Now that we have reached the centennial of Erwin Schrodinger's seminal paper introducing the wavefunction theory of matter, it is right and proper to inquire as to its legacy. It is undeniable that today every paper in atomic physics cites his 1926 equation in the first paragraph. But the philosophy undergirding the wavefunction seems to have fallen into the shadows. And Schrodinger left his program incomplete. I will argue here that recent developments in nonlinear mathematics, including so-called "chaos theory", permit finishing the task. It turns out that one nonlinear addition to his equation from 1926 can resolve both the Measurement Problem and the Randomness Problem. With this emendation, the wavefunction alone suffices to explain the outcomes of many experiments (and it is particles that can be relegated to the shadows).
Now that we have reached the centennial of Erwin Schrodinger's seminal paper introducing the wavefunction theory of matter, it is right and proper to inquire as to its legacy. It is undeniable that today every paper in atomic physics cites his 1926 equation in the first paragraph. But the philosophy undergirding the wavefunction seems to have fallen into the shadows. And Schrodinger left his program incomplete. I will argue here that recent developments in nonlinear mathematics, including so-called "chaos theory", permit finishing the task. It turns out that one nonlinear addition to his equation from 1926 can resolve both the Measurement Problem and the Randomness Problem. With this emendation, the wavefunction alone suffices to explain the outcomes of many experiments (and it is particles that can be relegated to the shadows).
While instructing an undergraduate physical chemistry course on quantum mechanics, I encountered difficulties in elucidating the concept of the particle-in-a-box thought experiment, initially formulated by Erwin Schrödinger. In particular, the node distribution of the wavefunction. Born's Copenhagen interpretation assigns the modulus of the wavefunction as the probability density distribution of particles. The developers of quantum mechanics, including Schrödinger, de Broglie, and Einstein, have openly expressed reservations about this interpretation. Schrödinger's original conception of the wavefunction is intrinsically linked to the particle's momentum rather than the statistical probabilities of particle locations. Because momentum can be encoded in different forms, it could be useful to create wavefunctions using different encoding equations, which is the major attempt of this manuscript. In addition to quantizing energy levels properly for several example systems, the proposed new equations have yielded a striking and alternative interpretation of Schrödinger's wavefunction that aligns well with classical mechanics. This proposed modification enhanced the simplicity of math and strengthened Schrödinger’s original interpretation. These equations could be baby steps to a brand-new quantum calculation algorithm that is simpler and faster than the established ones.
In this paper I address the most common objections to the claim that Schrodinger was right in 1926: the wavefunction provides the correct, and complete, description of atomic phenomena. I suggest that the line of droplets in the Wilson cloud chamber, the click of the ``photon detector", and ``that dot on the screen" can all be explained within a context of wavefunction models and Schrodinger's-type equations, albeit nonlinear. No auxiliary hypotheses about point particles or probabilities are required. The random locations of the triggered ``particle detectors" can be explained by ``chaos" (meaning sensitive dependence on initial conditions). Even Born's ad hoc invocation of probabilities may be justifiable in certain circumstances. As an illustration, I present simulations from a (toy) wavefunction ``particle-detectors" model.
Quantum mechanics is our most successful physical theory. It predicts the behavior of atoms, molecules, and light to ten decimal places. No experiment has ever contradicted it. It also contains a crack at its foundation. The theory’s central equation—the Schrodinger equation—describes a smooth, continuous, perfectly predictable evolution of the quantum state. But whenever anyone performs a measurement, they get a single, definite result. Which result they get appears to be random. The transition from smooth evolution to random outcome is called the “collapse of the wavefunction,” and after nearly a century of analysis, there is no agreed-upon explanation for what causes it, when it occurs, or whether it occurs at all. This is the measurement problem. The measurement problem is not a minor puzzle at the margins of the theory. It is a failure to explain the most basic operation in science: looking at the result of an experiment. Around it cluster other puzzles. Light behaves like a wave in some experiments and like a particle in others—two contradictory descriptions. Two particles that have interacted remain correlated across arbitrary distances, as though communicating faster than light. The probabilities that govern measurement outcomes appear as a postulate—the Born rule—inserted by hand with no deeper justification. And the theory offers no unified picture of reality: quantum mechanics, general relativity, and the standard model of particle physics speak incompatible mathematical languages. A century of effort has produced many interpretations of quantum mechanics. Copenhagen. Many-worlds. De Broglie-Bohm. Objective collapse. Quantum Bayesianism. None has achieved consensus. The measurement problem remains. This document proposes that the source of these puzzles is not quantum mechanics itself. It is a single, invisible assumption made at the very beginning of the theory in 1900—an assumption so deeply embedded that nobody noticed they were making it. Change that assumption, and the puzzles do not get solved. They dissolve. They stop being puzzles. They become predictable consequences of geometry.
On 13 March 1926, the first of six papers on wave mechanics by Erwin Schrödinger was published in the Annalen der Physik with the title Quantisation as an eigenvalue problem (Part 1) (Schrödinger, 1926b). The startling first paragraph reads: ‘In this paper, I wish to consider, first, the simple case of the hydrogen atom (nonrelativistic and unperturbed), and show that the customary quantum conditions can be replaced by another postulate, in which the concept of “whole numbers”, merely as such, is not introduced. Rather, when integralness does appear, it arises in the same natural way as it does in the case of node-numbers of a vibrating string. The new conception is capable of generalisation, and strikes, I believe, very deeply at the true nature of quantum rules.’ These papers were the fruits of an extraordinary burst of creativity on Schrödinger's part which resulted from his interactions with Einstein in the latter part of 1925. Central to these exchanges were de Broglie's remarkable researches which culminated in his famous PhD dissertation and published papers of 1924. These events have already be recounted in Chap. 9. The subsequent developments which led to Schrödinger's discovery of the equation which bears his name will be taken up in Sect. 14.2, but let us first understand more about Schrödinger's background. Schrödinger's background in physics and mathematics Education and career up to 1925 Unlike Heisenberg, Jordan, Pauli and Dirac, Erwin Schrödinger was not one of the young Turks who developed Knabenphysik , young man's physics.
Abstract In June 1925 Heisenberg made the breakthrough that would lead to matrix mechanics. The theory was abstract, offered no means for visualizing the mechanics of the atom, and its application required a lot of heavy lifting. Erwin Schrödinger subsequently made the breakthrough that would lead to wave mechanics, building on Louis de Broglie’s association of a wave with an electron. In Schrödinger’s wave equation the quantum numbers of the hydrogen atom arise naturally. He argued that his ‘wavefunction’ offered a realistic description of the electron in an atom, and sought to eliminate Bohr’s discontinuous quantum jumps. But in the spring of 1926, Max Born offered an enduring alternative. The modulus-square of the wavefunction is a measure of probability in an unvisualizable microphysics. In an intense debate with Bohr, Schrödinger demanded a theory based on logical concepts free from contradictions. Bohr was left to ponder what this might mean.
It had been known for decades that matter, like light, had both wave and particle properties, but there was no quantitative description relating the two. Schrödinger derived it. His equation begins with a wave description and leads to clusters of waves that behave like particles. His first success was to predict the average radii of the orbits of electrons in atoms. These matched the values postulated by Bohr. But there is a hitch, embedded in the word “average.” Different measurements of the same property of an object, such as position, give a variety of answers and not due to experimental error. There is at best a probability associated with each property. The Schrödinger equation has been interpreted to argue that every object actually has a mix of the values of a property until you measure it; hence, it “chooses” one. Schrödinger himself was not happy with this and hence the cat. Is it both alive and dead until you look and force it to choose one? Interpretations still raise controversy. The equation itself ranks as one of the most important in all of physics.