The procedure used to "do physics" in the macroscopic world is familiar: You take an object, start it off with a particular position and velocity, subject it to known forces (say gravity or friction, or both), and follow its trajectory. You then measure the dynamical properties (say position or energy) of that object at a later time and compare those measurements with the prediction using Isaac Newton's laws of motion. Newton's laws directly predict what those quantities should be at that later time, so the comparison is straightforward. However, the microscopic laws of physics, quantum mechanics, aren't so simple. The quantum concepts are so alien and counterintuitive that the language used to describe the mathematics and physics is often ambiguous and confusing. Therefore, for you to \emph{really} understand what is going on, some mathematics must be used. This is my attempt to explain some "weird facts" using a minimum of mathematics, yet eliminating as much ambiguity as possible.
The procedure used to "do physics" in the macroscopic world is familiar: You take an object, start it off with a particular position and velocity, subject it to known forces (say gravity or friction, or both), and follow its trajectory. You then measure the dynamical properties (say position or energy) of that object at a later time and compare those measurements with the prediction using Isaac Newton's laws of motion. Newton's laws directly predict what those quantities should be at that later time, so the comparison is straightforward. However, the microscopic laws of physics, quantum mechanics, aren't so simple. The quantum concepts are so alien and counterintuitive that the language used to describe the mathematics and physics is often ambiguous and confusing. Therefore, for you to \emph{really} understand what is going on, some mathematics must be used. This is my attempt to explain some "weird facts" using a minimum of mathematics, yet eliminating as much ambiguity as possible.
Those defile the purity of mathematical and philosophical truthy who confound real quantities with their relations and sensible measures. —ISAAC NEWTON, Principia Before we say anything about quantum mechanics, let us take a quick look at the Newtonian (or ‘classical’) view of the physical universe. According to that view, nature consists of an enormous number of particles. When Newtonian physics is combined with the theory of the electromagnetic field, it becomes convenient to think of these particles as dimensionless (even if there is a kind of conceptual strain involved in trying to think of something as having a mass but not any size ), and as possessing electrical properties – negative charge, or positive charge, or neutrality. This leads to the well-known ‘solar system’ view of the atom – with the electrons whirling around the nucleons (neutrons and protons) just as the planets whirl around the sun. Out of atoms are built molecules; out of molecules, macroscopic objects, scaling from dust motes to whole planets and stars. These latter also fall into larger groupings – solar systems and galaxies – but these larger structures differ from the ones previously mentioned in being held together exclusively by gravitational forces. At every level, however, one has trajectories (ultimately that means the possibility of continuously tracing the movements of the elementary particles) and one has causality (ultimately that means the possibility of extrapolating from the history of the universe up to a given time to its whole sequence of future states).
The difficulties of the old quantum theory were largely dispelled by the development of quantum mechanics in the 1920s. First Heisenberg (1925) developed the rather abstract matrix mechanics and soon after Schro¨dinger (1926; see also Eckart, 1926) developed the physically more appealing wave mechanics. Subsequently he showed them to be mathematically equivalent. The physicists were greatly relieved; now they had once again a mathematical formalism that enabled the results of experiments to be calculated. It was possible to move forward. It is certainly a very elegant and indeed beautiful formalism, and does enable the results of many physical measurements to be calculated with great accuracy. It is used daily by atomic and nuclear physicists; there is no other way to make calculations. From the earliest days there has been intense debate about the interpretation of quantum mechanics, and it shows no signs of being settled; if anything it is becoming more intense over the years. As Penrose has remarked, no one understands quantum mechanics. Feynman (1967; see also Sykes, 1994) also admitted that ‘nobody understands quantum mechanics’, and Einstein once remarked: ‘I have thought a hundred times as much about the quantum problems as I have about general relativity theory’ (Pais, 1994, p. 57). It should therefore be evident that the most that can be done here is to try to show where the real difficulties lie, and how the failure to recognize that quantum mechanics is an essentially statistical theory has led to a series of confusing quantum paradoxes. It is just these paradoxes that have often been used to support theological speculations. It is useful to ask what ‘understanding quantum mechanics’ actually means. Certainly it means being able to calculate the results of many measurements. The same applies to Newton’s theory of gravitation, which we understand in the sense that we can use it as a basis of calculation, although we do not really understand the gravitational force itself. Very often, especially in modern physics, the mathematical formalism does not give us the sort of intuitive physical feeling for what is going on that we often obtain from classical mechanics. Perhaps we can never achieve more than this for quantum mechanics. In classical mechanics we can assume that a property that is measured objectively exists prior to the interaction of the measuring apparatus with the observed system. Quantum mechanics, however, ‘is incompatible with the proposition that measurement discovers some unknown but pre-existing property’ (Peres, 1997, p. 14). There is an important distinction between mysteries and absurdities. We may be forced to live with mysteries, but there is no reason why we should put up with absurdities such as the wave-particle duality, tunnelling through potential barriers, acausal events and the collapse of the wave function. These are the quantum paradoxes that will be discussed later on. They arise within the Copenhagen interpretation of quantum mechanics and are solved by other interpretations. These alternative interpretations enable us to make conjectures about what is going on, even though we cannot understand it in detail. This is acceptable and a stimulus to further research. It is certainly difficult enough to understand quantum mechanics, but there is no need to try to visualize logical absurdities in the effort to do so. Quantum mechanics is a mathematical formalism that makes it possible to calculate the results of some experiments. Anything more than this, such as questions about the interpretation of what is measured and whether it tells us what is actually going on in the system being studied, belongs to philosophy. Thus we are concerned here with what is essentially a philosophical debate; if it were just a matter of physics the debate would have been settled long ago. One of the difficulties is that it invokes the results of experiments and uses the language of quantum mechanics. Inevitably this language uses philosophical terms and then it is easy to give the impression that the philosophical propositions are in some way entailed by the physical results. These in turn can influence theological thinking. In order to be quite clear which arguments are justified and which are not, it is essential to separate the three levels of discourse: physical, philosophical and theological. Our task is to relate these together, and this immediately raises the traditional problems of the philosophy of science, including the relation of experiment to theory, and of theory to explanation. We begin by discussing experiments, distinguishing between those that are real and those that are imaginary.
Quantum mechanics is the newest science and should therefore be the most precise, but unfortunately, it's quite the opposite. If you don't understand quantum mechanics, ask yourself if I'm right, because others don't either. The field of experimental quantum mechanics is so vast and confusing that its theoretical practitioners will always have to adopt new explanations from their own context to conform to the facts of quantum mechanics. Quantum mechanics was created in the early 20th century to establish mathematical and physical laws or rules for subatomic objects subject to an external potential field in order to explain so-called quantum systems. Studying the spatiotemporal evolution of a quantum system, defined as a microscopic subatomic system of low energy density subjected to a potential field, external or internal, is a rather complex subject. However, in 1927, the well-known Schrödinger PDE, valid for infinite free space subject to Dirichlet boundary conditions, was introduced. Schrödinger's PDE must be complemented by the Bohr/Copenhagen interpretation of the wave function Ψ involving the rules of quantum systems of instantaneous entanglement and superposition. We all know that the Schrödinger equation is incomplete because it lives and operates in the incomplete space R^4 and therefore the wave function Ψ is incomplete in itself and cannot be defined as scalar, a vector, nor a tensor. The complex wave function Ψ has never been properly defined. Moreover, the Schrödinger PDE is not Lorentz invariant and is not compatible with the special theory of relativity, and is obviously even less so with the general theory of relativity. However, quantum mechanics has come a long way, both in theory and practice, since 1927, nearly a century ago. This long journey has added even more illogical and confusing properties to the Ψ wave function, such as causality and wave function collapse. Many attempts have been made over the last century to reform the Schrödinger equation, the most notable being to combine it with the general theory of relativity, but all these attempts have been in vain. The breakthrough came in 2020-2024 [2,3,4], when the author of this article claimed to replace the classical Schrödinger equation Ψ with the PDE for its square Ψ^2, which is logical and physical. Ψ^2 is needed to express the quantum energy density flux (U(x,y,z,t) = Ψ^2 (x,y,z,t)) and we therefore propose a new Schrödinger equation for Ψ^2 which should have the form of the energy density diffusion PDE such as that of thermal conduction. Equation 3 should be supplemented by the advanced artificial intelligence proposed by the author [5,6]. Comparing Schrödinger's classical PDE of 1927 with the PDE proposed for Ψ^2 in 2020 shows that we currently have two different or distinct theories of quantum mechanics! Which one is more theoretically true and more practically useful? This is the purpose of this article.
Quantum mechanics allows a remarkably accurate description of nature and powerful predictive capabilities. The analyses of quantum systems and their interpretation lead to many surprises, for example, the ability to detect the characteristics of an object without ever touching it in any way, via "interaction-free measurement," or the teleportation of an atomic state over large distances. The results can become downright bizarre. Quantum mechanics is a subtle subject that usually involves complicated mathematics -- calculus, partial differential equations, etc., for complete understanding. Most texts for general audiences avoid all mathematics. The result is that the reader misses almost all deep understanding of the subject, much of which can be probed with just high-school level algebra and trigonometry. Thus, readers with that level of mathematics can learn so much more about this fundamental science. The book starts with a discussion of the basic physics of waves (an appendix reviews some necessary classical physics concepts) and then introduces the fundamentals of quantum mechanics, including the wave function, superposition, entanglement, Bell's theorem, etc., and applications to Bose--Einstein condensation, quantum computing, and much more. The interpretation of the mathematics of quantum mechanics into a world view has been the subject of much controversy. The result is a variety of conflicting interpretations, from the famous Copenhagen view of Bohr to the multiple universes of Everett. We discuss these interpretations in the chapter "What is a wave function?" and include some very recent advances, for example, quantum Bayesianism, and measurements of the reality of the wave function.
[I] think I can safely say that nobody understands quantum mechanics. Richard Feynman, The Character of Physical Law Quantum mechanics [is] that mysterious, confusing discipline, which none of us really understands but which we know how to use. It works perfectly, as far as we can tell, in describing physical reality, but it is a ‘counter-intuitive discipline’… Murray Gell-Mann, Questions for the Future [The quantum] postulate implies a renunciation as regards the causal space-time co-ordination of atomic processes. Niels Bohr, Atomic Theory and the Description of Nature It should be emphasized, however, that the probability function does not in itself represent a course of events in the course of time. It represents a tendency for events and our knowledge of events. The probability function can be connected with reality only if one essential condition is fulfilled: if a new measurement is made to determine a certain property of the system. [T]he idea of an objective real world whose smallest parts exist objectively in the same sense as stones or trees exist, independently of whether or not we observe them…is impossible … Werner Heisenberg, Physics and Philosophy [In] quantum theory it is the principle of causality , or more accurately that of determinism , which must be dropped and replaced by something else.… We now have a new form of the law of causality.… It is as follows: if in a certain process the initial conditions are determined as accurately as the uncertainty relations permit, then the probabilities of all possible subsequent states are governed by exact laws. Max Born, The Restless Universe
Unlike special relativity, quantum mechanics was not the inspired product of one mind. It developed, in fits and starts, over a period of 25 years (1900–1925), and many cooks stirred the broth. Even now, a century later, quantum mechanics raises profound conceptual questions. Every competent physicist can “do” quantum mechanics, and the predictions it makes are in spectacularly good agreement with experimental results; there can be little doubt that quantum mechanics is “right.” On the other hand, as Richard Feynman once remarked, “nobody understands quantum mechanics.” I'll tell the story in two parts. First the history (how the essential pieces of the puzzle were assembled), and then the implications (what the finished picture has to say about our world). Photons Like relativity, quantum mechanics began with the study of light. In both cases, this historical association is misleading. Relativity, really, has nothing to do with light – to be sure, it involves that magic speed limit, c , and of course the most familiar thing that travels at speed c is light. So light is a convenient vehicle for introducing the subject. But it is perfectly possible to imagine a universe in which there is no such thing as light, and yet relativity might still be valid. Quantum mechanics, too, starts with light – in this case the “quantum” of light (the photon). But, as it turns out, this is just one rather specialized example of much more general principles – in fact, a rather bad example, because photons are by their nature relativistic (they travel at the speed of light), and it would be more natural to begin with the quantum mechanics of slow-moving objects, in analogy with Newtonian mechanics.