Do Quantum Wave Functions Collapse?

In the last post, I showed you how a projectile in a superposition of moving to the left or of moving to the right can only be measured to be doing one or the other. But what happens to the wave function of the system when the measurement is made? Does it… does it… COLLAPSE!?

Sounds scary. But it is only scary when it is badly explained.

Today I’ll show you what wave function collapse would mean, what it would require, and what a couple of the alternatives are. Among other things, I’ll show you that:

• The standard way of explaining wave function collapse, which argues collapse is required to avoid a logical problem, is not legitimate;
• If the Schrödinger wave equation is correct, then wave function collapse can never happen (and anything resembling “collapse” is viewed not as a physical effect but as a user’s choice);
• Therefore, if wave function collapse really does occur, then the Schrödinger equation is wrong;
• But if the Schrödinger wave equation is correct, an understanding of why quantum theory predicts only probabilities for multiple possibilities, rather than definite outcomes, is still lacking.

Today’s post uses several previous posts and their figures as a foundation, so I’ll start with a review of the most recent one, with links to others of relevance.

Quick Review

[If this review does not make sense to you, you should definitely first read the posts that are linked below, where all these points are carefully explained.]

I’ll refer extensively to my last post, in which (see Fig. 1) we have

• a projectile initially located near x=0 and able to move only along the x-axis;
• a measuring device in the form of a simple microscopic ball to the projectile’s left at x=-1
  • which may in turn be connected to other, more stable measuring devices, such as a Geiger-Müller counter,
  • which may, in their turn, be measured by human ears and recorded in human brains
• an initial wave function which has the measuring device essentially stationary and the projectile in a superposition of going to the right OR to the left

The wave function of this system of two objects exists in the two-dimensional space of possibilities, with x₁ the position of the projectile and x₂ the position of the ball. Its behavior over time is as shown in Fig. 2. It has two peaks which initially travel in opposite directions, back-to-back, in the space of possibilities. However, the two peaks behave differently. One peak evolves through a collision of the projectile with the measuring ball, causing the ball to move to the left, while the other peak does not. As a consequence, the peaks do not remain back-to-back; one continues in steady motion to the right, while the other changes direction and ends up lower down (at more negative x2).

[Again, if any of this does not make sense to you, you should definitely first read the previous post, and perhaps more of the posts linked above, where all these points are carefully explained.]

How do we understand what this is showing us? As usual, we can refer to pre-quantum physics to guide us. In pre-quantum physics we would have said there are two possibilities, shown in Figs. 3a and 3b, with equal probability of occuring. Fig. 3a shows what happens if the projectile moves leftwards;

• the left panel shows the projectile and ball in physical space;
• the right panel shows the evolution of the system in the space of possibilities.

Notice that the system’s motion changes direction when it crosses the orange diagonal line, where the projectile and ball are in the same location and collisions can occur. Meanwhile, Fig. 3b shows us that if instead the projectile moves rightward, then its motion — and that of the system, also — is steady, and the ball remains largely stationary.

OR

Compare the right panels of Figs. 3a and 3b with the wave function in Fig. 2; the motion of the two peaks of the latter combines the motion of the stars in the former.

The Classic, Wrong View of Collapse

If you’ve ever read about wave-function collapse, you may well have seen it explained as follows. This approach is badly misleading, but I need to review it because I know many readers will have seen it. Later we’ll see both what’s right and what’s wrong with it.

Here we focus on the projectile alone, and ignore (unwisely) the ball that measures it. Before the measurement takes place, the wave function for the projectile looks like this, with its two peaks, telling us the ball may be to the left of center OR it may be to the right of center.

But what does it look like after the measurement carried out by the measuring ball?

Suppose our measurement did not detect the projectile to the left of the center — the measuring ball remained stationary. Then, given the two-peaked structure of the initial wave function in Fig. 4, the projectile must now be located to the right of center, and now has essentially zero probability to be found to the left of center.

Next, suppose we decide, immediately after the first measurement, to make a second measurement of where the projectile is. Based on the result of the first measurement and the highlighted conclusion in the previous paragraph, the second measurement should also find the projectile’s position to the right of center.

But this tells us that the projectile’s wave function, after the first measurement but before the second, can no longer look like Fig. 4. If the wave function still looked like Fig. 4, it would predict that our second measurement, like our first one, would have a 50% chance of finding the projectile left of center and a 50% chance of finding it right of center. That’s in complete contradiction to the highlighted conclusion that we drew from our first measurement, which instead implies that our second measurement, if done immediately thereafter, has a 100% chance of finding the projectile to the right of center.

In short, keeping the wave function of Fig. 4 would lead to a logical inconsistency!

Instead, the wave function after the first measurement must presumably now look more like Fig. 5, where there is a nearly 100% chance of finding the projectile to the right of center. Again, this form of the wave function seems logically required during the period after the first measurement is performed but before the second measurement.

How, exactly, did simply measuring the projectile cause its wave function to suddenly change — or “collapse”, as people say — from the form in Fig. 4 to the form Fig. 5, with the left peak disappearing and all its probability transferred the right peak? And if we want to say that it didn’t collapse in this way, how is a logical contradiction avoided?

This becomes even more worrisome if we imagine that we had waited a long time before the first measurement, so that the two peaks of the wave function were a mile apart, or a million miles apart, when it was performed. How does measuring (which, in this case, involves failing to detect) the projectile at the location of the ball cause a drastic change in the ball’s wave function a million miles away?

Stop, stop, stop! This entire line of argument, and the logical inconsistency that it points to, is completely unconvincing and totally misleading. That’s because particles don’t have wave functions — systems do. And our projectile is not an isolated system which can be treated as having its own wave function; if it were, we could never have performed the first measurement in the first place!

Instead, if we want to have a hope of understanding this properly, we need to include the projectile and the device that made that first measurement in a single system, with a combined wave function. Fortunately, we’ve already done this — in fact we did it in the last post, which I reviewed in Figs. 1-3 above — so we just have to look at the answer and think about it. We’ll see that there are still conceptual issues, but no logical contradictions.

Toward The Correct View

We have already seen that once we include the measurement device in the wave function, and even start to take seriously the fact that we must amplify the initial measurement (as in a Geiger-Müller counter) to assure that it can be recorded permanently, we need to be using a wave function for a much larger space of possibilities than the one used in Figs. 4 and 5; at a minimum we need the two-dimensional space used in Figs. 1-3. I’m going to ignore the amplification step here; it’s too complicated, and we won’t really need to look at it in detail. But remember that it is there, because it will have some conceptual implications.

Specifically, Fig. 2 shows us that if Schrödinger’s equation is correct, the wave function does not collapse when we include a measurement process.

• The wave function initially has two peaks representing the projectile moving in one direction (to positive x₁) or the other (to negative x₁), all while the ball sits still at left, ready to measure, at x₂=-1.
• The peak that moves toward positive x₁ represents the fact that if the projectile moves right, it misses the ball and the ball does not respond — nor does the Geiger-Müller counter, our ears, or our brains.
• The peak that moves to negative x₁ represents the fact that if the projectile moves left, something striking happens when x₁ and x₂ are both at -1; this is roughly where the collision of the projectile and ball occurs.
  • The collision causes the second peak to move toward more negative x₂, representing the motion of the ball to the left in physical space.
  • Then, although not shown, the ball’s motion gets drastically amplified as sketched here — perhaps in a Geiger-Müller counter, during which a billion or so electrons are liberated from their atoms and create a macroscopic electric current
• The response (or lack of response) of the Geiger-Müller counter can leave a permanent record in the world, as a mark upon a piece of paper, or in an electronic recording device, and/or in a person’s brain.

To understand this, we need both to understand Fig. 2 and to think beyond it. During the amplification process following the collision of the projectile with the ball, the second peak of the wave function isn’t merely moving across the measly two-dimension space of possibilities depicted in Fig. 2. Instead it is moving across a truly gigantic space of possibilities with billions (at least!!) of dimensions! This high-dimensional space has many, many axes, representing the positions of all the liberated electrons in the counter and all of the objects used to record the information.

But still, within that gigantic space, the peak remains a peak — because even though there’s only a 50% chance that the projectile went to the left, there’s a near-100% chance that if it goes left, then the ball will respond AND amplification will occur AND the Geiger-Müller counter will click AND my eardrums will respond AND my brain will record the event in memory.

[This way of saying things is too glib. Because of the complexity of the interactions involved, the original peak, carrying 50% probability, actually breaks up into a set of many mini-peaks, thanks to the liberated electrons taking many different paths in physical space. But all of these mini-peaks will cause the counter to click, and their total probability taken together is 50% — and so the measurement outcome, at a macroscopic level, is just as I originally stated it.]

The other peak tells us that there’s a 50% chance that the projectile went to the right. If so, then neither the ball nor the Geiger-Müller counter will respond, and my brain will record the counter’s silence.

In this simple situation with a perfect measuring ball, we start with two peaks in the wave function, and after the initial measurement there are still two peaks, as you can see in Fig. 2. The wave function has certainly not collapsed; although the measurement can redirect peaks (or break them into multiple peaks), the Schrödinger equation does not eliminate peaks in the wave function and and does not move the probability they carry to other, distant peaks.

However, this doesn’t necessarily address the following issue. It’s all fine and good to have an evolving wave function which gives me probabilities for the result of the first measurement, and then goes on to tell me the probabilities for the second measurement once the result of the first measurement is determined. But why can’t quantum physics just get serious for once, and actually predict the result of the first measurement? Why can it only give us probabilities? And where do those probabilities really come from?

If we want to have answers to these questions, we might need wave function collapse, or something like it.

In the following three sections, I’m going to discuss three of the various different approaches to dealing with this problematic issue, and interpreting what the equations might be telling us. I’m not going to advocate for any one of them; they’re all consistent with current data, at least in principle, and the first and last are consistent with the Schrödinger equation. One way that I think is useful to keep them straight is to look at how they deal with

• potentialities — those paths through possibility space for which the wave function is not vanishingly tiny and therefore have have a reasonable probability of actually occuring — and
• actualities — the one or more potentialities that actually do occur in the real world.

Interpretation	Standard Probabilistic	Wave Function Collapses	Many Worlds
Potentialities	Many	One	Many
Actualities	One	One	Many
Conceptual Issues (incomplete list)	Maybe an incomplete description of nature? Is there an underlying non-probabilistic reality that explains where the probabilties come from?	When and how does wave function collapse occur, and how do the expected probabilities emerge? What equations should replace the Schrödinger equation?	What does having many “universes” mean and can it be verified? When do two “universes” actually separate? Where do the expected probabilities of predictions come from?

Consistent but Incomplete?

Let’s start by considering a minimalist view. You may well find it unsatisfying. [But to quote a certain teenager I know, “maybe that’s a you problem.” 😉 ]

We’ll just accept that the wave function is a device for computing probabilities. That’s all it does and all it is supposed to do. Results of individual measurements on a physical system are random, but repeated attempts to perform the same experiment will give results that are distributed according to the probabilities assigned by wave function calculations. And we don’t worry about the details in between one measurement and the next; we just know what the system is doing when we make measurements, and in between we don’t and presumably can’t know.

Importantly, in contrast to the earlier argument for wave function collapse (see Figs. 4 and 5), no logical inconsistency arises here. The wave function’s two peaks in Fig. 2 imply that after the first measurement but before the second, the following logical statements are true:

• if the ball did not respond, then the projectile is on the right; and
• if the ball did respond, then the projectile is on the left.

The first of these if…then… statements is recorded in the peak on the right (see Fig. 6), while the second correponds to the peak at bottom left. As we see in the wave function’s shape, these logical statements about the initial measurement are correct, no matter what its outcome actually was.

And this means that if we quickly do a second measurement, then

• we will only find the projectile on the right (x₁ > 0) if the first measurement observed the ball where it originally was (x₂ very near -1).
• we will only find the projectile on the left of center (x₁ < 0) if the first measurement observed the ball recoiling from a collision (x₂ < -1).

Thus we will never find a logical contradiction.

We would indeed have a contradiction if the ball remained where it was (at x₂ near -1) AND the projectile is found on the left (x₁ < 0). But that situation, whose location in the space of possibilities is circled in red in Fig. 7, lies well outside either peak — and thus it has probability that is exceedingly small, effectively zero.

Compare that with the argument around Figs. 4 and 5, which claimed there was a logical inconsistency. That inconsistency comes from using a wave function which did not include the measuring device in the wave function. Without the measuring ball, the space of possibilities had only one dimension (x₁), not two dimensions (x₁ , x₂); but it’s the second dimension that, by keeping track of the ball’s position x₂, cleanly separates the circled area in FIG from the peak that lies below it. If you can’t distinguish these two regions on the plane because you’ve compressed the plane to a line, as in Figs. 4 and 5, then you will naturally become confused. The confusion is only avoided if one includes the measuring device in the wave function.

What Is the Apparent Collapse of the Wave Function?

Staying in this two-dimensional plane gives us a way to think about how we might fix the invalid wave function collapse argument of Figs. 4 and 5.

Suppose that we assume, or that we know, that the measuring ball did not respond to the projectile, and thus remains at x₂ near -1. In this case it is perfectly fair for us to ignore everything in the space of possibilities that has x₂ much larger or much smaller than -1, because those possibilities are hypotheticals that could have happened but are known to us (or assumed by us) not to have happened. Ignoring possibilities that are known (or assumed) not to have occurred is perfectly okay for the purpose of calculating the probabilities of future events that we might observe. [In basic logic, the question is: given known (or assumed) facts F, what is the probability of a specific possibility P?]

This logic then restricts our attention to the region between the two blue lines in Fig. 8. Only one of the two peaks in the wave function lies in this region. So if we write only the part of the wave function which remains relevant to us, given our knowledge or assumption about the measuring ball, we can ignore the other peak. The resulting wave function needed to calculate the system’s future probabilities will look similar to the one in Fig. 5, not the one in Fig. 4, although in principle it remains two-dimensional and contains the additional information that the measuring ball did not move from x₂=-1.

It might seem as though the wave function really has collapsed from Fig. 4 initially to Fig. 5, since there is now only a peak localized at positive x₁, with no remnant at negative x₁. But the wave function did not in fact collapse as a matter of physics. We simply chose, for our own justifiable reasons, to ignore most of the wave function in Fig. 8.

In this view, wave function “collapse” is logic, not physical evolution. It isn’t a natural process that happens at a particular moment in time, or that literally transfers information from the far left to the far right of the x axis. We ourselves “collapsed” — or rather, “restricted” — the wave function by declaring a lack of interest in some of the possibilities that the original wave function describes, thereby cutting away most of the space of possibilities. We ignore the peak at the lower left of Fig. 8 because it applies to a set of possibilities that we have already rejected.

It’s perfectly legitimate for us to make choices of this kind. Indeed, if you look back over my recent dozen posts, I have been making such choices over and over again, picking which wave function to study for one purpose or another. We shouldn’t get upset that the Schrödinger equation didn’t predict our choices; after all, it didn’t predict what experiments we were going to do, either.

And so, from this perspective, there’s a logically consistent Schrödinger equation that tells us how wave functions evolve. We ourselves can always choose to declare assumptions or learn facts, and use those to restrict the space of possibilities to a smaller one, in which the wave function may have a more restricted shape. When this happens, the wave function naively might seem to have collapsed, but in fact it is we who have changed our questions.

But even if you accept this viewpoint, it’s not the end of the story. There’s are two more puzzles well worth thinking about.

If Many Things Can Happen, Why Do We Observe Only One?

Actually the first one really isn’t a puzzle; it only sounds like one at first.

Let’s say we’re fine with the wave function telling us the probabilities for one thing or another to happen, and agree that it does so without logical contradiction. Nevertheless, you and I only experience one thing happening. Doesn’t the Schrödinger equation fail to explain that? It gives two peaks in Fig. 2, and seems to have no explanation for why we only experience one or the other.

But in fact, it does explain it. Superposition is an OR, not an AND. The fact that you and I only experience one thing happening is exactly what the Schrödinger equation and the wave function say: that when you measure something that has multiple possibilities under a wave function — that is, a superposition of different possibilities — only one result will be observed. We can already see this with the measuring ball, which only “experiences” being struck or not being struck; it doesn’t experience both, even though the wave function’s two peaks describe the possibility of both. If you include the “macroball” of my 2nd post on measurement in the system, and/or maybe an entire Geiger-Müller counter and even our ears and brains, this would remain true:

• either the system is in one peak which gives a high probability for the projectile to go left and for us to hear the counter click,
• or the system is in the other peak which gives a high probability for the projectile to go right and for us not to hear the counter click.

So in fact, rather than this issue being a bug, it’s a feature; the Schrödinger equation and the simple probabilistic interpretation do indeed tell us that out of all the possible realities that the wave function tells us are probable, we will only experience one.

Why Can’t Quantum Physics Do Better Than Probabilities?

A much more serious concern, dating back to 1911 and Einstein, who (yet again!) was the first to notice the potential problem, is that nothing in the equations can tell us whether the projectile goes left or right, and whether the ball consequently does or does not respond. The direction of the projectile really is purely random, according to the Schrödinger equation.

That, in turn, would seem to suggest that quantum physics is a theory that is missing information. This is typical in situations where we can only predict probabilities.

If I flip a coin, I usually view it as having a 50-50 chance of landing tails or heads. But that’s not really true. The actual process of tossing the coin in a particular way and in a particular location, and the details of the floor in which it lands, could be measured carefully, and a sufficiently motivated physicist who measured all the forces exerted on the coin by the hand that tossed it and by the floor on which it landed, as well as that of any air resistance, could figure out which tosses of the coin would land on heads and which on tails. The flip of the coin seems random, but only because we’re ignorant and/or careless; it’s not really random at all.

Probability is useful, normally, when there is a completely knowable chain of events, but we happen not to know everything about it. In quantum physics we seem to have a logically consistent theory of probabilities. Is there a completely knowable chain of events — a deterministic process — that lies behind its probabilistic exterior?

If so, then the current quantum theory is incomplete. Schrödinger’s equation might be exactly (or largely) correct, but perhaps there is something missing from it (“hidden variables”, in the jargon) that needs to be added — missing information that lies behind the probabilities and that, if known, would make precise predictions possible, rather than probabilistic ones. This idea, that quantum physics is correct but incomplete, would have gotten Einstein’s vote, I believe.

Collapse: Schrödinger is Wrong?

However, everything I have written in the last couple of sections assumes that the Schrödinger equation is correct. Maybe it’s not.

I’ve shown you that wave function collapse is not required for a consistent theory of quantum physics; there’s no logical problem that makes it necessary. Nevertheless, wave function collapse might still occur, and might conceivably explain the probabilistic nature of the current quantum theory.

Maybe the evolution shown in Fig. 2 is not correct. Even if the wave function at the start of Fig. 2 remains the same, with its two peaks indicating two possibilities with 50% probability, perhaps the true equation of quantum physics causes the wave function to lose one the two peaks shown in Fig. 9. With only one peak, a single possibility with 100% probability, the wave function would then make a far less ambiguous prediction. In other words, maybe the correct equations do cause the wave function to collapse naturally, sometimes to the peak on the right of Fig. 9, and sometimes to the other peak at bottom left.

Perhaps hidden variables of some type help the equation choose which peak to select in each run of the experiment, so that the results come out 50-50 overall but are predictable each time. Or maybe the outcome is still random, but there’s still only one peak in the final wave function, not two, which then affects future probabilities.

Such suggestions are not logically inconsistent, nor proven to be inconsistent with data. They require that the Schrödinger equation be only an approximation to the truth; but lots of famous equations have turned out to be imperfect when examined with sufficient care. Newton’s second and third laws of motion and his law of gravity are all in that category.

Unfortunately, the notion of collapse raises many difficult questions:

• If this collapse happens, when does it happen?
  • when the measurement begins?
  • when the measurement ends?
  • halfway through the measurement?
• What if the measurement isn’t completed immediately? Suppose I’m on vacation when my Geiger-Müller counter records a passing electron, and I don’t find out about it for weeks? Does the collapse happen before I get home, or only when I look at the readout?
• What equations govern the collapse?
  • Does the collapse happen suddenly, or does it take a certain amount of time?
  • If the latter, how long?
  • How far must a measurement proceed before collapse begins?
  • Can we measure the partial collapse when it is halfway done?
  • Does collapse sometimes fail to complete, and can we see signs of this?
  • If the parts of the wave function are millions of miles apart, does the collapse happen faster than the cosmic speed limit (a.k.a. the speed of light) would allow, or not?
  • Since different observers who move relative to one another have different notions of time, and can even disagree as to which of two events happens first, can they disagree on the causes and effects of collapse?
• Does nature collapse the wave function only when an actual measurement is being performed? (and what is an “actual measurement”?)
• If an ordinary non-measurement interaction occurs, such as a collision among two atoms in a gas, how does nature know not to collapse the wave function in this case?
• Does the collapse still happen when a measurement fails partway, perhaps due to a breakdown in the device, such as an interruption of the amplification?
• What if we ourselves decide, halfway through a measurement, to abandon it? For instance, what if we refuse to read out the Geiger-Müller counter, so that its result is ultimately not recorded?

Some of these questions overlap. But this is not the complete list, either.

None of these questions can be answered without a detailed, consistent theory of how wave functions might collapse — i.e. a set of precise equations and concepts which make predictions for experiments. There have been attempts, but I have not seen one that I find particularly appealing. Also, so far there have been no experimental indications that there is anything incorrect about the Schrödinger equation (other than the very fact that it predicts only probabilities and not specific outcomes.)

Oh, and by the way, if we interpret wave function “collapse” as I described above in the context of Fig. 8 — as a restriction that we ourselves place based on our own assumptions or knowledge — then none of the questions listed above even need to be asked, much less answered. There are no new equations; we restrict whenever we want; doing so doesn’t affect the measured objects; there’s no worry about something happening faster than the cosmic speed limit; etc.

Because of this — and because there’s no great theory of wave function collapse that has attractive math, is consistent with Einstein’s special relativity and with quantum field theory, and answers the majority of the long list of questions that I’ve listed above — I’m personally pretty skeptical that the idea of a physical collapse of a wave function can provide a workable alternative to existing quantum theory. [But that might just be a “me” problem.]

Everything, Everywhere, All At “Once”?

Then there’s the many-worlds interpretation, which originates with Hugh Everett. In this interpretation of Fig. 2, the Schrödinger equation is correct, and both peaks that appear in the wave function represent things that do happen: the projectile goes to the right, leaving the measuring ball alone AND it goes to the left, hitting the measuring ball. It does this in two different “universes” — two strands of history that split at the time of the interaction between the projectile and the ball, or perhaps even earlier, when the ball is set in a superposition and starts moving in one direction or the other. Each of these strands carries off one of the possibilities and runs that possibility’s future.

It’s almost as though nature is a chess computer that takes the current board position and runs all the possible moves and all the possible responses and all the counter-responses — runs, in short, all possible chess matches — and views them all as equally real, rather than viewing them as a set of possible matches of which only one will actually be played. It takes all the reasonable potentialities and claims they are truly actualities, each one in its own version of reality.

In this view, superposition now becomes both AND and OR. In any one universe, it’s an OR; in our universe, our strand of history, the projectile goes left OR it goes right. But viewed across all the “simultaneous” or “parallel” universes, it’s also an AND; if in our universe it goes right, there’s another comparable universe in which it goes left. (Admittedly these universes are neither simultaneous nor parallel nor universes, so we could use better terminology here.)

In any one such world, a measurement device, including a human brain, will still never experience two contradictory measurement outcomes. The set of all strands of history will include all the possible outcomes, but no person’s brain will ever remember anything but a logically consistent past, the story of one particular strand. That’s because we ourselves live the OR in one strand, even though the wider universe/quantum-multiverse/meta-universe — the combination of all those strands — lives out the AND.

Is this picture logically consistent? Possibly, though actually counting strands is a nasty business; what does it mean for a strand to split, when does it split, etc? And how many strands are actually found in Fig. 10? To say there are only two, one for each peak, is presumably far too simple; after all, the peaks in a mountain range cannot be counted unambiguously. (Do we count every protruding rock? every ant hill?) What about the regions between the peaks, where the wave function is very small, but not zero? What was the situation at the start of the universe — was there just one strand to begin with, or were there always many?

Some of the conceptual questions that are naturally raised seem to me rather similar to those that arise for wave function collapse, though they don’t necessarily require the invention of new equations. [That said, one might wonder if grounding the idea might require a quantum theory with gravity, for which we don’t yet have a complete set of equations and concepts.]

But beyond these issues, a serious limitation on many-worlds is that it risks being metaphysics rather than physics. First of all, it says all physically possible worlds exist, although many are highly improbable — that potentiality almost guarantees actuality. Second, it’s impossible to verify the idea experimentally, because by the very construction of this interpretation, we can’t observe any strands of the universe except the one we’re living within. So I’m not sure whether this interpretation really helps us out compared to the one we started with; it might be a matter of personal preference rather than of any physical difference.

——-

In this brief tour of just a few of the issues with superposition in quantum physics, I hope I have helped clarify what some of the questions are, even though I have posed far more questions than I have answered. I look forward to your further questions, proposed answers, and criticisms in the comments.