by James V Stone, University of Sheffield, UK.

“There is a theory which states that if ever anyone discovers exactly what the Universe is for and why it is here, it will instantly disappear and be replaced by something even more bizarre and inexplicable.
There is another theory which states that this has already happened.”
Douglas Adams.

Introduction

Isaac Newton (1642–1726) derived equations that work extraordinarily well in describing how planets orbit the sun and how apples fall from trees, so these equations have a clear physical interpretation in terms of everyday experience. The equations of quantum mechanics also work well, but unlike Newton’s equations they do not have any obvious physical interpretation.

Image for post
Image for post
Figure 1: The authors of three equivalent theories of quantum mechanics.

For philosophers, the monumental achievements of quantum mechanics are both a blessing and a curse. They are a blessing because they provide the raw material for endless debates regarding the nature of reality, and they are a curse because the language of mathematical physics cannot always be translated into any spoken language. As one of the pioneers of quantum mechanics said:

We must be clear that when it comes to atoms, language can be used only as in poetry. (Bohr, around 1920, as later recalled by Heisenberg).

For physicists, the equations of quantum mechanics provide an account of the physical world that is without equal in terms of pure numerical accuracy. But physicists do not judge a theory on accuracy alone. Like all scientists, physicists seek a coherent, detailed theory that is both numerically accurate and intuitively satisfying. However, quantum mechanics is the most successful and also the most counter-intuitive theory imaginable. Precisely because the equations of quantum mechanics work so well, a common response (originally stated by David Mermin) is to “shut up, and calculate”.

Quantum mechanics, and especially Schrödinger’s framework, describes everything in terms of waves. How, then, does quantum mechanics make contact with the macroscopic, too-solid world of objects?

The short answer is that no-one really knows. The longer answer is that the probabilities defined with such perfection by quantum mechanics yield singular events in the macroscopic world. The equations of quantum mechanics do not specify which events will occur, but they do specify the exact probability of each possible event. For example, we do not know the future location of a particle, but we do know the exact value of the probability that it will be in any given location. These probabilities are implicit in Schrödinger’s wavefunction. Given such a wavefunction, what causes a particle to coalesce out of the miasma of quantum probabilities defined by that wavefunction?

Again, the short answer is that no-one really knows. The longer answer is that something disturbs the wavefunction, which somehow forces it to yield up one unique realisation out of the infinitude of possibilities it represents. This disturbance is usually referred to as the collapse of the wavefunction. A wavefunction collapse can be caused by something as simple as a photon materialising onto a photographic plate. Because this represents a kind of physical measurement, wavefunction collapse has come to be associated with the process of measuring the state of a physical system. The following sections provide a brief overview of the main interpretations of quantum mechanics.

The Copenhagen Interpretation

After a gestation period of a quarter of a century, quantum mechanics was born in 1926. With three fathers, Heisenberg, Schrödinger and Dirac (Figure 1) and many uncles, it was not long before the proud family began to worry about how to interpret the equations that defined the wondrous child they had created. Over the next few years, Niels Bohr and Werner Heisenberg (among others), wrote a series of papers discussing the problem, resulting in the Copenhagen interpretation.

The Copenhagen interpretation assumes that the values of all physical quantities are undefined until they are measured, at which point the process of measurement induces the wavefunction to collapse. It is only this wavefunction collapse that forces each of the physical quantities measured to adopt a single value.

A key feature of the Copenhagen interpretation is that Schrödinger’s wavefunction is not a property of the universe; rather, it is a descriptive tool that is useful inasmuch as it provides predictions regarding the state of the physical world, but the wavefunction itself cannot be said to correspond to anything physical. For all practical purposes, this seems to be the position adopted by Schrödinger. While developing ideas about his famous ambivalent cat (Figure 2), Schrödinger wrote in a letter to Einstein:

I am long past the state where I thought that one can consider the wavefunction as somehow a direct description of reality.

Image for post
Image for post
Figure 2: Schrödinger’s cat — wanted, dead or alive (or both).

The Copenhagen interpretation is reminiscent of the outcome of a dispute between the Catholic church and Galileo (1564–1642). Galileo famously supported Copernicus’ heliocentric theory that the Earth orbits the Sun. This view contradicted holy scripture and (more importantly) the Catholic church. Armed with Copernicus’ theory, Galileo developed formulae for predicting the positions of heavenly bodies, which were widely used to aid navigation on the high seas. But after Galileo was threatened with torture for his heretical views, and (implicitly) the opportunity to be burned at the stake, he had to ‘admit’ that he had never supported the heliocentric theory. The compromise, or fudge, forced on Galileo was that his equations merely served the purpose of predicting the motions of the heavens; in the language of the time, Galileo’s equations were merely a tool that could be employed to ‘save the phenomena’. The difference is that, whereas Galileo was forced to downgrade his equations to mere ‘hypothetical’ descriptions of reality, Schrödinger voluntarily downgraded his wavefunction to a practical tool that could be employed to ‘save the phenomena’.

Image for post
Image for post
Figure 3: Geometric representation of Heisenberg’s uncertainty principle. For a particle, position uncertainty is represented on the horizontal axis and momentum uncertainty on the vertical axis, so the total uncertainty is the product of these uncertainties, which is represented by the grey area. a) Uncertainties in position and momentum have equal precision. b) Small position uncertainty means large momentum uncertainty. c) Small momentum uncertainty means large position uncertainty. The grey area, and hence the minimum uncertainty is the same in a–c.

A key outcome of the discussions between Bohr and Heisenberg was the complementarity principle, proposed by Bohr in 1928. This states that the values of certain pairs of physical quantities cannot both be known at the same time. For example, Heisenberg’s uncertainty principle states that the position and momentum of a particle cannot both be known exactly (Figure 3). More generally, Bohr’s complementarity principle implies that the entities in a double-slit experiment (Figure 4) behave like waves or particles depending on the type of measurement imposed on the apparatus.

Image for post
Image for post
Figure 4: The double-slit experiment. Waves travel from the source (top) until they reach the first barrier, which contains a slit. A semi- circular wave emanates from the slit until it reaches the second barrier, which contains two slits. The two semi-circular waves emanating from these slits interfere with each other, producing peaks and troughs along radial lines that form an interference pattern on a screen (bottom).

Objective Collapse Theories

Objective collapse theories treat the Schrödinger wavefunction as if it were a physical wave, which spontaneously collapses once the number of particles in an object becomes sufficiently large. A prominent version of this idea was published in 1986 by Ghirardi, Rimini and Weber, who proposed that each wavefunction collapses spontaneously on average every hundred million years. This makes it sound as if a wavefunction collapse is a rare event. However, a 2g cube of carbon (for example) contains about 10 to the power 23) atoms, and if we associate a wavefunction with each atom then 10 to the power 15 carbon atom wavefunctions in the cube will collapse each year, which comes to about 32 million wavefunction collapses per second. In other words, macroscopic objects comprise collections of wavefunctions that are collapsing pretty much all the time. Each wavefunction collapse acts like a measurement, causing a cascade of further collapses, which forces the entire object to remain as an object in the world of classical physics (i.e. a conventional object). In a sense, spontaneous collapse involves the system behaving as if it continuously measures itself.

Because larger objects contain more particles, objective collapse depends on the size of the object under consideration. This, in turn, means that there should be an upper limit on the size of object that can exist in a superposition of quantum states. Thus, objective collapse theories are not merely abstract ideas — they make predictions that can be tested experimentally. If nothing else, spontaneous wavefunction collapse saves Schrödinger’s cat from dangling in an undefined state, somewhere between life and death.

The philosophical problems generated by the mysterious wavefunction collapse clearly infuriated Schrödinger. As early as 1929, six years before he attempted to dispense with the whole idea by creating his famously half-dead cat, Schrödinger declared:

If this damned quantum leaping is to remain, I regret having dealt with quantum theory at all.

Bohmian Mechanics

When de Broglie proposed in 1924 that particles, as well as light, behave like waves, he envisaged each particle being guided through space by a pilot wave. With the advent of Schrödinger’s wave mechanics (in 1926), it soon became clear that Schrödinger’s wavefunction was a prime candidate for de Broglie’s pilot wave. This general notion was developed by Bohm in 1952 into what is now called Bohmian mechanics.

An unusual aspect of Bohmian mechanics is that it is deterministic. However, removing the random element from quantum mechanics requires an additional equation for a particle’s position. In practice, despite this apparent non-randomness, Bohmian mechanics makes the same predictions as standard quantum mechanics. Bohmian mechanics is historically important because it played a key role in the development of Bell’s theorem.

The Many-Worlds Interpretation

As inscrutable as it is, the problems of wavefunction collapse seem trivial compared with the alternatives. For example, the many-worlds interpretation, which derives from the work of Hugh Everett, requires the creation of many entire new universes every time a measurement is made. Specifically, it is assumed that when a measurement with N possible outcomes is made, this causes the universe to split into N versions of itself. In each of these universes, life goes on as if a different one of the N outcomes had occurred.

An attractive feature of the many-worlds interpretation is that it is simple. However, this simplicity comes at a price, and it has been noted that the many-worlds interpretation is cheap on assumptions but expensive on universes.

The von Neumann–Wigner Interpretation

Because the human retina effectively measures the arrival of photons as efficiently as a photographic plate, it has been suggested that human consciousness is necessary for wavefunction collapse to occur. In the limit, this seems to suggest that any event in the physical world exists as a mere potentiality until a sentient being observes it. At a single stroke, this solves the age-old riddle of whether or not a tree falling in an uninhabited forest makes a sound, or (as Einstein noted) whether the moon ceases to exist if no-one is looking at it. However, by analogy with the many-worlds interpretation, the von Neumann–Wigner interpretation is cheap on assumptions but expensive on sentient beings.

Summary

Plato proposed that our perceptions of reality are like the shadows cast on a wall; all we have are these shadows of a world that we can never see directly, cast by people and objects that we can never know with certainty. Even in this modern era, Plato’s prosaic analogy serves as a cautionary tale for science, with a clear message that we should treat our observations of the world not as solid facts, but as provisional clues. When considered in the context of quantum mechanics, Plato’s prescient analogy is more accurate than he could ever have imagined.

Note: This is an edited extract from the final chapter of The Quantum Menagerie by James V Stone (published December 2020). Topics mentioned here are explained more completely in previous chapters of the book. Interpreting quantum mechanics is an academic field in its own right, and only the main interpretations are included above.

James V Stone is an Honorary Associate Professor at the University of Sheffield, UK.

Chapter 1, the table of contents and book reviews can be seen here The Quantum Menagerie.

James V Stone is an Honorary Associate Professor at the University of Sheffield, England. Published books: jim-stone.staff.shef.ac.uk/books.html

Get the Medium app

A button that says 'Download on the App Store', and if clicked it will lead you to the iOS App store
A button that says 'Get it on, Google Play', and if clicked it will lead you to the Google Play store