But then, surprisingly, on April 12, 1926, after a very careful perusal of both matrix mechanics and Schrödinger’s wave mechanics,
Pauli sent a lengthy letter to Jordan in which he proved that the two approaches were identical [or more accurately stated mathematically equivalent]. Schrödinger himself proved the same thing, a little less completely, a month later. . . . In the equivalence paper, Schrödinger mentions pro forma, that it was really impossible to decide between the two theories—and then went on to argue fiercely the merits of wave mechanics. (p. 57; brackets added)
Yet despite his attraction to the more traditional approach of Schrödinger’s wave mechanics compared to matrix mechanics, during his investigations Max Born made a discovery described in two papers titled (in translation), “Quantum Mechanics of Collision Phenomena,” published in the Zeitschrift für Physik in June and July of 1926 that challenged Schrödinger’s claim that wave mechanics, based on measurements of actual waves, was closer to classical physics than matrix mechanics, which dealt only with abstract numerical matrices. The June paper discovered an indeterminacy or uncertainty in Schrödinger’s method of determining the position of alleged particles by measuring the density of wave packets.
Calling it the “measurement problem,” Born found that the impact of the measurement would actually produce a “scattering” of the waves in the “wave packet” causing an indeterminacy in the measurement. Producing an unavoidable uncertainty or probability in the measurements in wave mechanics contrary to the strict causality and determinism in classical mechanics, he concluded that this showed it was not closer to traditional physics as Schrödinger claimed. According to Pais: “It is the first paper to contain the quantum mechanical probability concept.”99 In his June paper Born described the scattering by a wave function Ψmn’ where the label n symbolizes the initial beam direction, while m denotes some particular direction of observation of the scattered particles. At that point Born introduced quantum mechanical probability: “Ψmn determines the probability for the scattering of the electron . . . into the direction [m].” (p. 286)
In the second paper, published in July, he interpreted Schrödinger’s wave function |Ψ|2 as the probability for locating the “particle” at the point of greatest density in the wave packet, adding to the measuring probability the probability of quantum states. Although Born had originally believed that Schrödinger’s wave mechanics led back to a more traditional interpretation of subatomic physics, his probabilistic interpretations convinced him otherwise. As stated in his autobiography:
Schrödinger believed . . . that he had accomplished a return to classical thinking; he regarded the electron not as a particle but as a density distribution given by the square of his wave function |Ψ|2.. He argued that the idea of particles and of quantum jumps be given up altogether; he never faltered in this conviction. . . . I, however, was witnessing the fertility of the particle concept every day in . . . brilliant experiments on atomic and molecular collisions and was convinced that particles could not simply be abolished. A way had to be found for reconciling particles and waves.100
Just as Newton’s conception of absolute space and time that were based on measurements made by rods and clocks that were unaffected by the relatively slight velocities of the earth had to be revised when Einstein discovered that when approaching the velocity of light measuring rods contract, clocks slow down, and mass increases (to account for the invariant velocity of light), so measurements of the subatomic or quantum world, assuming that they would follow the same Newtonian calculation method, when actually measured, would have to be radically revised.
As usual, physicists were confounded when they encountered the wave-particle duality, the statistical nature of quantum mechanics, and the uncertainty principle due to the interacting measurements at the subatomic level of inquiry that refuted the Newtonian assumption of the universality of the laws of nature at all levels or scales of inquiry. Here again we encounter a further aspect of the third radical revision in our conceptions of reality at different dimensions or levels of inquiry. The bewilderment decreased somewhat when it was discovered that the formalisms of Dirac’s theory, Schrödinger’s wave mechanics, and Heisenberg’s matrix mechanics were equivalent: according to Emilio Segrè, “[f]or all three the essential relation that produces the quantification is pq – qp = h/2πi . . . [while] for Heisenberg p and q are matrices; for Schrödinger q is a number and p the differential operator p = h/2πi ∂/∂q . . . [and] for Dirac p and q are special numbers obeying a noncommutative algebra. . . .”101 But the dispute continued with Bohr inviting Schrödinger to his Institute in Copenhagen on October 27, 1926, to discuss their theoretical differences with such intensity that Schrödinger became ill from the tension during the exchange, even though Bohr had the reputation of being a “very considerate and friendly person by nature.” Yet no resolution was reached.
After Schrödinger left Copenhagen Bohr carried on his dispute with the same intensity with Heisenberg, who was an associate at his Institute at the time. Trying to resolve their differences with Bohr defending the view that the solution depended on forging the correct conceptual framework and Heisenberg insisting, as usual, that the resolution would depend upon devising the correct mathematical formalism, they, too, arrived at an impasse. Frustrated and exhausted by these intense discussions, Bohr decided to take a skiing trip to Norway to relax leaving Heisenberg at the Institute to pursue his investigation.
Concentrating on his measurement problem, as a result of his discussion with Bohr, Heisenberg decided to investigate the difficulty involved in measuring the position and momentum of a particle under a gamma ray microscope. The latter is used because its short wavelength provides great accuracy in determining the position, but according to Planck’s formula ε = hv, a short wavelength also has a high frequency with high energy such that the interaction between the wave and the particle adversely affects the precision of the momentum measurement. To reduce the inaccuracy of the latter a longer wavelength is required, but that produces less certainty in the position measurement.
Rather than trying to remove the discrepancy, by the end of February 1927, Heisenberg had decided to accept it as unavoidable and devise a formula to state what initially came to be known as the famous “uncertainty or indeterminacy relations.” An appreciation of the radical change involved is seen if contrasted with what was taken for granted in classical mechanics as stated by Pierre-Simon Laplace in 1886.
An intellect which at a given instant knew all the forces acting in nature, and the position of all things of which the world consists . . . would embrace in the same formula the motions of the greatest bodies in the universe and those of the slightest atoms; nothing would be uncertain for it, and the future, like the past would be present to its eyes.102
It was this assurance that nature is governed by exact laws that would disclose a final knowledge of the universe at all dimensions that Heisenberg was rebutting. Having accepted the conjugate indeterminacy, Heisenberg sought a mathematical formula that would describe the resultant uncertainty. Although the conditions necessary for measuring the conjoined values of the conjugate magnitude’s position and momentum, along with energy and time, could not be precisely measured, either of the dimensions alone could be exactly determined, but the more precise the measurement of one the less precise the measurement of the other. As Heisenberg expressed this mathematically: if the uncertainty in accuracy of the measurement of each of the interdependent conjugate attributes is represented by the delta symbol (Δ), then the product of the conjoined magnitudes momentum p and position q cannot be reduced to less than Planck’s constant barred, Δp × Δq must be equal to or greater than ħ. The second uncertainty states that in the time interval Δt the energy can only be measured with an accuracy equal to or greater than ħ.
Heisenberg published the results in the April 1927 issue of the Zeitschrift für Physik. Having received the proofs of the article, Bohr sent a copy to Einstein “adding i
n an enclosed letter that it ‘represents a most significant . . . exceptionally brilliant . . . contribution to the discussion of the general problems of quantum theory.’”103 What makes it exceptional is not just the calculated mathematical equation, as significant as that is, but that it reversed the age-old assumption that for the mathematics to be correct it must accurately represent the experimental results. Heisenberg affirmed that it is the mathematics that limits or sets the possible experimental outcome! As he states:
Instead of asking: How can one in the known mathematical scheme express a given experimental situation? the other question was put: Is it true, perhaps, that only such experimental situations can arise in nature as can be expressed in the mathematical formalism? The assumption that this was actually true led to limitations in the use of the concepts that had been the basis of classical physics since Newton.104
As an indication of the influence Heisenberg’s paper had on Bohr, when a famous article by Einstein, Podolsky, and Rosen (known as the EPR article), titled “Can Quantum Mechanical Descriptions of Physical Reality Be Considered Complete?” was published in the Physical Review in 1935,105 claiming that although Heisenberg’s formalism was consistent with all the known quantum data it was “incomplete” because it did not allow precise measurements of the conjugate attributes’ position and momentum and energy and time, Bohr had a ready reply.
In the following issue of the Review, in an article with exactly the same title, he replied that in quantum mechanics
we are not dealing with an incomplete description characterized by the arbitrary picking out of different elements of physical reality at the cost of sacrificing other elements, but . . . the impossibility, in the field of quantum theory, of accurately controlling the reaction of the object on the measuring instruments. . . . Indeed we have . . . not merely to do with an ignorance of the value of certain physical quantities, but with the impossibility of defining these quantities in an unambiguous way.106
In the draft of a paper in July 10, 1927, Bohr had used the term ‘complementarity’ for the first time that became a famous designation for the conjugate uncertainty measurements. Then in a collection of articles published later in his life, there is a clear statement of how he believed quantum mechanics has changed our method and understanding of the subatomic quantum domain in contrast to the macroscopic and atomic level of experience, an explanation that had assumed that a precise description and definite explanation of the external world was always possible, even if out of reach at the time:
Within the scope of classical physics, all characteristic properties of a given object can in principle be ascertained by a single experimental arrangement. . . . In quantum physics, however, evidence about atomic objects obtained by different experimental arrangements exhibits a novel kind of complementary relationship. Indeed, it must be recognized that such evidence which appears contradictory when combined into a single picture is attempted, exhausts all conceivable knowledge about the object. Far from restricting our efforts to put questions to nature in the form of experiments, the notion of complementarity simply characterizes the answers we can receive by such inquiry, whenever the interaction between the measuring instruments and the objects forms an integral part of the phenomena.107
Just as we found that our sensory system modifies what we observe, so we have learned that what unobservable properties the world discloses experimentally at a certain dimension also partially reflects the methods and instruments used in investigating it. This realization that all experience and knowledge is due to an interaction with the world, not just an immediate awareness or disclosure of it as usually appears to be the case, brought about a radical transformation in our conception of reality and how we come to know it—a reversion especially imposed by Heisenberg’s discovery of the “uncertainty principle” and Bohr’s “Copenhagen interpretation” that has been a crucial feature of the third scientific transformation of our conception of reality.
What is surprising is that Einstein, in his article with Podolsky and Rosen, did not realize that the uncertainties encountered when investigating a deeper domain of particle physics did not permit the same exact measurements as those made on a larger scale was analogous to his special theory of relativity. His theory also claimed that the exact measurements of space, time, and motion made within the lesser velocities of ordinary experience cannot be made when the velocities are so extreme they effect the measuring devices such that the measurements are relative to the velocities of the measurer rather than being absolute as Newton claimed.
For millennia humans believed that the picturesque world as ordinarily experienced was the actual world. Even as late as the nineteenth century Ernst Mach declared that “Atoms cannot be perceived by the senses . . . they are things of thought” implying they did not exist. Yet at the beginning of modern classical science, with the introduction of the telescope and the microscope, scientists began to realize that the existence of the ordinary world, as objective, determinate, and independent as it appears to be, really depends on very complex, unseen underlying conditions. With every discovery of a new dimension of the world the assumption usually has been that this must be the final reality, not just another level of inquiry.
But even if all existence and knowledge is conditional, it is equally erroneous to infer that we do not know anything about the world or that it has any objective properties as concluded in the article by Einstein, Podolsky, and Rosen if the quantum mechanical worldview were accepted. But were their view true, how could we account for the corrective and progressive advances in scientific knowledge and its extraordinary technological consequences before and since their time?
Supposing that whatever knowledge of the universe and human existence we acquire depends upon the physical conditions within which they exist, along with the method of investigation used, this does not preclude their being actual within those conditions, otherwise we would have to deny that the ordinary world we live in exists and that the independent subatomic world does not have any of the physical properties it has because their existence is dependent on a more extensive background physical context. Consider water existing as vapor or ice under different conditions.
What we have to realize is that the meaning of ‘existence’ has changed with the acquisition of greater knowledge, just as Bohr argued that the meaning of ‘understanding’ has changed. Just because particles are so minute that they prevent the measuring of certain conjugate properties does not mean that these properties do not exist in the object conjointly—that the particle does not have a simultaneous position and momentum or energy and time just because the conditions prevent their being measured conjointly. How could it exist without these conjugate properties?
As added evidence of this conception of “contextual realism” that I referred to in a previous book bearing that title and again in this book, I’ll continue the review of additional scientific discoveries showing the limits, not necessarily the negation, of Newton’s corpuscular-mechanistic view of reality at a deeper level of inquiry, along with illustrating that additional physical or quantitative properties of subatomic particles have been discovered despite their existence being dependent upon the type of measurement used to identify them.
The first is the property of spin. It is common knowledge that microscopic particles are not defined by sensory qualities, but by their primary properties of mass, charge, energy, and momentum. At about the time the previously described quantum mechanical discoveries were being made, two physics students in their mid-twenties from the University of Leyden in Holland, Samuel S. Goudsmit and George E. Uhlenbeck, suggested that on Bohr’s model, electrons like planets, in addition to having an orbital motion around the nucleus, also revolve on their axes with an invariant angular momentum called “spin,” whose value is ½h/2π. This value remains constant “even when the electron is outside the atom, and is totally independent of the linear speed or environment of the electron,”108 indicative of its inherent though co
nditional nature. Furthermore, because the electron is electrically charged and follows the laws of quantum mechanics, it has two additional properties.
First, having an electrical charge it acts as a tiny magnet whose movement creates a magnetic moment that creates an electromagnetic field. Second, in quantum mechanics an entity with the properties of angular momentum has just two possible spin orientations: “If we perform any measurement whatsoever to determine the angle between the direction of the electron spin and any given direction in space, we find that the angle is always either 00 or 1800—in other words, the spin is either parallel or antiparallel to the chosen direction” (pp. 53, 55). Moreover, the spin of certain pairs of subatomic particles are such that measuring the spin of one particle will instantly cause its twin particle to begin spinning in the opposite direction at the same rate, however great the distance between them, a discovery made by John Stewart Bell and published in the Review of Modern Physics in 1966. Given these perplexing features of quantum mechanics, it has been claimed that the concept of spin as an actual rotary motion of the electron should not be taken literally: “it is more accurate to say that the electron has an intrinsic angular momentum of ½h/2/π, called spin, as if it were rotating about its axis” (p. 57). But how is it possible to use “as if states” in scientific theorizing?
For example, the concept of spin is considered an additional important property of electrons and other particles, having explanatory as well empirical consequences that have been experimentally confirmed. The electric charge along with its spin gives the electron its magnetic moment that helps explain the emission and reabsorption of photons. In addition, the two possible spin orientations imply two energy states that help explain “the peculiar pattern of close lines or doublets in the Balmer series of the hydrogen spectrum.” Called quantum electrodynamics or QED for short, the theory measured the magnetic moment of the electron “in a unit called the Bohr magneton, denoted µe” that has “a very great accuracy . . . found to be 1.001 159 652 µe” (p. 58).
Three Scientific Revolutions: How They Transformed Our Conceptions of Reality Page 20
