Quantum puzzles. Of the various quantum puzzles, the most basic is probably wave-particle duality. The atom itself is, however, the most important, puzzling consequence for the ordinary world. The traditional way of summing up what is most puzzling about quantum mechanics is the Heisenberg uncertainty principle, but recently the most discussed is called “Bell’s inequality.” All of them are described here as a way of introducing quantum mechanics as it is currently understood, and after explaining the spatiomaterialist theory of quantum matter, I will show how they can be solved.

Wave-particle duality. According to Bohr, the basic puzzle of quantum mechanics is the dual nature of the basic entities it describes. They all appear to be like both particles and like waves. What classical physics took to be waves turn out to have a particle-like nature as well, and what classical physics took to be particles turn out to have a wave-like nature as well. Bohr thought that both appearances of the underlying reality are due in part to our measuring apparatus and the classical expectations on which they are constructed. But wave-like and particle-like natures are apparently incompatible, and since both of these classical conceptions of reality are needed to make all of the possible predictions, he called the basic puzzle of quantum mechanics “complementarity.” Bohr was the originator of the so-called "Copenhagen interpretation" of quantum mechanics, which holds that the reality behind these complementary phenomena is incomprehensible to us.

The particle-like nature of electromagnetic waves. Light has long since been thought to have wave-like nature in classical physics. Early in the nineteenth century, Thomas Young showed that light passing through two narrow, closely spaced holes (or slits) produces a pattern of light and dark lines on the screen that finally intercepts them, and he explained it by the wavelike nature of light. The places on the screen where the waves emerging from each slit interfere constructively are bright, while the places where they interfere destructively are dark. When one slit is blocked, the interference pattern disappears.

Diffraction phenomena also indicate the wave like nature of light. Light rays passing through a hole that approximates its wavelength will be spread out as it leaves the hole, with the range of the spread varying inversely with the size of the hole. When the hole is large, the light hitting the distant screen is like a point, but when the hole is small, the diffraction is great.

Moreover, as we have seen, light had been explained by Maxwell as the wavelike coupling of the electric and magnetic forces described by his equations. Indeed, it enabled him to predict the velocity of light.

The particle-like nature of light was the first of the discoveries that eventually culminated in quantum mechanics. Instead of propagating like a wave in an elastic medium, as the classical model assumed, it became clear that light is actually made up of distinct particles, which are now called “photons”. This particle-like nature means that the energy and momentum carried by light do not combine continuously, as they do in ordinary waves, but come in separate units, called “quanta”. The size of the quantum of light is now represented by Planck’s constant, h, which is part of every new equation used in quantum mechanics. It appears in the new equations for the energy and momentum of light. The energy, E, is given by E = hf  (where E is energy, f is frequency), and the momentum, p, is given by p = h/l (where p is momentum, and l is the wavelength).

Max Planck first discovered the particle-like nature of light in 1900, though he did not fully understand what he was on to. He discovered the constant named after him by tinkering with a classical equation for calculating the amount of energy given off at each frequency in so-called blackbody radiation, that is, a hot body in which no frequency of light should be favored. (It is best approximated by a box with mirrored interior walls in which light of all possible wavelengths for a box with certain temperature are being reflected back and forth.) The classical equation assumed that the frequencies of light being given off varied continuously from the lowest to the highest, with the peak intensity depending on the temperature. That assumption worked well enough for the low frequencies, but at high frequencies, it led to the conclusion that the total energy given off should be infinite. This absurdity was called the “ultraviolet catastrophe.” Planck discovered a formula that avoided the catastrophe and predicted the total quantity of energy given off at each frequency by introducing a constant, h, into the formula which restricted the frequencies of light. That is the source of the equation for the energy of light: E = hf.  (Though its meaning is still obscure, it can, perhaps, be seen as requiring the photons to differ from one another by that constant amount.)

Albert Einstein made it clearer that what Planck had discovered was the particle-like nature of light by using Planck’s constant is his own explanation of the photoelectric effect (in 1905, the same year that he published his special theory of relativity). It had been known that light being intercepted by material objects could release electrons from the material objects, but it was found that the release of electrons did not depend on the total energy of the light waves (the intensity of the light), as one would expect on the wave hypothesis. It depends on the frequency of the light. Below a certain frequency, no electrons are released, regardless how intense the light may be at that frequency. Whereas light with a higher frequency would release electrons even though the intensity was much less. Einstein showed that the release of the electrons depended on the absorption of single photons, each of whose energy depended on Planck’s constant: E = hf.

Much later (in 1923), Arthur Compton showed that photons also have a momentum like particles. He shot high energy photons (x rays) at electrons and used arguments based on the conservation of momentum and energy to predict correctly the amount by which their energies would be changed by such scattering.

The particle-like nature of the light does not change its wave-like properties. Indeed, it turns out that interference effects still occur when light is sent through the two-slit apparatus one photon at a time. Over time, they still accumulate in fringes on the distant wall.

The wave-like nature of particles with rest mass. Material objects are understood in classical physics as having definite locations in space at each moment and to follow definite trajectories as they move from one place to another. But the behavior of objects with rest mass on the smallest scale is peculiar in the opposite way from photons, according to quantum theory. Just as light waves have a particle-like nature, so material objects have a wave-like nature.

The wave-like nature of particles with rest mass was predicted in 1923 by de Broglie. What Einstein’s special theory of relativity implies about the relativistic increase in mass leads to the conclusion that the energy of a photon is equal to the product of its momentum and the velocity of light, or E = pc. Since the velocity of light is equal to the product of the frequency and wavelength, or c = fl, it follows that the momentum of a photon is p = h/l. De Broglie went on to suggest that the same relationship holds of particles with kinetic energy. He argued that particles, such as electrons, protons and material objects with mass generally would also have a wave length that varied inversely with their momentum in the same way. 

Interference and diffraction phenomena were the kind of empirical evidence that was taken as showing that light has a wave-like nature, and soon after de Broglie’s prediction, it was shown that the electrons forced to pass through very small holes do exhibit diffraction, that is, the smaller the hole, the more they spread out. Eventually, even interference phenomena were demonstrated with electrons. When electrons moving at a certain velocity are projected through narrow, closely spaced, parallel slits at a screen (where the distance between the slits approximates their de Broglie wave length), they also form an interference pattern on the far wall, as if they were waves. Even when the electrons were sent one at a time, they tended to land on the distant screen only along certain fringes, leaving lines between them without any hits. Thus, each particle is like a wave. The same has been show to hold for neutrons, though in the case of ordinary sized objects, the wavelengths are so small that interference effects are undetectable.

The structure of the hydrogen atom. The laws of quantum mechanics were discovered mainly by attempting to explain the structure of the hydrogen atom. It had been established by Ernest Rutherford that the atom is composed of a massive, positively charged nucleus surrounded by far less massive electrons, and Niels Bohr hoped to explain the chemical properties of atoms by the nature of the interactions between the electrons and the nucleus. It was clear that atoms could not be explained in classical terms on the model of the solar system, since according to Maxwell’s equations, the orbital motion of an electron would generate (as the acceleration of a negatively charged particle) an electromagnetic wave which would drain its energy until the electron was located at rest with the nucleus. In fact, atoms with electrons located around it are quite stable, and when such atoms are excited (by supplying energy to them), they give off electromagnetic radiation at a certain set of distinctive frequencies.

Bohr explained the frequencies of the spectrum of hydrogen atoms (in 1913) by assuming that electrons can have only certain orbits, each characterized by an energy level that corresponds to the total energy of an electron with kinetic energy in a force field with potential energy imposed by the nucleus. (The total quantity of energy is negative, because the kinetic energy of the particle is not great enough to replace all the negative potential energy that would be required to free it, and according to our assumption about the nature of potential energy, the negative sign for potential energy indicates that the nucleus and electron have less rest mass.) The energies of the possible orbits were determined as a function of Planck’s constant, and a number was assigned to each possible orbit, starting with the lowest energy orbit and counting upwards (n = 1, 2, 3, . . ). Bohr showed that the spectral lines of the hydrogen atom could be explained by the differences in the energies of these permitted electron orbits.

The basic puzzle of quantum mechanics is the structure of the atom itself, that is, what is going on that only certain energy levels are possible for electrons bound to a nucleus by electromagnetic forces.

Given the structure of the atom, however, there is another problem, for it does not seem possible that electrons could be jumping from one orbital to another. When a photon is absorbed or emitted by an atom, an electron changes from one permitted orbital to another (so that the atom changes from one energy state to another). But the photon has a particle-like nature, and the particle seems to change its position and motion in an instantaneous, step-like change, that is, without accelerating nor even moving continuously from one state to the next. It hard to see how the electron’s change of orbital can be explained as the motion of a material object, since a material object can change location only by moving across space continuously as time passes time.

Another puzzle has to do with the timing of the emission of photons. When an atom or molecule is in an energy state that can decay into a lower energy state, it is not possible, even in principle, to say exactly when it will decay. The timing can be assigned a probability, but the theory has nothing to explain why it happens at one moment rather than another within that range.

Electron jumps also seem to be involved in the phenomenon of tunneling. “Tunneling” refers to situations in which electrons seem to jump across barriers imposed by force fields. On classical principles, crossing such a force field would require more energy than the electron has. Nevertheless, some electrons do jump across. Only a few electrons do so, and there is no way to predict which ones will jump. But it is so regular that this phenomenon is used as a kind of microscope for mapping the surfaces of material objects.

Erwin Schrödinger thought that it would be possible to avoid these puzzles about electron jumps and explain everything deterministically by following up on de Broglie’s suggestion and explaining the behavior of the electron in an atom as a wave. Using the model of the classical equation for waves and taking the electron wave to be in a potential field, Schrödinger presented an equation in 1925 that explained the energy levels of the permitted orbitals of electrons in the force field imposed by the nucleus of the hydrogen atom. The time-independent Schrödinger equation (with the temporal changes factored out so that it represents only the spatial structure of the wave) portrays the electron bound to the nucleus of the hydrogen atom as a standing wave, like a plucked string on a guitar. This made it possible for Schrödinger to explain the numbers that Bohr had assigned to the permitted orbitals of electrons as the energy states in which the electron could be such a stable, standing wave. The lowest energy level corresponds to the string with no nodes (that is, half the wave length for its energy), the next one to a string with one node, and so on. The problem of quantum jumps seemed to be solved, because the transitions between such energy states of atoms were explained as smooth and continuous transitions of waves. 

Schrödinger believed that his wavefunction showed that electrons were not particles at all, but could be explained purely as waves in an electromagnetic field. This did not explain why electrons appear to be particles, for example, how they leave vapor trails in a Wilson cloud chamber or interact at a certain point on the distant wall in the two-slit interference experiment. But it is possible to explain why electrons seem to have a determinate location by holding that they are a "superposition" of waves with slightly different wavelengths, because in regions where such wave interfere constructively, they clump together in what are called “wave packets.” Since the locations where such a set of waves interfere constructively have more or less precise locations in space and seem to move through the space occupied by the waves, the Schrödinger wavefunction could explain the appearance that electrons move like particles. (This was not a fully adequate explanation, however, because such wave packets also tend to disperse over time, and yet electrons actually turn up later at definite locations.)

However, it was not possible to interpret the Schrödinger wavefunction as the description of a classical wave. One problem was that it contained complex numbers. There is no way to measure quantities multiplied by the square root of minus one, and yet those complex numbers are essential to the wavefunction, since they describe the phases of the waves that are superimposed in the quantum system and, thereby, determine the interference phenomena.

Furthermore, the Schrödinger wavefunction described a wave in a space that can have more than three dimensions (or what is called “configuration space). When more than one particle is involved, the space occupied by the wave described by Schrödinger’s wavefunction has three times as many dimensions as there are particles. There is no obvious way to relate such an equations to the actual three dimensional world.

What is now the orthodox interpretation of the Schrödinger wavefunction was first proposed by Max Born in 1926. He took the square of the (time-independent) wavefunction in some region of configuration space to be a measure of the probability of finding that the particle located in that region of configuration space (thereby predicting a measurable property, such as location, momentum or kinetic energy). The predictions are confirmed by measurement.

Since the predictions are merely probabilistic predictions, however, Born took the Schrödinger wavefunction to be a representation, not of the world itself, but of what we can know about it. This avoided the problems of quantum jumps and wave packets that spread out, because what really happens is not knowable. And insofar as it is taken realistically, it implies that what happens is not fully determined by the state that precedes it.

Heisenberg uncertainty principle. An entirely different mathematical representation of these same quantum phenomena was developed by Werner Heisenberg. His “matrix mechanics” is basically an algorithm for making predictions of measurements without any attempt to explain what is going on beneath the observable surface. Though Schrödinger showed that Heisenberg’s matrix mechanics and his own wavefunction are mathematically equivalent, matrix mechanics makes the limitations on what can be known about the classical properties of the entities described by quantum mechanics clear. In arguing against Schrödinger, he defended what has come to be known as the Heisenberg uncertainty principle.

In matrix mechanics, there are pairs of variables called “complementary” or “conjugate” variables, because the measurement of one affects the measurement of the other. That is, the results of measuring one variable and then the other would be different if they were measured in the opposite order. The position and momentum of an electron are complementary variables, meaning that the position and momentum of an electron cannot both be measured with arbitrarily high precision But the more precise one measurement is, the less precise the other is. Using Born’s probabilistic interpretation of the wavefunction to express the “uncertainties” in such measurement, Heisenberg derived a general principle about complementary variables: the product of the uncertainty about the position and the uncertainty about the momentum cannot be less than Planck’s constant divided by four pi.

Heisenberg’s uncertainty principle holds in a parallel way for other conjugate variables, such as energy and time, angular momentum and orientation, and cycle and phase. In each case, one variable is more particle-like and the other is more wave-like, and thus, the variables are said to be complementary.

Heisenberg apparently took his uncertainty principle to be a basic postulate from which all of quantum mechanics could be developed. He rejected talk about the wave-particle duality and took a purely instrumentalist approach which simply denied that there is any aspect of the world that is not described by his matrix mechanics (or by their equivalents in using the Schrödinger wavefunction).

The equivalence of Heisenberg’s matrix mechanics and Schrödinger’s equation means that the Heisenberg uncertainty principle can be derived in a similar way from Schrödinger’s equation.

The solution of Schrödinger’s equation for a given situation yields a wavefunction, which is a complete description of the quantum system. But in order to predict a measurable property, it is necessary to apply an appropriate mathematical operator to the wavefunction. The operator yields an “expectation value” for that property, which may be a precise value or an average value.

But some pairs of operators are not commutable, such as the position and momentum of a particle. Though it is often possible to make precise predictions of these properties, the prediction of one makes it impossible to predict the other. That is, when one property is predicted by one operator, the mathematical operation changes the wavefunction and so the prediction made for the other property is not the same as it would have been if the second property had been predicted first. Since the order in which the operators are applied to the wavefunction makes a difference in what they predict, it is impossible to predict both properties at once. Thus, the conjugate variables to which Heisenberg’s uncertainty applies turn out to be the pairs of properties predicted by non-commutable operators.

When the operator yields an expectation value that is just the average result for an entire series of experiments, it can often be represented as a superposition of different wavefunctions for each of which the operator gives an expectation value. When the measurement is made and one of them turns out to be true, the wavefunction is said to “collapse,” because the system turns out to have one or another of precise predicted outcomes. This is called the “collapse of the wavefunction,” because it is assumed that prior to the measurement, what actually existed was a superposition of different wavefunctions.

This interpretation of the measurement of a quantum system exacerbates the problem, for the superposed states of the system can evolve in radically different ways. In the most famous example, a cat is locked in a box with a devise triggered by an unpredictable beta decay that will, with 50% probability, release a poison that kills the cat within a certain period of time. But until someone looks to see what has happened, there is a superposition of the two states, one with a dead cat and another with a living cat, and reality only resolves itself into one or the other possibility at the moment someone looks. This implausible implication of measurement being the collapse of the wavefunction is called the problem of "Schrödinger’s cat."

The Heisenberg uncertainty principle is, perhaps, the most general statement of the puzzles of quantum mechanics, and a genuine ontological explanation of quantum mechanics, if there is one, should reveal the source of this limitation on our knowledge.

Bell correlations. Recently, attention has focused on a final quantum mystery, called “Bell’s Theorem” or “Bell’s Inequality.”^[1] John Bell showed that quantum mechanics entails, in certain circumstances, a statistical correlation between events occurring at a distance that seems to be possible only if the events have effects on one another that travel faster than the velocity of light. It holds for interactions in which particles move away from one another in opposite directions with opposite orientations of a “spin”.

Spin. Spin is a quantum property that was first recognized with the discovery of quantum field theory. The Schrödinger wavefunction is the law of non-relativistic quantum mechanics, and a more complete law was discovered by Paul Dirac when he combined the Schrödinger wavefunction with Einstein’s special theory of relativity, that is, taking the relationship it describes between space and time into account.

There was an asymmetry between the time-dependent and time–independent wavefunctions derived by solving Schrödinger equation. The time-independent wavefunction, describing the spatial aspects of the standing wave, is a second order differential equation, whereas the time-dependent wavefunction, describing how the quantum system unfolds in time, is a first order differential equation. Dirac derived a time-dependent wavefunction that was a second order differential equation, making time and space symmetrical, as they are in the special theory of relativity.

It is puzzling just what makes Dirac's derivation work, but it involved several profound discoveries.

Dirac discovered that there are twice as many solutions for the wavefunctions than had been thought, half of them corresponding to negative energy. This was the discovery of antimatter, such as, for example, the positively charged electron as the negative partner of the negatively charged electron, called the "positron."

Dirac discovered that quantum particles have another property, called “spin,” which was a  new quantum number that was needed for wavefunctions to describe fully any quantum situation. That is, spin is a new quantum number (namely, s) needed to describe the atom (along with Bohr’s numbers for the energy states of atoms (n), a number for the orbital angular momentum of the electron (i), and a number for its magnetic moment (m)).

It is believed that the intrinsic spin of an electron has little to do with a spinning electrical charge. The spin of a particle is defined operationally as the strength of the magnetic force that results when a magnetic field is imposed on the particle. Particles, such as the electron, that have ½ spin (called “fermions”) have one of only two possible magnetic moments (positive and negative). Since there is no way for them not to have a magnetic moment, it is hard to see how they could be a classical material object with a charge that is somehow actually spinning.

Bell’s Inequality. John Bell discovered a curious consequence of quantum mechanics involving spin. The spin of a particle (either a rest mass or a photon, which has a spin of 1) would seem to a property that the particle carries with it, but a prediction made on this assumption contradicts quantum mechanics. And it seems to have been disproved empirically. This suggest that spin is a property that depends, not on the particle itself, but on what happens elsewhere in a much more inclusive system involving both particles.

The system is one in which two objects are generated in a way that requires them to have opposite orientations of spin, and they move away from one another in opposite directions. Since space is three dimensional, the spin of a particle can be measured from three different, mutually perpendicular directions. If one particles is measured as having as having spin, say, up, in some direction, then the other particle will never turn out to have anything but the opposite, down, orientation of spin when it is measured in the same direction. This holds regardless which of the three independent directions in space the magnetic field is oriented, and quantum mechanics does not permit one to infer from its spin in one direction what its spin in any other direction is. Thus, if spin is a property that the particles already have when they part from one another, the outcome of measuring the spin of the particles that moved off one way from their creation from one direction should not enable us to predict the spin of the other particle when measured from a different direction. Bell showed that, on this assumption, a certain inequality must hold about the frequency with which measurements of spin in one particle in one direction would correlate with measurements of the spin of the other particle in one of the other directions.

However, quantum theory predicts and experiments have confirmed that this inequality will be violated. When two objects are generated in this way, and the spin orientation of one of these objects is measured in one direction, it is possible to predict the outcome of a measurement of the spin orientation (up or down) of the other object in an independent direction of three dimensional space more often than the Bell inequality allows. It is not a reliable prediction in any particular case, but statistically it is more frequent than would be possible, if the spin orientations of both objects were already determined when they parted and they were simply carried away with them, as the principle of local action would require.

Though the two measurements can be made as far apart in space as one likes, it seems that the only way the measurements could be correlated is if the measurement of one object were somehow affecting the state of the other. And since the two measurements can be made to occur as near to one another in time as one likes, there are instances of this phenomenon in which such an effect could hold only if something travels between them faster than the velocity of light. This puzzling correlation is not only a consequence of quantum theory, but has also been confirmed experimentally, and thus, it seems that we must give up the principle of local action. But it seems to violate the principle of local action. Since the different outcomes are a superposition of different wavefunctions, this is seen as just another puzzles about the so-called "collapse of the wavefunction."

The puzzles of quantum mechanics have to do with understanding what in the world corresponds to the Schrödinger equation. The “Copenhagen interpretation” of quantum mechanics, developed by Bohr, is the received view. It simply denies that it is possible to describe the nature of what exists except by applying the classical conceptions of particles or wave, which if not strictly speaking incompatible, are, at best, complementary. Defenders of the Copenhagen interpretation see the puzzles of quantum mechanics as deriving from its departures from classical physics, as if classical physics were based on intuitions ( or a form of imagination) that is anthropocentric and, thus, merely subjective. And some go on to insist that the uncertainty is a real indeterminism about what happens in the world.

The chief opponent of this view was Einstein. He was resisting the reification of quantum uncertainty as indeterminism when he claimed, “God does not play dice with the universe.” A view of the world as being constituted by substances of some kind is what kept Einstein from accepting quantum mechanics as the complete description of what exists. His acceptance of spacetime as a substance made him most sympathetic to Spinoza, for Spinoza believed that the world is a single substance. But what seems to have kept Einstein from admitting that such a substance could have indeterminism as a basic property were his ontological instincts.

In what follows, I will elaborate the the assumptions of spatiomaterialism in a way that explains ontologically why quantum mechanics is true. It is, as I have warned, more speculative than the rest of the argument of ontological philosophy. But it may suggest the power of an ontological approach and vindicate Einstein’s view of the nature of the world in at least one respect.