Schrödinger’s probability wave equation (1926)
If the double-slit experiment encapsulates the essence of quantum mechanics in one basic experiment, Schrödinger’s probability wave equation has been described as its central ingredient, its hallmark feature, its mathematical engine[1]. "Schrödinger’s equation has been universally recognized as one of the greatest achievements of 20th century thought, containing much of physics and, in principle, all of chemistry".[2]
It was Schrödinger’s equation that enabled physicists to calculate how so-called quantum probability waves move, and therefore make precise predictions that could be compared with experiment. Yet, as Richard Feynman said, it came from nowhere else other than Schrödinger’s brain. Such waves are permanently and completely unobservable. It should also be noted that Schrödinger’s equation yields the same results as Heisenberg's matrix mechanics' mathematical formulism, though in a totally different format.
The equation itself
Schrödinger started off with the classical Newtonian concept that a particle's total Energy is comprised of not only of the energy deriving from its motion, its so-called kinetic energy, E ∝ mv2, but also its potential energy (V). However, in light of de Broglie's discovery that matter also behaves like a wave, he then described the particle's kinetic energy in terms of its wave function, which he chose to express by reference its momentum.
Schrödinger then traded the classical energy equation for a wave with a differential wave equation which showed how the individual states of the quantum system and their amplitudes varied in relation to each other at any given point in time, demonstrating the relative importance of each state to the system as a whole as an infinite series.[2.1]
Energy passes continuously from one vibration pattern to another
The term on the left hand side is the key to Schrödinger’s equation because it gives the wave function in time, which was what Schrödinger was trying to derive. The kinetic energy term appears on the right hand side of the equation.
The equation postulates that the wave function ψ of the energy state of the system as a whole ultimately reveals itself to be a manifestation of an infinite series of the system’s internal periodic functions. In the process the system’s underlying quantisation and internal harmonies are revealed - imagine the natural frequencies and the number of nodes of a one-dimensional standing wave inherent in a violin string. Similar analogies may be drawn for two and three dimensional waves. Under Schrödinger’s system, energy passes continuously from one vibration pattern to another rather than by means of quantum jumps, and what looks like a particle becomes a superposition of thousands of waves, much as the French mathematician and physicist Louis de Broglie had described. Soon it was shown that Schrödinger’s theory, which became known as wave mechanics, gave a complete description of the spectral lines in the hydrogen atom.[3]
As observed, Schrödinger' equation has been described as encapsulating probabilities, although Schrödinger himself hated the idea of his equation being portrayed in this fashion. Thus, it has been said that the wave function ψ permeates all of space and evolves in accordance with the equation, ψ encoding the probability of finding the particle within any given region, as well as the probabilities for its momentum, energy and so on. It is essentially a list of numbers, one number for every possible configuration of the system[4]. Probability waves thus give rise to predictions, but the waves themselves are outside the arena of everyday reality. However, the theory is accepted by scientists even though the central tenets are unobservable, and no one really knows how it works.
Working from that equation published in 1926, physicists can input the details of how things are now, and then calculate the probability that they will be one way or another or another still at any moment in the future, and across a broad sweep the probabilistic predictions of quantum mechanics arising from the equation match experimental data - always [5].
As James Gleick says, "Quantum mechanics taught that a particle was not a particle but a smudge, a travelling cloud of probabilities, like a wave in that the essence was spread out. The wave equation made it possible to compute with smudges and accommodate the probability that a feature of interest might appear anywhere within a certain range." Classically electrons should seek their state of lowest energy and spiral into the positively charged nuclei. That was impossible in quantum mechanics "because it would give the electron a definite pointlike position. Quantum-mechanical uncertainly was the air that saved the bubble from collapse. Schrödinger’s equation showed where the electron clouds would find their minimum energy, and on those clouds depended all that was solid in the world".[6]
Yet paradoxically (nothing unusual about paradoxes in quantum mechanics), as Steven Weinberg points out, the Schrödinger equation, which governs the way that wave functions change over time, does not involve probabilities. It is just as deterministic as Newton’s equations of motion and gravitation. “That is, given the wave function at any moment, the Schrödinger equation will tell you precisely what the wave function will be at any future time. There is not even the possibility of chaos, the extreme sensitivity to initial conditions that is possible in Newtonian mechanics. So if we regard the whole process of measurement as being governed by the equations of quantum mechanics, and these equations are perfectly deterministic, how did probabilities get into quantum mechanics in the first place?”, inquires Weinberg. [7]
Recall however that the idea of probability was introduced into quantum mechanics by Max Born in a 1926 paper in which he found a way to reconcile particles and waves by postulating that the wave ψ determines the likelihood that the electron will be in a particular position. This was not a probability due to ignorance as in the case of the Maxwell and Boltzmann's averaging techniques (because it was impossible to calculate exact values for so many particles), but a concession to the understanding that probabilities are all we are ever likely to know about an atomic system [8]
The Copenhagen interpretation
These ideas were taken one step further in the Copenhagen interpretation of quantum mechanics by Niels Bohr and others, an interpretation which postulates that whenever you try to see a probability wave experimentally, the very act of observation or measurement thwarts your attempt, and that when you look at an electron’s probability wave (for these purposes, look or observe equals measure, generally with the use of appropriate technology), the electron responds by snapping to attention and coalescing at one definite location. Correspondingly, the probability wave surges to 100% at that spot, while collapsing to 0% everywhere else. Look away and the needle-thin probability wave rapidly spreads, indicating that there’s once again a reasonable chance of finding the electron at a variety of locations. On the other hand, every time you attempt to see the probability haze, it disappears (collapses) and is supplanted by familiar reality. This interpretation is actually corroborated by experiments, but the whole imagery of a “collapse” may be seen as misleading.
Bohr's expedient
The Copenhagen analysis does not accord mathematically with Schrödinger’s equation (but it does accord with experiments in the laboratory), a conundrum prompting Bohr to come up with a unique solution: evolve probability waves according to Schrödinger’s equation when you’re not looking or performing any kind of measurement, but when you look, ignore Schrödinger’s equation and simply declare that your observation has caused the wave to collapse. Before you "look", the state of the atomic system is undefined, having only the potentiality of certain values with certain probabilities. Whatever constitutes “looking” or “measuring” is not defined. "What does the "measurement" really mean, and why does the act of measurement change the state of the system from a superposition of possibilities to a single certainty? Does the change of state occur when a dog or even a fly observes the system? What about when a molecule in the air interacts with the system, which we expect to be occurring all the time, yet which we do not usually treat as a measurement that can interfere with the outcome? Or is there some special physical significance in a human consciously learning the state of the universe?" [9] Bohr said that he was drawing a line in the sand separating small things (atoms and their constituents) and big things (such as observers or experimenters or their equipment), but he didn’t say where that line was. He couldn’t.
Bohr also came to regard the wave/particle duality as the central core of understanding quantum mechanics in the sense that whether an object behaves like a wave or a particle depended on your choice of apparatus for looking at it. For example, if you view a particle through a particle detector, it will present as particles, but if you view it through a wave detector it will exhibit both wave and particle properties. He called this idea complementarity.[9.1]
The two states of Schrödinger's cat
In 1935, Schrödinger responded to the Copenhagen line of thinking that quantum physics consists of what happens when nobody is looking with a thought experiment. A living cat is placed inside a steel chamber along with a device containing a vial of hydrocyanic acid. In the chamber, there is a tiny chunk of radioactive material. If a single atom of the substance decays during the test period, a relay mechanism will trip a hammer, which will in turn break the vial and kill the cat. "Suppose the radioactive source is such that quantum theory predicts a 50% probability of one decay particle each hour. After an hour has passed, there is an equal probability of either state - the live cat state or the dead cat state".[9.2] So to an onlooker on the outside, the radioactive substance both has and has not decayed, and we do not know whether the vial has been broken, the hydrocyanic acid released, and the cat killed. The cat is therefore left in a limbo of being simultaneously dead and not dead. It is only when we break open the box and learn the condition of the cat that that superposition of states (dead and not dead) is lost, and the cat becomes one or the other.
From the Copenhagen point of view, the observation or measurement itself affects the outcome, and there is no outcome until the act of looking or measurement has been made. At that moment the superposition of states which previously existed collapses and the cat suddenly becomes either fully alive or fully dead. But until someone looks, the situation inside the container is akin to “having in it the living cat and the dead cat (if you’ll excuse the expression) mixed or smeared out in equal parts”[10].
Robert Crease says that the image is “a brilliant demonstration of the flaws of extending the theories of the microworld to those of the macroworld”[11], and although there remain many theorists who interpret the idea of the superposition of states when applied to macroscopic scales literally, the absurdity of the cat paradox is obvious. However, with each passing year, experiments seem to have confirmed that Schrödinger’s equation works without modification for increasingly large collections of particles, such as you and me and the equipment we may use to make relevant observations.
A superposition of "two entire worlds"
In 1957, Hugh Everett developed his “many worlds” interpretation of quantum mechanics, the central insight of which was that the state of a quantum system reflects the state of the whole universe around it, with the effect that we must include the observer in a complete description of the measurement. Under this scenario, the quantum state after the measurement is still a superposition, but a superposition of two entire worlds. A corollary of this is that everything in nature obeys the laws of quantum mechanics, whether large or small. [12]
According to Brian Greene, the way to think about the problem is this: you and I and computers and bacteria and viruses and everything else (material) are made of molecules and atoms, which are themselves composed of particles like electrons and quarks. Schrödinger’s equation works for electrons and quarks and all evidence points to its working for things made from these constituents, regardless of the numbers of particles involved. This means that Schrödinger’s equation should still continue to apply during a measurement. After all, a measurement is just one collection of particles (the person, the equipment, the computer…) coming into contact with another (the particle or particles being measured.[13] In other words, Schrödinger’s equation doesn’t allow waves to collapse, and an essential element of the Copenhagen interpretation is undermined.
Other hypotheses
In any event, as Robert Crease has also been moved to comment, the whole scenario of the “collapse” of the wave function is misleading. The Schrödinger wave waves in multidimensional “configuration space” – three dimensions for every particle in the system. The particles are observable but have lost their predictability; the waves are predictable but have lost their observability. Observing the position and momentum of something does not allow you to make predictions about where you will see it next. The wave is used to predict the probability of another event, but after the event is observed, the wave has no more value and has to be discarded or “reset”, in other words modified, to incorporate new information.
The collapse imagery captures the idea that before the event happened – before you detected the particle, say – it could be anywhere, so one thinks that the event or particle is everywhere. So the image conjures up the idea of a structure, extended through space, suddenly getting drawn up into a single point, which is misleading. Nothing collapses. The wave function, whose purpose is only to give probabilities has flowed along – “predictably, deterministically” – but once an event happens, the function has exhausted its purpose, and must be reset. The whole conundrum is encapsulated in the aphorism, “[t]he wave tells the particle where to go, the particle tells the wave where to start and stop”[14].
Entanglement and decoherence
Another hypothesis explaining how and why this happens is that the particle’s wave function, which comprises all the possible physical states a particle might have, becomes enmeshed or entangled as it mingles with the wave functions of other quantum systems around it, causing the particle’s many quantum possibilities to collapse into a single classical reality. Shades of the "many worlds" approach. This is known as decoherence [15].
Here, Weinberg again: "[i]n a measurement, the spin (or whatever else is measured) is put in an interaction with a macroscopic environment that jitters in an unpredictable way. For example, the environment might be the shower of photons in a beam of light that is used to observe the system, as unpredictable in practice as a shower of raindrops. Such an environment causes the superposition of different states in the wave function to break down, leading to an unpredictable result of the measurement. (This is called decoherence.) It is as if a noisy background somehow unpredictably left only one of the notes of a chord audible. But this begs the question. If the deterministic Schrödinger equation governs the changes through time not only of the spin but also of the measuring apparatus and the physicist using it, then the results of measurement should not in principle be unpredictable". [16]
Weinberg sees this so called "instrumentalist approach" as a descendant of the Copenhagen line of thinking, and he is far from happy with human beings being bought into the laws of nature at the most fundamental level [17]. An ideal of what a physical theory should be “should be something that doesn't refer in any specific way to human beings", he said. "It should be something from which everything else—including anything you can say systematically about chemistry, or biology, or human affairs—can be derived. It shouldn't have human beings at the beginning in the laws of nature". “And yet", he says" I don't see any way of formulating quantum mechanics without an interpretive postulate that refers to what happens when people choose to measure one thing or another thing.” [18].
Weinberg's putative answer is to the effect that "a new theory might be designed so that the superpositions of states of large things like physicists and their apparatus even in isolation suffer an actual rapid spontaneous collapse, in which probabilities evolve to give the results expected in quantum mechanics. The many histories of Everett would naturally collapse to a single history. The goal in inventing a new theory is to make this happen not by giving measurement any special status in the laws of physics, but as part of what in the post-quantum theory would be the ordinary processes of physics".[19]
Continuous spontaneous localisation
In this context, the collapse of the wave function to a single possibility is a random event without any contribution coming from human or environmental interference. This is known as “continuous spontaneous localisation” (CSL) by means of which quantum probabilities randomly collapse into classical certainties. The theory predicts that the action of collapse imparts a slight jiggle to particles, "quantum nudges" as it were, creating an omnipresent background vibration that might be detectable in experiments.
Where lies the divide between the very big and the very small?
But in all this, isn't there something which has been overlooked: where lies this so-called distinction between the "very small" and the "very large"? Where does the quantum world end and the so-called classical world of Newtonian physics begin, and vice-versa, and what is the true nature of matter in between the micro and the macro? An experiment involving a millimetre-size membrane resembling a tiny trampoline tethered to a silicon chip, just barely visible to the naked eye, is currently underway in Delft in the Netherlands, to test the plausibility of such a hypothesis. The membrane can be jolted into long-lasting vibration, and ultimately scientists plan to use a laser to push the membrane into a quantum combination or superposition of multiple waves.[20].
With the membrane vibrating at say two different amplitudes at once, in much the same way as two or more musical notes make up a chord, researchers will then watch to see what happens if the membrane settles into a single amplitude, much the same as a single musical note when the other notes stop playing [21]. If all proceeds according to plan, future tests may involve a half-millimetre long bug called a tardigrade traversing the membrane as a passenger, but that is some considerable time off yet and researchers must await the device first being placed into superposition.
What would Einstein have said
And what would Einstein have thought of all this? Notwithstanding his “God does not play dice” aphorism, Einstein did not dispute the indeterministic nature of quantum mechanics[22],[23]. He had little choice given his 1905 papers: the photoelectric effect (On a Heuristic Viewpoint Concerning the Production and Transformation of Light) which depended upon the very notion of quanta – discrete chunks of energy - and his work on Brownian motion accounting for the jiggling motion of particles in a fluid in the same year. What he did dispute was that quantum mechanics was the end point of the inquiry. Einstein was of the view that there was a deeper reality yet to be discovered lurking behind the randomness, and asserted that a deterministic sub-realm of the universe would lead to the randomness of the quantum realm. To fill in this deeper level of reality, Einstein sought a unified field theory, in which particles derive from structures that look nothing like particles.
Although Einstein was not anti-quantum, he was definitely anti-Copenhagen interpretation – the view that it is only when you observe or measure a particle that it “collapses’ and materialises somewhere. He recoiled from the idea that a simple act of measurement should cause a break in the continuous evolution of a physical system, and this was the context of his dice-rolling jibe. In Einstein’s view, collapse could not be a real process, since it would require instantaneous action at a distance faster than the speed of light thereby violating general relativity theory. And what was an act of measurement anyway? Einstein would have been happy to accept indeterminism if only someone could spell out what an act of measurement was and how it was possible for particles to stay in sync while acting at a distance. On this view wave function collapse was not a physical process, but the acquisition of knowledge.
[1] Brian Greene, The Hidden Reality – Parallel Universes and the Deep Laws of the Cosmos, Publisher: Alfred A Knopf, New York (2011), 168, 201.
[2] JP McEvoy, Introducing Quantum Theory: A Graphic Guide, Icon Books, Kindle Edition.
[2.1] Michael Box, “The Fundamental Nature of Light”, WEA course, 16 August 2016, 4.1; also:
https://www.physlink.com/education/askexperts/ae329.cfm and https://journeymanphilosopher.blogspot.com/2011/05/trying-to-understand-schrodingers.html from which the equation derives.
[3] This is a heavily edited summary of the material appearing in JP McEvoy's Introducing Quantum Theory: A Graphic Guide, Icon Books, Kindle Edition, locations 1287-1383
[4] Steven Weinberg, “The trouble with Quantum Mechanics”, The New York Review of Books, January 19, 2017.
[5] Ibid, 192.
[6] James Gleick, Genius - Richard Feynman and modern physics, Little, Brown and Company, London, 1992, 89.
[7 ] Steven Weinberg, “The trouble with Quantum Mechanics”, op cit.
[8] JP McEvoy, Introducing Quantum Theory: A Graphic Guide, Icon Books Ltd. Kindle Edition, location 1420-1425.
[9] Questions posed by Yasunori Namura in "The Quantum Multiverse", Scientific American, 22 at 25, noted on the page Branes and multiple universes
[9.1] JP McEvoy, op cit.
[9.2] Ibid.
[10] Schrödinger’s quantum terminology.
[11] Greene, op cit, 228.
[12] Discussed in Yasunori Namura's article, op cit, 25 ff. Everett's theories are analysed in some detail in Brian Greene's 2011 publication, op cit, pp 189-209, 321.
[13] Greene (2011), 202-3.
[14] Robert P. Crease, “The basic equation of Quantum Theory: Schrödinger’s Equation”, Ch 9 in The Great Equations – Breakthroughs in Science form Pythagoras to Heisenberg, Norton, New York, 2008, 227 – 228.
[15] This is an edited summary of Tim Folger's article “Crossing the Quantum divide”, Scientific American, July 2018, 20-27. The concepts of decoherence, superposition and quantum entanglement are considered in another context at Black holes, the information paradox and quantum entanglement: the link between quantum physics and general relativity?
[16] Weinberg, "The trouble with Quantum Mechanics", op cit.
[17] Ibid
[18] Cited in Folger, op cit, at 24
[19] "The trouble with Quantum Mechanics", op cit.
[20] Folger, op cit.
[21] Weinberg, "The trouble with Quantum Mechanics", op cit.
[22] From George Musser’s article, “Is the cosmos random?”, Scientific American, Special Issue, 100 years of General Relativity, September 2015, 78-83, building upon the work of Don A Howard, a historian at the University of Notre Dame.
[23] And as it turned out, God not only played dice, he ran a casino: Stephen Hawking, Brief Answers to the Big Questions, John Murray, London, 2018, 52. Furthermore, "we are the product of quantum fluctuations in the very early universe. God really does play dice": ibid at 63.