In what sense can something be, at one and the same time, both a discrete particle (Werner Heisenberg) and a continuous wave (Erwin Schrödinger)?

The information interpretation of quantum mechanics explains that the wave function is purely abstract immaterial information about where concrete material particles will be found statistically when a large number of particles are measured.

Quantum waves are never seen. They are not "observables," which Heisenberg made his chief criterion for the new quantum mechanics. He declared that the electron orbits of the "old" quantum theory of the Bohr atom simply do not exist because they are not observable. Only the spectral lines of light given off by transitions between energy levels are observable, he said.

Following the traditional Copenhagen Interpretation, many physicists today describe a quantum object as either a wave or a particle, depending on the free choice of the experimenter.

Calculations by a physicist describing the evolution of a quantum system, an electron or a photon, for example, goes in two stages.

Between measurements there is a wave stage in which the wave function explores all the possibilities available, given the configuration of surrounding particles, especially those nearby, which represent the boundary conditions for the Schrödinger equation of motion for the wave function. Because the space where the possibilities are non-zero is large, we say that the wave function (or "possibilities function") is nonlocal. Albert Einstein always hoped for a local "objective reality."

The quantum process raises deep metaphysical questions about possibilities, with their calculable probabilities, and the actualities.

Information about the new interaction may or may not be recorded. If the new information is irreversibly recorded, it may later be observed. It must be recorded before it can be observed. A "conscious observer" is not involved in the recording of the measurement. The recording of a measurement happens before the observer makes an observation. In modern physics, that can be days or weeks before the observation which requires lengthy calculations and "data reduction."

When you hear or read that electrons are both waves and particles, you might think "either-or" - between measurements a wave of possibilities, then an actual particle.

Or you might accept Einstein's deeply help belief that it is always a particle with a path, despite the practical impossibility of making measurements along the path.

That a light wave might actually be composed of quanta (later called photons) was first proposed by Albert Einstein as his "light-quantum hypothesis."

He wrote in 1905:

In accordance with the assumption to be considered here, the energy of a light ray spreading out from a point source is not continuously distributed over an increasing space but consists of a finite number of energy quanta which are localized at points in space, which move without dividing, and which can only be produced and absorbed as whole units.

("A Heuristic Viewpoint on the Production and Transformation of Light," Annalen der Physik, vol.17, p.133, English translation - American Journal of Physics, 33, 5, p.368)

In 1909, Einstein speculated about the connection between wave and particle views:

This is wave-particle duality fourteen years before Louis deBroglie's matter waves and Erwin Schrödinger's wave equation and wave mechanics

When light was shown to exhibit interference and diffraction, it seemed almost certain that light should be considered a wave...A large body of facts shows undeniably that light has certain fundamental properties that are better explained by Newton's emission theory of light than by the oscillation theory. For this reason, I believe that the next phase in the development of theoretical physics will bring us a theory of light that can be considered a fusion of the oscillation and emission theories...

Even without delving deeply into theory, one notices that our theory of light cannot explain certain fundamental properties of phenomena associated with light. Why does the color of light, and not its intensity, determine whether a certain photochemical reaction occurs? Why is light of short wavelength generally more effective chemically than light of longer wavelength? Why is the speed of photoelectrically produced cathode rays independent of the light's intensity? Why are higher temperatures (and, thus, higher molecular energies) required to add a short-wavelength component to the radiation emitted by an object?

The fundamental property of the oscillation theory that engenders these difficulties seems to me the following. In the kinetic theory of molecules, for every process in which only a few elementary particles participate (e.g., molecular collisions), the inverse process also exists. But that is not the case for the elementary processes of radiation.

Einstein's view since 1905 was that light quanta are emitted in particular directions. There are no outgoing spherical waves (except probability amplitude or "possibilities" waves). Even less likely are incoming spherical waves, never seen in nature

According to our prevailing theory, an oscillating ion generates a spherical wave that propagates outwards. The inverse process does not exist as an elementary process. A converging spherical wave is mathematically possible, to be sure; but to approach its realization requires a vast number of emitting entities. The elementary process of emission is not invertible. In this, I believe, our oscillation theory does not hit the mark. Newton's emission theory of light seems to contain more truth with respect to this point than the oscillation theory since, first of all, the energy given to a light particle is not scattered over infinite space, but remains available for an elementary process of absorption.

(On the Development of Our Views Concerning the Nature and Constitution of Radiation)

Also in 1909, Einstein’s imagined an experiment in which the energy of an electron (a cathode ray) is converted to a light quantum and back.

Consider the laws governing the production of secondary cathode radiation by X-rays. If primary cathode rays impinge on a metal plate P1, they produce X-rays. If these X-rays impinge on a second metal plate P2, cathode rays are again produced whose speed is of the same order as that of the primary cathode rays.

As far as we know today, the speed of the secondary cathode rays depends neither on the distance between P1 and P2, nor on the intensity of the primary cathode rays, but rather entirely on the speed of the primary cathode rays. Let’s assume that this is strictly true. What would happen if we reduced the intensity of the primary cathode rays or the size of P1 on which they fall, so that the impact of an electron of the primary cathode rays can be considered an isolated process?

If the above is really true then, because of the independence of the secondary cathode rays’ speed on the primary cathode rays’ intensity, we must assume that an electron impinging on P1 will either cause no electrons to be produced at P2, or else a secondary emission of an electron whose speed is of the same order as that of the initial electron impinging on P1. In other words, the elementary process of radiation seems to occur in such a way that it does not scatter the energy of the primary electron in a spherical wave propagating in every direction, as the oscillation theory demands.

(On the Development of Our Views Concerning the Nature and Constitution of Radiation)

Extending his 1905 hypothesis, Einstein shows energy can not spread out like a wave continuously over a large volume, because it is absorbed in its entirety to produce an ejected electron at P2, with essentially the same energy as the original electron absorbed at P1.

Rather, at least a large part of this energy seems to be available at some place on P2, or somewhere else. The elementary process of the emission of radiation appears to be directional. Moreover, one has the impression that the production of X-rays at P1 and the production of secondary cathode rays at P2 are essentially inverse processes...Therefore, the constitution of radiation seems to be different from what our oscillation theory predicts.

The theory of thermal radiation has given important clues about this, mostly by the theory on which Planck based his radiation formula...Planck’s theory leads to the following conjecture. If it is really true that a radiative resonator can only assume energy values that are multiples of hν, the obvious assumption is that the emission and absorption of light occurs only in these energy quantities. On the basis of this hypothesis, the light-quanta hypothesis, the questions raised above about the emission and absorption of light can be answered. As far as we know, the quantitative consequences of this light-quanta hypothesis are confirmed. This provokes the following question. Is it not thinkable that Planck’s radiation formula is correct, but that another derivation could be found that does not rest on such a seemingly monstrous assumption as Planck’s theory? Is it not possible to replace the light-quanta hypothesis with another assumption, with which one could do justice to known phenomena? If it is necessary to modify the theory’s elements, couldn’t one keep the propagation laws intact, and only change the conceptions of the elementary processes of emission and absorption?

This conception seems to me the most natural: that the manifestation of light’s electromagnetic waves is constrained at singularity points, like the manifestation of electrostatic fields in the theory of the electron. I imagine to myself, each such singular point surrounded by a field that has essentially the same character as a plane wave, and whose amplitude decreases with the distance between the singular points. If many such singularities are separated by a distance small with respect to the dimensions of the field of one singular point, their fields will be superimposed, and will form in their totality an oscillating field that is only slightly different from the oscillating field in our present electromagnetic theory of light.

(On the Development of Our Views Concerning the Nature and Constitution of Radiation)

Einstein thus imagines many singular points (his light quanta) whose average behavior has the shape of a light wave.

Just as a large number of randomly distributed discrete points approaches the smooth continuous appearance of the normal distribution, Einstein imagines the “totality” of points would look like an oscillating light wave or field. Einstein never published the implicit idea that the light wave would be stronger where there are many particles, less where there are few. But he described to many friends, including Max Born, the idea of a “ghost field” (Gespensterfeld) or “guiding field” (Führungsfeld) that represents the probability of finding a particle at different positions.

Our modern view of the relationship between waves and particles is straightforward. The wave is a complex function with values at every place in space whose absolute square gives us the probability of finding a discrete particle there. The wave (later the wave function ψ) is similar to the continuous gravitational or electromagnetic fields that specify the force on a test particle at any place in space and time. These “probability” fields are not substantial or as Einstein called them “ponderable.”

The “fusion of wave and emission theories of light” that Einstein expected is now seen to consist of a theoretical continuous field that provides abstract information (probabilities) about the outcomes of experiments on localized discrete particles.

Before either of these theories was developed in the mid-1920's, Einstein in 1916 showed how both wave-like and particle-like behaviors are seen in light quanta, and that the emission of light is done at random times and in random directions. This was the introduction of ontological chance (Zufall) into physics, over a decade before Heisenberg announced that quantum mechanics is acausal in his "uncertainty principle" paper of 1927.

As late as 1917, Einstein felt very much alone in believing the reality (his emphasis) of light quanta:

I do not doubt anymore the reality of radiation quanta, although I still stand quite alone in this conviction

(Letter to Besso, quoted by Abraham Pais," "Subtle is the Lord...", p.411)

Einstein in 1916 had just derived his A and B coefficients describing the absorption, spontaneous emission, and (his newly predicted) stimulated emission of radiation. In two papers, "Emission and Absorption of Radiation in Quantum Theory," and "On the Quantum Theory of Radiation," he derived the Planck law (for Planck it was mostly a guess at the formula), he derived Planck's postulate E = hν, and he derived Bohr's second postulate
E_m - E_n = hν. Einstein did this by exploiting the obvious relationship between the Maxwell-Boltzmann distribution of gas particle velocities and the distribution of radiation in Planck's law.

The formal similarity between the chromatic distribution curve for thermal radiation and the Maxwell velocity-distribution law is too striking to have remained hidden for long. In fact, it was this similarity which led W. Wien, some time ago, to an extension of the radiation formula in his important theoretical paper, in which he derived his displacement law...Not long ago I discovered a derivation of Planck's formula which was closely related to Wien's original argument and which was based on the fundamental assumption of quantum theory. This derivation displays the relationship between Maxwell's curve and the chromatic distribution curve and deserves attention not only because of its simplicity, but especially because it seems to throw some light on the mechanism of emission and absorption of radiation by matter, a process which is still obscure to us.

("On the Quantum Theory of Radiation," Sources of Quantum Mechanics, B. L. van der Waerden, Dover, 1967, p.63; Physikalische Zeitschrift, 18, pp.121–128, 1917)

If light quanta are particles with energy E = hν traveling at the velocity of light c, then they should have a momentum p = E/c = hν/c. When light is absorbed by material particles, this momentum will clearly be transferred to the particle. But when light is emitted by an atom or molecule, a problem appears.

Does the molecule receive an impulse when it absorbs or emits the energy ε? For example, let us look at emission from the point of view of classical electrodynamics. When a body emits the radiation ε it suffers a recoil (momentum) ε/c if the entire amount of radiation energy is emitted in the same direction. If, however, the emission is a spatially symmetric process, e.g., a spherical wave, no recoil at all occurs. This alternative also plays a role in the quantum theory of radiation. When a molecule absorbs or emits the energy ε in the form of radiation during the transition between quantum theoretically possible states, then this elementary process can be viewed either as a completely or partially directed one in space, or also as a symmetrical (nondirected) one. It turns out that we arrive at a theory that is free of contradictions, only if we interpret those elementary processes as completely directed processes.

("On the Quantum Theory of Radiation, p.65)

An outgoing light particle must impart momentum hν/c to the atom or molecule, but the direction of the momentum can not be predicted! Neither can the theory predict the time when the light quantum will be emitted.

Such a random time was not unknown to physics. When Ernest Rutherford derived the law for radioactive decay of unstable atomic nuclei in 1900, he could only give the probability of decay time. Einstein saw the connection with radiation emission:

It speaks in favor of the theory that the statistical law assumed for [spontaneous] emission is nothing but the Rutherford law of radioactive decay.

(Pais," "Subtle is the Lord...", p.411)

Einstein clearly saw, as none of his contemporaries did, that since spontaneous emission is a statistical process, it cannot possibly be described with classical physics.

The properties of elementary processes required...make it seem almost inevitable to formulate a truly quantized theory of radiation.

(Pais, ibid.)

Planck had assumed that energy levels were discrete (compare Bohr's stationery states in the old quantum theory). Einstein saw that transitions between those levels should be discrete quanta. When Bohr formulated his atom theory (and for the next dozen years), he ignored Einstein's light quanta

Planck's theory leads to the following conjecture. If it is really true that a radiative resonator can only assume energy values that are multiples of hν, the obvious assumption is that the emission and absorption of light occurs only in these energy quantities. On the basis of this hypothesis, the light-quanta hypothesis, the questions raised above about the emission and absorption of light can be answered. As far as we know, the quantitative consequences of this light-quanta hypothesis are confirmed. This provokes the following question. Is it not thinkable that Planck's radiation formula is correct, but that another derivation could be found that does not rest on such a seemingly monstrous assumption as Planck's theory? Is it not possible to replace the light-quanta hypothesis with another assumption, with which one could do justice to known phenomena? If it is necessary to modify the theory's elements, couldn't one keep the propagation laws intact, and only change the conceptions of the elementary processes of emission and absorption?

To arrive at a certain answer to this question, let us proceed in the opposite direction of Planck in his radiation theory. Let us view Planck's radiation formula as correct, and ask ourselves whether something concerning the composition of radiation can be derived from it.

("On the Development of Our Views Concerning the Nature and Constitution of Radiation", Physikalische Zeitschrift, 10, 817-825, 1909)

Eight years later, in his paper on the A and B coefficients (transition probabilities) for the emission and absorption of radiation, Einstein carried through his attempt to understand the Planck law. He confirmed that light behaves sometimes like waves (notably when a great number of particles are present and for low energies), at other times like the particles of a gas (for few particles and high energies).

Quantum mechanics is able to effect a reconciliation of the wave and corpuscular properties of light. The essential point is the association of each of the translational states of a photon with one of the wave functions of ordinary wave optics. The nature of this association cannot be pictured on a basis of classical mechanics, but is something entirely new. It would be quite wrong to picture the photon and its associated wave as interacting in the way in which particles and waves can interact in classical mechanics. The association can be interpreted only statistically, the wave function giving us information about the probability of our finding the photon in any particular place when we make an observation of where it is.

Note that the information about the possibility of a photon at a given point does not have to be "knowledge" for some conscious observer. It is statistical information about the photon, even if it is never observed

Some time before the discovery of quantum mechanics people [viz., Einstein!] realized that the connexion between light waves and photons must be of a statistical character. What they did not clearly realize, however, was that the wave function gives information about the probability of one photon being in a particular place and not the probable number of photons in that place.

(Principles of Quantum Mechanics, 4th ed., Chapter 1, p.9)

But if we accept that Einstein always conceived the particle as indivisible and located at a given point in space and time (his local "objective reality"), we can agree with Dirac that the wave function gives us the probability of the individual particle "being in a particular place."