If one would like to know of a readable account of the turmoil facing young physicists in the early twentieth century owing to Bohr’s introduction of the solar model of the atom, the dualism between the wave and the particle properties of electrons, and the necessity of using both Newtonian and quantum laws, I know of no more fascinating book than Heisenberg’s biographical account, Physics and Beyond: Encounters and Conversations (1971), recounting his intellectual development and outstanding contributions to the current revolutionary developments in physics quoted earlier.
Owing to his decisions to study atomic physics, enroll at the University of Munich, and attend the lectures of Arnold Sommerfeld from 1920 to 1922, he was privileged to have been invited by Sommerfeld to hear the lectures by Bohr, then age thirty-seven, in the summer of 1922 at what has come to be known as the “Göttingen Bohr Festival” at Göttingen’s famous school of mathematics. Having been introduced to Bohr by Sommerfeld and then, during the question period following a lecture titled “Advanced Objections,” to Hendrik “Hans” Kramer’s speculations on quantum mechanics discussed by Bohr during the lecture, Bohr was so impressed by Heisenberg’s comments that at the end of the discussion period he asked Heisenberg to join him on a walk that afternoon over the Hain Mountain. As Heisenberg wrote: “This walk was to have profound repercussions on my scientific career, or perhaps it is more correct to say that my real scientific career only began that afternoon.”92
During their walk Bohr began the conversation by indicating what initially had been his main concern in physics, saying that it was not what one might think, the inner structure of atoms, but the “stability of nature,” an insight also as to how a great scientist thinks. Giving examples to show that classical Newtonian science took for granted the normal stability and uniformity of nature in formulating the laws and mathematical equations that describe the causal connections and inner structure of the Newtonian corpuscular-mechanistic worldview, Bohr’s basic motivation was not merely to describe their stability and uniformity but to explain what accounts for it.
As quoted by Heisenberg, Bohr stated:
My starting point was not at all the idea that an atom is a small-scale planetary system and as such governed by the laws of astronomy. I never took things as literally as that. My starting point was rather the stability of matter, a pure miracle when considered from the standpoint of classical physics. (p. 39)
He then explained what he meant by the stability, persistence, and recurrence of the physical and chemical properties of the elements in terms of their combining in definite proportions to constitute the molecular structures of certain substances while retaining their original properties when decomposed, which forms the basis of the classifications in Mendeleev’s Periodic Table. Still, the connection between the stability in nature and Bohr’s fascination with atomic structure is revealed in a statement he made, again quoted by Heisenberg:
The existence of uniform substances, of solid bodies, depends on the stability of atoms; that is . . . quite inexplicable in terms of the basic principle of Newtonian physics, according to which all effects have precisely determined causes, and according to which the present state of a phenomenon or process is fully determined by the one that immediately preceded it. This fact used to disturb me a great deal when I first began to look into atomic physics. (p. 39)
This also reveals the radical distinction between the scientific orientation of Bohr and Einstein, despite their close friendship and enduring admiration for each other, the latter searching for a formula that would describe a four-dimensional space-time field that would precisely unify all the laws and structure of the universe while the former was trying to determine the nature of the subatomic structures that produce the underlying probabilistic causal interactions in nature. Accordingly, Bohr describes all the developments in the past few decades that led to the realization of the limitations of classical physics and the need to replace or supplement it by an understanding, however tenuous, of the deeper subatomic realm (pp. 39-40).
Bohr then presents his position that seems to have had a lasting effect on Heisenberg. This position affirms that since classical physics deals with our familiar macroscopic world it can provide visual or verbal representations of that world while subatomic or quantum physics investigates a domain that is so unique that it does not allow such ordinary descriptions. As Bohr clearly states in a quotation by Heisenberg:
We know from the stability of matter that Newtonian physics does not apply to the interior of the atom; at best it can occasionally offer us a guideline. It follows that there can be no descriptive account of the structure of the atom; all such accounts must necessarily be based on classical concepts which, as we saw, no longer apply. You see that anyone trying to develop such as theory is really trying the impossible. For we intend to say something about the structure of the atom but lack a language in which we can make ourselves understood. (p. 40)
In response to these skeptical reflections, Heisenberg bluntly asks Bohr what was the point of his introducing “all those atomic models” in his lectures and what did he expect to show by them, to which Bohr replies: “These models have been deduced, or if you prefer guessed, from experiments, not from theoretical calculations. I hope that they describe the structure of the atoms as well, but only as well, as is possible in the descriptive language of classical physics” (p. 41).
Heisenberg next asks how can we ever expect to understand the nature of atoms if a description of their inner structure is not clearly defined? Hesitating momentarily, Bohr replies, “I think we may yet be able to do so. But in the process we may have to learn what the word ‘understanding’ really means” (p. 41). His answer foretells the eventual “uncertainty” in the conception of the interior structure of the atom as well as those inherent in quantum mechanics in general.
Impressed by the conversation with Heisenberg, Bohr invited him to Copenhagen to attend Bohr’s Institute, but finding Bohr’s attempt to construct the inner structure of atoms on the experimental evidence unappealing, Heisenberg decided instead to go to Göttingen in the fall of 1924 to study with Max Born whose approach to atomic physics was based more on the mathematical calculations. Owing to his father being a professor of Greek at the University of Munich, Heisenberg had learned to read the Greek classics in the original when he was just sixteen years old. Thus, like Kepler, he became attracted to atomic physics by reading Plato’s Timaeus, which equated the four basic elements of fire, earth, air, and water, along with the cosmos, to the five Pythagorean geometric solids. Despite his realization that the endeavor seemed to be “wild speculation,” Heisenberg “was enthralled by the idea that the smallest particles of matter must reduce to some mathematical form (p. 8).
Suffering from a severe attack of hay fever in Munich the following summer, Heisenberg sought shelter in the small island of Heligoland on the North Sea where the sea breeze dispersed any pollen-laden air. It was there he began writing a paper on “electron states” based entirely on the measured light frequencies absorbed or emitted by the atom that was published in Zeitschrift für Physik (Writing on Physics) in September 1925. As he indicated, despite rejecting any attempt to visualize the interior structure of the atom, like Plato he seemed to find in the numbers a kind of abstract mathematical reflection of its interior.
Within a few days . . . it had become clear to me what precisely had to take the place of the Bohr-Sommerfeld quantum conditions in an atomic physics working with none but observable [or measurable] magnitudes. . . . The energy principle had held for all the terms, and I could no longer doubt the mathematical consistency and coherence of the kind of quantum mechanics to which my calculations pointed. At first, I was deeply alarmed. I had the feeling that, through the surface of atomic phenomena, I was looking at a strangely beautiful interior, and felt almost giddy at the thought that I now had to probe this wealth of mathematical structures nature had so generously spread out before me. (p. 61; brackets added)
The paper
did contain peculiar mathematical functions when the matrices were added, subtracted, and especially when multiplied, unlike those of traditional mathematics, that he was unable to explain. But given the fact that such famous mathematicians as David Hilbert, Richard Courant, and Max Born were among his colleagues at Göttingen University, when shown to Born the latter recognized the incongruities, especially the violation of what is known as the commutative law. The law states that when multiplying two numbers the order of the numbers is irrelevant: 9 × 8 = 8 × 9, but Heisenberg did not find this to be true of his calculations.
When shown the paper Born was not perplexed because he recognized it as an example of matrix mechanics. As authors Crease and Mann state:
Born was one of the few physicists in Europe—perhaps the only one—with a good knowledge of matrix mathematics. He realized that Heisenberg’s quantum-theoretical series were nothing more, nothing less, than awkward manipulations of frequency matrices. . . . Born was delighted. Rewriting Heisenberg’s equations as matrices led to a whole new world of applications he could explore.93
Yet it still required some interpretation as the continuing quotation indicates:
The first thing he figured out was that the matrix q for position and the matrix p for momentum are noncommutative in a very special way: that is, pq is not only different from qp, but the difference between pq and qp is always the same amount, no matter what p or q you chose. Mathematically he wrote this pq - qp = ħ/i, where ħ, as usual, is Planck’s constant divided by twice pi, and i is the special symbol mathematicians use for the square root of minus one. (p. 50)
But still not satisfied Born, with the aid of Pascual Jordan (who wrote most of the article because Born had suffered a nervous collapse), published a paper also in the Zeitschrift für Physik that described the basic mathematics of what later became known as matrix mechanics. As described by Ne’eman and Kirsh:
They arranged the measurable quantities in square arrays of numbers (such arrays are called “matrices”) and by defining mathematical operations between these matrices, they created a consistent quantum mechanical theory. This version of quantum mechanics, which has been known as matrix mechanics, succeeded in explaining certain experimental facts in atomic physics and even predicted unknown phenomena which were verified later experimentally.94
For their contributions Heisenberg won the Nobel Prize in 1932, when he was only thirty-one years old; Dirac, together with Schrödinger, in 1933; and Born in 1954. Ne’eman and Kirsh add that the “number of important contributions” Heisenberg “made to physics exceeded that of any other physicist in the twentieth century, except Einstein” (p. 44). I would state, and Bohr.
As an indication of how intriguing the mathematics of quantum mechanics was in those early days, Paul Dirac, a close friend of Heisenberg with whom he had discussed atomic theory in their student days, and who collaborated with Born in writing several articles, also favored a mathematical construction of the interior of the atom rather than Bohr’s solar model. After receiving a copy of Heisenberg’s paper and studying it for little over a week, he “sat down and wrote an alternative formulation, which presented quantum mechanics as a coherent axiomatic theory” (p. 44).
In my discussion I may have quoted statements that gave the impression that Heisenberg remained very satisfied with his introduction of matrices in place of Bohr’s pictorial electron orbits, but later I read a statement by him quoted by Crease and Mann that apparently shows his misgivings in working with a purely numerical system such as matrices despite its earning him the Nobel Prize. It was written to Pauli after Heisenberg wrote his famous article in collaboration with Jordan and Born published in the Zeitschrift für Physik.
I’ve taken a lot of trouble to make the work physical, and I’m relatively content with it. But I’m still pretty unhappy with the theory as a whole and I was delighted that you were completely on my side about [the relative roles] of mathematics and physics. Here I’m in an environment that thinks and feels exactly the opposite way, and I don’t know whether I’m just too stupid to understand the mathematics. Göttingen is divided into two camps: one, which speaks, like [the prominent mathematician David Hilbert] (and [another mathematical physicist, Herman Weyl ], in a letter to Jordan), of the great success that will follow the development of matrix calculations in physics; the other, which, like [physicist James Franck], maintains that the matrices will never be understood. I’m always annoyed when I hear the theory going by the name of matrix physics. For awhile, I intended to strike the word “matrix” completely out of the paper and replace it with another [term] — “quantum-theoretical quantity,” for example.95 (italics, brackets, and parentheses are in the original)
How different his present dissatisfaction with matrices is from his youthful intention of replacing physical objects with mathematical symbols.
But as might be expected, not all the physicists at the time were disdainful of visual model building, the Viennese-born Erwin Schrödinger (1887–1961) being an outstanding example. Attracted to de Broglie’s discovery that particles also have wave properties that are physical and observable, he was the physicist Einstein consulted when asked by the committee examining de Broglie’s doctoral dissertation of its credibility, but also being undecided Einstein did not give his approval until Schrödinger assured him that the thesis had merit.
Schrödinger, influenced by Einstein’s belief that fields should replace material particles as the fundamental reality along with de Broglie’s discovery that particles have wave properties, decided to see if a complete wave theory could be formulated to replace Bohr’s electronic solar model. De Broglie had shown that although mass and momentum were considered properties of particles, they also could be depicted as functions of waves. Such equivalences had previously been discovered: Einstein showing the equivalence of mass and energy (E = mc2) and Planck that energy could be equated with frequency (e = hv). Since waves have energy and energy has mass, it was possible that material particles could be depicted as waves as well as particles.
Starting with the classical wave equation that describes the spatial properties of electromagnetic waves, Schrödinger began investigating whether a wave equation could be found to describe the wave properties of subatomic particles, one that would supplement the equations of Newtonian mechanics. In four papers he presented his new theory of “wave mechanics” that appeared in Annalen der Physik (Annals of Physics) from January to April 1926 with the title “Quantization as an Eigenvalue Problem.”
Among his publications he proposed that waves should be considered the basic reality and introduced what is now known as the “celebrated” Schrödinger’s wave equation that contains the famous scalar wave function Ψ(psi), along with the Hamiltonian H, “which is simply the observable corresponding to the energy of the system under consideration.”96 As explained by Pais, in replacing particles with waves he
suggested that waves are the basic reality, particles are only derivative things. In support of this monistic view he considered a wave packet made up out of linear harmonic oscillator wavefunctions . . . a superposition of eigenfunctions so chosen that at a given time the packet looks like a blob localized in a more or less small region. . . . He examined what happened to his packet in the course of time and found: “Our wave packet holds permanently together, does not expand over an ever greater domain in the course of time.” This result led him to anticipate that a particle is nothing more nor less than a very confined packet of waves, and that, therefore, wave mechanics would turn out to be a branch of classical physics, a new branch, to be sure, yet as classical as the theory of vibrating strings or drums or balls.97
According to the equation, between measurements the state vector known as the “wave function” moves in an undisturbed, regular way as described in classical physics until it is measured, which then causes it to collapse into an eigenvalue or single value that produces the observation. Thus its state is uncertain until measured. Owing to its being an extens
ion of classical wave theory, formulated in a mathematics more familiar than matrix mechanics, and providing a visualizable explanation, it was acclaimed by most physicists. As described by Crease and Mann,
the mathematics Schrödinger used was much easier for physicists to understand. . . . If it was hard to imagine how a solid object like an atom could really be made out of waves—what was making the waves?—many physicists had confidence that Schrödinger, a clever fellow, would figure out the answer.98
Moreover, as also explained by Crease and Mann, Schrödinger even proposed an explanation as to how particles could be considered as waves.
A particle was in reality nothing but “a group of waves of relatively small dimensions in every direction,” that is, a sort of tiny clump of waves, its behavior governed by wave interactions. Ordinarily, the bundle of waves was small enough that one could think of it as a dot, a point, a particle in the old sense. But in the microworld, Schrödinger argued, this approximation broke down. There it became useless to talk about particles. At very small distances, “we must proceed strictly according to the wave theory, that is, we must proceed from the wave equation, and not from the fundamental equation of mechanics, in order to include all possible processes.” (p. 56)
Unfortunately, these expectations turned out to be illusory. Instead of determinate or distinct portrayals of the electron states of the atom, the solutions to Schrödinger’s wave equation produced small cloudlike images reminding one of Rorschach ink blots. Yet even Born, who contributed to the article creating matrix mechanics, after reading Schrödinger’s first paper wrote that he was drawn to the traditional aspects of Schrödinger’s wave mechanics, a view that angered Heisenberg.
Three Scientific Revolutions: How They Transformed Our Conceptions of Reality Page 19