a measure of the probability that a particle would actually be located at that particular point. In this way he would retain the ideas of both waves and particles and resolve the differences between Heisenberg's more particulate theory and Schrödinger's more wavelike theory. Born's probabilistic view would be used to interpret Schrödinger's results and would be endorsed by Bohr to become a part of what would be called the Copenhagen interpretation of quantum mechanics (to be further explained later on). Born, in 1954, received the Nobel Prize in Physics “for his fundamental research in quantum mechanics, especially for his statistical interpretation of the wavefunction.”
But Born's probabilities are not “statistical” in the classical physics sense of indicating the probable behavior of a collection of particles. Classical statistics, for example, would show that atoms of helium gas released from a popped balloon are unlikely, in bouncing around the room, to find themselves some time later all back in the region that had been occupied by the balloon. Born's probabilities in the case of electron states are statistical only in “not being definite” about location. They are more akin to the probabilities of getting a seven or “snake eyes” in the roll of a pair of dice.
(Note, Schrödinger's equation applied to free particles also results in wavefunctions, and these evolve with time in such a way as to indicate the probable motion of the particle. His equations can, in principle, be applied to whole systems of particles and objects, with resulting wavefunctions that describe the evolution of the entire system, even the evolution of the entire universe.)
A Way to Visualize the Atom
The most common way of describing the spatial form of the quantum atom today comes from Born's probability interpretation of Schrödinger's standing-wave solutions Ψ for hydrogen. In the best representation of the atom that I have seen anywhere, the probability of hydrogen's one electron being at any particular point in space is marked as a spot of certain brightness in an otherwise three-dimensional blackness. If for each solution to Schrödinger's equation for hydrogen we visualize all points in space as having a brightness in proportion to the magnitude of Ψ, we create a fuzzy white probability cloud with inherent symmetries. The cross sections of five of these clouds, representative of five of the lowest energy states in hydrogen, are shown magnified one hundred million times in Figure 3.8.
These states are in fact, mathematically, standing-wave solutions to Schrödinger's equation. They provide rigorously derived three-dimensional representations in furtherance of de Broglie's idea of a standing wave closed on itself (which was illustrated in Fig. 3.2). The lobed spatial states in Figure 3.8 may be thought of as standing waves where the bright areas are where the amplitude of wave fluctuation is highest and the dark areas are places of near zero amplitude.
Each cloud in Figure 3.8 is a separate possible pattern of probability for the electron. With exceptions to be described later, the electron will be found in (i.e., will occupy) only one standing-wave spatial state, represented by only one cloud (or, in two dimensions, by one cloud cross section) at any particular time. The single-proton nucleus, which would be located at the geometrical center of each cloud, is too small to be visible even at the very high magnifications for the clouds shown in Figure 3.8.
For clarity of viewing, I have shown each cloud cross section separately, but all states, each represented by its particular cloud, collectively surround the same nucleus in a single atom, so all states (and their cloudlike representations and the cross sections of them) are actually superimposed one over the next, each symmetrically surrounding the same tiny nucleus of the hydrogen atom. These probability clouds are our best visual description of the hydrogen atom.
As you will see, straightforward modifications of this visual model for hydrogen can be used to describe the possible states of the electron in the atoms of the rest of the elements. When physicists set up to solve for the presence of two or more electrons in the states of these atoms, with the inclusion of spin for each electron, they found that no two electrons would be allowed to occupy the same total spin and spatial state. They were thus able to derive Pauli's exclusion principle! It was no longer an assumption.
Fig. 3.8. Relative sizes of the spatial states of the hydrogen atom. (Images from Fig. 5-5 of Leighton, Reference F, with permission from Margaret L. Leighton.)
Each cloud cross section in Figure 3.8 is labeled by numbers and letters identifying it as representing that Schrödinger solution with a specific n, ℓ, and m quantum number (spin will be included later). The number at the top indicates n, the Bohr energy level; the letter afterward indicates the nature of a spectral line and an angular momentum (to be defined later) corresponding to a quantum number for ℓ; and the number for m at the bottom indicates a magnetic characteristic (also to be defined later). The properties corresponding to these quantum numbers for each state have all been observed experimentally, directly or indirectly.
The two m = 0 clouds at the lower left of the figure look in three dimensions like a fuzzy ball and a fuzzy ball within a hollow fuzzy ball. The three-cloud cross section marked m = +1 or m = –1 have lobes of probability extending out in different directions. The smallest of the cross sections, the cross section of the “fuzzy ball” shown at the lower left, is for the 1s lowest-energy “ground state.” Everything in nature, including the electron in hydrogen, tends toward a lowest-energy state. So, what we see to the lower left in Figure 3.8 is the most likely size and shape of the probability cloud for the hydrogen atom (of course magnified 100 million times). The atoms of the rest of the elements have similar, solved-for ground states of minimum size.
By contrast, in classical physics there is no reason to expect any minimum size for the atom. And, as noted earlier, the radiation of energy from an accelerating electron in classical orbit would collapse the atom down to the size of the nucleus, as much as 100,000 times smaller than the size that we observe for atoms. Quantum mechanics explains why this collapse does not actually take place: (as noted earlier) there simply is no solution to Schrödinger's equation that allows a state smaller than the ground state. And, as you will see in Part Four and Appendix D, the minimum and maximum sizes of the entire complex of many states occupied by the many electrons in the atoms of the other elements can be roughly predicted by extension from Schrödinger's solutions for the hydrogen atom. These predictions explain why we (ourselves composed of atoms), like everything around us, are the size that we are and not 100,000 times smaller. There simply are no smaller states and atomic sizes for the electron: they don't exist.
Finally, note that Schrödinger's solutions have come to be called “orbitals” because of their correspondence with the orbits of the Bohr-Sommerfeld model of the atom. But, as noted above, these standing-wave solutions for the electron in any of these solutions is no way in the form of orbits. Nor do these probability clouds rotate. There is nothing orbital about them. Thus, so as not to mislead, in this book I refer to Schrödinger's solutions simply as “spatial states.” The probability clouds are “representations” of these spatial states.
The probability clouds are “fuzzy,” spread out and diffuse. Although for any state the electron is most likely to be found in the bright regions of high probability, it has some small probability of being found even in most of the darker regions, with the probability eventually fading off to zero at infinity in any direction. The location of the electron is thus uncertain: though the electron is most likely found where the probability clouds are brightest, it actually could be almost anywhere. The distance over which the probabilities get to be very small is very, very small for large objects compared to their size. We know their location with relative certainty. But for the electron in the atom, this distance and the uncertainty in the electron's position are as large as the atom itself. (More on uncertainty later on.)
Analogy to the Vibrations in a Drum
What occurs in Schrödinger's atom is in some ways analogous to what happens in musical instruments, and specifically in the operation of the drum. The ground state of the hydrogen atom and its infinite number of higher energy states mathematically resemble the fundamental mode of vibration of the drum head and all of its infinite number of higher-level harmonic vibrations.
Like the discrete set of patterns of vibration in a drum head, Schrödinger's equation applied to the natural tuning inherent in the physics of the atom results in fundamental and additional standing-wave solutions for the electron. But the standing waves in this case are Schrödinger's spatial-state solutions. They are three-dimensional, and they have nothing to do with physical vibrations. Instead they provide information about the probability of location of the electron around the nucleus of the atom, and about other properties that this electron may have.
By their superposition around the nucleus, there is analogy of electron states to the superimposed vibration states of the drum head. But in the drum head all vibration states can be to some degree simultaneously active, whereas the electron, with only a couple of exceptions to be described later, occupies and has the properties of only one state at any given time. While the patterns of the drum head are confined to exist only within the drum and with zero amplitude of vibration at the rim, the probability patterns of the electron in the atom are constrained only to have zero value at infinity in any direction. Like each pattern of harmonic vibration in the drum head, each Schrödinger wavefunction solution fluctuates. But it does so in a complex way, and the amplitude of these fluctuations at any location is a measure of the probability that the electron can be found at that location. Said another way, the brightness in the probability clouds for the possible states of the electron is always a positive measure, in each case the absolute positive magnitude of one of the wavefunction solutions for Ψ.
Schrödinger's spatial-state solutions for the hydrogen atom collectively provide most of the information needed to predict the physical and chemical properties of hydrogen. Each spatial-state solution has its own set of properties. These states and their associated energies not only describe the hydrogen atom. They are the key to visualizing and understanding the makeup and properties of the atoms of the rest of the elements, and the atoms of the elements are the building blocks of our bodies and all of the substances of the universe around us.
As will be explained in Part Four and Appendix D, the atoms of the other elements are “tuned” differently by having more protons in the nucleus and an equally greater number of electrons around it. The greater number of protons exerts a greater attraction on each electron and pulls down the energies and sizes of the states for each electron. And the energy levels and patterns of the probability-cloud representations of the various states also differ from those of hydrogen because of the manner in which the electrons repel and interact with each other. Ultimately, it is the numbers of states at each energy level and the manner in which the states are populated with electrons that determines the properties of the elements and all of chemistry.
At this point I want to give you a preview to the realization that quantum mechanics is not just “all theory.” So I interrupt our historical narrative to next, in Chapter 4, explain the operation one of the most extensively used of quantum devices, the laser. Its operation is made possible by the quantum nature of our world, by the separate and distinct quantized energy levels of the electron in the atom. Through the work of scientists and engineers we have learned how to make use of this quantum arrangement, not just for the laser, but also for the many inventions and developments to be described in Part Five.