Schrödinger’s Paradox Revisited
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Again we return to the infamous Schrödinger experiment. In the essay preceding this one I replaced Schrödinger’s cat and a vial of poison with a light bulb, a battery, and a toggle switch. In this essay I want to conduct an even more elaborate version of that imaginary experiment. First let’s review that original experiment.
We have enclosed our experimental apparatus within a perfectly opaque, completely soundproof, thermally insulated box. Simply put, the apparatus consists of three parts: the lower part consists of a light bulb, a battery, a toggle switch, and the wires that connect them into a circuit; the middle part consists of a hammer mounted on a pivot and a solenoid against which the hammer rests in such a way that if the solenoid’s core moves, the hammer will lose its balance, swing down, and hit the toggle switch in a way that opens the circuit and turns the light off; and the upper part consists of an unstable particle, such as an atom of tritium, inside the tube of a Geiger counter that feeds its output into an amplifier, whose output goes to the solenoid. With that apparatus assembled inside the box we turn on the light, close up the box, and wait.
An atom of tritium consists of one proton and two neutrons stuck together with a single electron revolving around them. With a half-life of 12.32 years, it decays by spitting out from its nucleus an electron and a quantum of the weak force (an electron anti-neutrino) and becoming an atom of helium-3. We can re-conceive the tritium atom, rather crudely, as a self-jostling quantum system comprising an atom of helium-3 (which we conceive as a potential-energy cage) containing an electron in its nucleus. This system exists in one of two states - the electron inside the cage (the un-decayed tritium atom) and the electron outside the cage (the decayed tritium atom). In the un-decayed state the system conforms to a state function that has two parts, the probability wave confined inside the cage and the much smaller probability wave outside the cage.
When the atom decays, the state function describing the system must change into one with a very small amplitude inside the potential-energy cage that the helium-3 nucleus represents and a larger amplitude outside the cage. Thus the electron has essentially zero probability of existing inside the nucleus and a probability close to unity of existing outside the nucleus. Taking energy with it, the electron moves rapidly and generates a faint electrical pulse in the Geiger counter. The amplifier strengthens the pulse, which then goes to the solenoid, where it generates a magnetic field that moves the solenoid’s core. That moving core nudges the hammer out of balance and the hammer, in consequence, swings down on its pivot and hits the toggle switch. Flipped by the hammer, the toggle switch breaks the electrical circuit feeding electricity from the battery to the light bulb and thus turns the light off. And there we have a simple nuclear decay detector.
If we assemble this apparatus and turn the light on, we will see when the tritium atom decays by noting when the light goes out (assuming, of course, that no part of the apparatus fails). In that case, although we interpret the state function associated with the tritium atom as representing a mixed state, we can see what state the atom actually occupies. If we put the apparatus into the box and close up the box, we no longer have that certainty.
In Schrödinger’s interpretation of the experiment, once we put the apparatus into the box the tritium atom goes into an actual mixed state; that is, the atom exists as both decayed and un-decayed at the same time. That proposition necessitates that the Geiger counter exist in a mixed state of receiving/not receiving an electrical pulse. Every component of the apparatus must then exist in two different states simultaneously. The hammer must exist as both balanced and fallen; the toggle switch must exist as both flipped and un-flipped; and the light bulb must exist as both off and on (and note that this does not mean that a 100-watt bulb shines out 50 watts: it means that the bulb shines out both 100 watts and zero watts at the same time).
This seems to give us an easy reductio ad absurdum, but we need to go deeper into the analysis of the experiment before we can resolve this paradox. We need to ask why does putting the apparatus into a box change the way we must describe it. Why does making a system un-observable in some way alter the very existence of the system? That question brings to mind George Berkeley’s proof of the existence of God.
At the beginning of the Eighteenth Century George Berkeley (1685 Mar 12 - 1753 Jan 14) developed his doctrine of immaterialism (now called subjective idealism), which he summed up in the statement "esse est percipi" (to exist is to be perceived). He asserted a belief that matter does not exist as a thing-in-itself, does not possess the property of reality, but rather that only perception is real, so that things exist only to the extent that they are perceived. He then argued that the continued and consistent existence of things necessitates the existence of an omnipresent observer to perceive them at all times.
In like manner, the København Interpretation of the quantum theory, usually associated with Niels Bohr, asserts that a quantum system does not achieve a definite state until someone subjects it to an observation. Until an observation collapses it onto a single, definite state, the state function describing the system evolves as the description of a mixed state.
But Schrödinger’s apparatus extends the presumed mixed state of the tritium atom into the realm of actions so vastly greater than the Planck unit that the single Planck unit of indeterminacy fades into complete insignificance. It pushes the blurry physics of the quantum realm into the classical, sharply focused realm of perfectly deterministic physics. In that latter realm the law of the excluded middle, the basis for the reductio ad absurdum, reigns: if we have put a system into one state (hammer balanced upright, light bulb glowing), then the system cannot at the same time manifest itself in a contrary state (hammer fallen, light bulb dark).
If, on the other hand, we interpret the mixed state as representing nothing more than our ignorance of which state the system occupies, then the paradox dissolves readily. In that case the tritium atom always exists in a definite state (un-decayed or decayed) so all of the other components of the apparatus do as well. We simply don’t know what state that is until we open the box and look at the light bulb. And that makes sense in light of our belief that putting the system into a box should diminish our knowledge of the system but certainly should not alter the system itself.
How did we get into this paradox? We misinterpreted the state function that we use to describe the tritium atom. In describing the state function of the electron trapped inside the un-decayed tritium atom’s nucleus I noted that it divides neatly into two parts – the high amplitude part inside the potential-energy cage that represents a helium-3 nucleus that will become manifest when the atom decays and the low amplitude part outside the cage. We can use those two parts via Enrico Fermi’s Golden Rule to calculate the probability that the atom will decay within a certain time interval and thence calculate the atom’s half life. But the fact that we can divide the state function into two parts that we can treat more less independently does not mean that it represents a genuine mixed state, a combination of un-decayed atom and decayed atom.
We can see the truth of that proposition in the fact that when the atom actually decays, the state function of the electron changes. Inside the cage the amplitude of the state function goes from high to minuscule and outside the cage the amplitude goes from low to high. An actual mixed state representing a true quantum stochastism would have a description consisting of a linear sum of that state function (both parts) with the original state function (both parts).
By reductio ad absurdum, then, we infer that the atom does not exist in a mixed state. At some time it decays and causes the apparatus to turn off the light and it doesn’t need any witnesses (feline or otherwise) to help it along. In accordance with Heisenberg’s indeterminacy principle, we cannot anticipate when that event will occur, but we can say with complete confidence that it will definitely occur. Thus, we do not need to add to our description of Reality any bizarre contortions, such as the many-worlds hypothesis. We simply accept the fact that the uncertainty that we calculate for this experiment merely reflects our ignorance of its outcome (until we open the box) and nothing more. For all of our theoretical intents and experimental purposes the Schrödingerisch mixed state does not exist.
Consider what Schrödinger had to say:
"It is typical of these cases that an indeterminacy originally restricted to the atomic domain becomes transformed into macroscopic indeterminacy, which can then be resolved by direct observation. That prevents us from so naively accepting as valid a "blurred model" for representing reality. In itself it would not embody anything unclear or contradictory. There is a difference between a shaky or out-of-focus photograph and a snapshot of clouds and fog banks." – E. Schrödinger, 1935.
So does the quantum theory express our own limitations, the imprecision inherent in our apparatus (the blurry photograph), or does it display a fuzziness inherent in Reality (the snapshots of clouds)? If we take the latter possibility as true to Reality, then by bringing that fuzziness into the macroscopic realm, Schrödinger’s thought experiment reduces it to an absurdity. Thus Schrödinger dismissed the "blurred model" of the very theory that he helped to create. In that act he stood with Albert Einstein, who commented to him in a letter he wrote in 1950 that "...one cannot get around the assumption of reality...as something independent of what is experimentally established."
In fact, Schrödinger presented his 1935 paper describing the cat experiment as part of a discussion of the incompleteness of quantum theory asserted by Einstein, Podolsky, and Rosen earlier that year. Where the København Interpretation echoes Berkeley in asserting that nothing is definite until it is perceived/observed, EPR asserted that if we can predict with certainty the outcome of an observation, then we can say that the data in that observation emanate from some element of physical reality. But if things must be perceived in order to have a definite existence, then what do we perceive that gives us the sensations that tell us about the object of our perception? What anchors those sensations in Reality? Modern science does not accept Berkeley’s theological solution of the problem, so we must perforce accept as a fact of Reality the existence of matter independent of any interaction we may have with it. But that necessitates in turn that we accept the existence of the events in which that matter participates, again independent of any interaction we may have with the system in which those events occur. The quantum theory blurs the locations of those events in space and time, but it does not blur the fact that they occur.
Physicists disagree little over the mathematical and operational side of the quantum theory: they refer to quantum mechanics as the most successful theory in modern physics so long as one does not seek to determine the deeper meaning of the theory. But the proponents of several different schools of thought seek to go further and understand the underlying reality, to discern the deeper meaning of the theory. That effort at gaining understanding generally begins with what physicists call the measurement problem, which centers on the question, When does the wave function collapse?
That question marks the boundary between the quantum-mechanical world and the classical-dynamics world. At its most basic the quantum realm involves objects of a scale near that of the atoms, while the realm of classical physics contains all the objects that we can perceive more or less directly. Schrödinger’s imaginary experiment lets us explore that boundary. As an aid in that exploration we refer to an analysis devised by a mathematician.
In 1932 Hungarian-born John von Neumann (1903 Dec 28 – 1957 Feb 08), considered by many to be the greatest mathematician of the twentieth century, published Mathematische Grundlagen der Quantenmechanik (Mathematical foundations of quantum mechanics). Regarded as a classic presentation of the quantum theory and often called simply Die Grundlagen, the book’s contents presented an all-quantum picture of the world. Even macroscopic objects, such as Newton’s apple or the stars themselves, have their own aleatric waves with their associated quantum numbers. On the one hand, von Neumann demonstrated that electrons and the other fundamental particles that make up matter do not exist as real entities because we cannot say that they possess any dynamical attributes before we measure them. That demonstration offered support to the philosophical doctrine of the Copenhagen interpretation. On the other hand, von Neumann did not endow measuring instruments with a special status, as the Copenhagen interpretation did, but regarded them also as quantum-mechanical phenomena.
Von Neumann analyzed the step-by-step process of the measurement act, thereby producing what we call a von Neumann chain. If we take the two-slit experiment as an example, we see that the von Neumann chain consists of nine steps: (1) a photon emerges from a source, (2) the photon passes through one of two slits cut into a opaque screen, (3) the photon triggers a detector, (4) the signal from the detector goes to a meter, (5) a needle or other registration device on the meter moves, (6) light from the meter goes to the eye of the observer, (7) a nerve impulse goes from the observer's retina to the observer's brain, (8) neural nets in the observer’s brain process the signal, and (9) the signal registers in the observer's consciousness.
Where in that series of steps does the collapse of the aleatric wave occur? We can illustrate the problem of locating the collapse by returning to our light-bulb version of the story of Schrödinger's cat with a slightly different apparatus. We place a single-photon source, a solid screen with a pair of slits cut into it, and a photo-detector behind one of the slits into a sealed, soundproof, lightproof box with our original solenoid, hammer, and light bulb apparatus. Inside the box, we so arrange the apparatus that if the photon passes through one slit, the hammer drops and the light turns off. But if the photon passes through the other slit, it does not hit the photocell, does not thus trigger an amplified signal to the solenoid, so the hammer does not drop and the light remains on. According to the Copenhagen interpretation of quantum mechanics, the bulb is neither lit nor dark until someone opens the box. Can we accept this as a reasonable proposition?
Von Neumann showed that we may place the collapse at any point on the chain that we like; the choice is purely arbitrary. But only one step has anything like a privileged position in the chain. Von Neumann identified that step as the signal registering in the consciousness of the observer's mind.
Too many people have taken this hint that consciousness may play a special role in the collapse of the wave function and reasoned to some rather silly conclusions. One school of thought has even come to the conclusion that human consciousness literally creates the world. But such absurdities come from people who don’t fully understand what the quantum theory tells us about the nature of Reality. So let’s look at what the theory tells us.
Any event is indeterminate as seen from the past, but is wholly determinate as seen from the future. This implies that we can describe the event with a spread-out state function that collapses into a Dirac delta, which then evolves again via Schroedinger’s Equation. This, in turn, implies that what causes the collapse comes from the future, traveling backward in time.
Consider what happens to the state function. Futureward from the magic instant called Now the state function spreads out as a wave function (indeterminate), but pastward from that instant the state function has the form of a Dirac delta (determinate).
So imagine a kind of seascape hovering over a Cartesian grid displaying only the x- and y-coordinates. The gently rolling swells of the seascape provide a graphic display of the probability density of finding the particle at a given location. But those features move, like swells on the ocean, until the instant that we call Now, when the entire seascape goes flat at sea level, except at one point, where the seascape becomes a single spike of infinitesimal width and infinite height (a Dirac delta). How does that collapse happen; more specifically, how does it happen without violating the laws of Relativity and of causality?
One other factor comes into play in this model. We actually have two ætherial seascapes – one for the location of the particle and one for the particle’s linear momentum or one for when an event occurs and one for the energy involved in the event. Heisenberg’s indeterminacy principle necessitates that when the probability distribution on one seascape collapses into a Dirac delta, the other must expand to cover its area uniformly.
Now we can do something that Schrödinger could not have done: we can improve his imaginary experiment. Just as real observations improve with advances in technology, so do imaginary observations. Start by replacing the light bulb with a stopwatch. In this version of the experiment we might also remove the hammer and have the solenoid click the stopwatch directly. In this experiment the stopwatch will record when the atom decays, so when we open the box we have no ambiguity as to when the atomic decay event occurred. Alternatively, we could merely modify our original experiment by putting a photocell inside the box and using its output to move the pen on a chart recorder outside the box. A time stamp on the chart tells us when the light went out and, thus, when the atom decayed.
Again, we have a perfectly indeterminate event before it occurs and a perfectly determinate event after it occurs. A certain number of particles riding an aleatric field described by a state function exist in a state of limbo, reflecting the indeterminacy encoded in the state function. An interaction with another set of particles, in order to exist as an event, selects one of many possible states to come real. In the current example the collection of particles constitutes an atomic nucleus and the decay of that nucleus marks the event that converts an indeterminate state function into a determinate one. But when the state function can’t decide which state to manifest, how do the particles exist?
If a particle can’t exist in both states, and can’t exist in either state, then it must exist in neither state. It must exist in a state of limbo until something happens that needs its properties manifest. For example, it might participate in a collision with another particle, which needs the particle’s mass manifest. Indeed, the only fact of which we have any certainty tells us that the particles must possess properties that obey conservation laws. In subsequent essays we will have to explore how that fact obliges us to describe matter.
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