# Configurations and Amplitude

26 10 April 2008 07:41AM

Previously in series:  Quantum Explanations

So the universe isn't made of little billiard balls, and it isn't made of crests and troughs in a pool of aether...  Then what is the stuff that stuff is made of?

(Diagrams stolen from qubit.org and edited for my purposes.)

In Figure 1, we see, at A, a half-silvered mirror, and two photon detectors, 1 and 2.

Early scientists, when they ran experiments like this, became confused about what the results meant.  They would send a photon toward the half-silvered mirror, and half the time they would see the detector at 1 click, and the other half the time they would see the detector at 2 click.

The early scientists—you're going to laugh at this—thought that the silver mirror deflected the photon half the time, and let it through half the time.

Ha, ha!  As if the half-silvered mirror did different things on different occasions!  I want you to let go of this idea, because if you cling to what early scientists thought, you will become extremely confused.  The half-silvered mirror obeys the same rule every time.

If you were going to write a computer program that was this experiment—not a computer program that predicted the result of the experiment, but a computer program that resembled the underlying reality—it might look sort of like this:

At the start of the program (the start of the experiment, the start of time) there's a certain mathematical entity, called a configuration.  You can think of this configuration as corresponding to "There is one photon heading from the photon source toward the half-silvered mirror", or just "A photon heading toward A."

A configuration can store a single complex value—"complex" as in the complex numbers (a + bi).  At the start of the program, there's already a complex number stored in the configuration "A photon heading toward A."  The exact value doesn't matter so long as it's not zero.  We'll let the configuration "A photon heading toward A" have a value of (-1 + 0i).

All this is a fact within the territory, not a description of anyone's knowledge.  A configuration isn't a proposition or a possibility or a way the world can be.  A configuration is a variable in the program—you can think of it as a kind of memory location whose index is "A photon heading toward A"—and it's out there in the territory.

As the complex numbers that get assigned to configurations are not positive real numbers between 0 and 1, there is no danger of confusing them with probabilities.  "A photon heading toward A" has complex value -1, which is hard to see as a degree of belief.  The complex numbers are values within the program, again out there in the territory.  We'll call the complex numbers amplitudes.

There are two other configurations, which we'll call "A photon going from A to Detector 1" and "A photon going from A to Detector 2."  These configurations don't have a complex value yet; it gets assigned as the program runs.

We are going to calculate the amplitudes of "A photon going from A toward 1" and "A photon going from A toward 2" using the value of "A photon going toward A", and the rule that describes the half-silvered mirror at A.

Roughly speaking, the half-silvered mirror rule is "Multiply by 1 when the photon goes straight, and multiply by i when the photon turns at a right angle."  This is the universal rule that relates the amplitude of the configuration of "a photon going in", to the amplitude that goes to the configurations of "a photon coming out straight" or "a photon being deflected".

So we pipe the amplitude of the configuration "A photon going toward A", which is (-1 + 0i), into the half-silvered mirror at A, and this transmits an amplitude of (-1 + 0i)*i = (0 + -i) to "A photon going from A toward 1", and also transmits an amplitude of (-1 + 0i)*1 = (-1 + 0i) to "A photon going from A toward 2".

In the Figure 1 experiment, these are all the configurations and all the transmitted amplitude we need to worry about, so we're done.  Or, if you want to think of "Detector 1 gets a photon" and "Detector 2 gets a photon" as separate configurations, they'd just inherit their values from "A->1" and "A->2" respectively.  (Actually, the values inherited should be multiplied by another complex factor, corresponding from the distance from A to the detector; but we will ignore that for today, and suppose that all distances traveled in our experiments happen to correspond to a complex factor of 1.)

So the final program state is:

Configuration "A photon going toward A": (-1 + 0i)
Configuration "A photon going from A toward 1": (0 + -i)
Configuration "A photon going from A toward 2": (-1 + 0i)
and optionally
Configuration "Detector 1 gets a photon": (0 + -i)
Configuration "Detector 2 gets a photon": (-1 + 0i)

This same result occurs—the same amplitudes stored in the same configurations—every time you run the program (every time you do the experiment).

Now, for complicated reasons that we aren't going to go into today—considerations that belong on a higher level of organization than fundamental quantum mechanics, the same way that atoms are more complicated than quarks—there's no simple measuring instrument that can directly tell us the exact amplitudes of each configuration.  We can't directly see the program state.

So how do physicists know what the amplitudes are?

We do have a magical measuring tool that can tell us the squared modulus of a configuration's amplitude.  If the original complex amplitude is (a + bi), we can get the positive real number (a2 + b2).  Think of the Pythagorean theorem: if you imagine the complex number as a little arrow stretching out from the origin on a two-dimensional plane, then the magic tool tells us the squared length of the little arrow, but it doesn't tell us the direction the arrow is pointing.

To be more precise, the magic tool actually just tells us the ratios of the squared lengths of the amplitudes in some configurations.  We don't know how long the arrows are in an absolute sense, just how long they are relative to each other.  But this turns out to be enough information to let us reconstruct the laws of physics—the rules of the program.  And so I can talk about amplitudes, not just ratios of squared moduli.

When we wave the magic tool over "Detector 1 gets a photon" and "Detector 2 gets a photon", we discover that these configurations have the same squared modulus—the lengths of the arrows are the same.  Thus speaks the magic tool.  By doing more complicated experiments (to be seen shortly), we can tell that the original complex numbers had a ratio of i to 1.

And what is this magical measuring tool?

Well, from the perspective of everyday life—way, way, way above the quantum level and a lot more complicated—the magical measuring tool is that we send some photons toward the half-silvered mirror, one at a time, and count up how many photons arrive at Detector 1 versus Detector 2 over a few thousand trials.  The ratio of these values is the ratio of the squared moduli of the amplitudes.  But the reason for this is not something we are going to consider yet.  Walk before you run.  It is not possible to understand what happens all the way up at the level of everyday life, before you understand what goes on in much simpler cases.

For today's purposes, we have a magical squared-modulus-ratio reader.  And the magic tool tells us that the little two-dimensional arrow for the configuration "Detector 1 gets a photon" has the same squared length as for "Detector 2 gets a photon".  That's all.

You may wonder, "Given that the magic tool works this way, what motivates us to use quantum theory, instead of thinking that the half-silvered mirror reflects the photon around half the time?"

Well, that's just begging to be confused—putting yourself into a historically realistic frame of mind like that and using everyday intuitions.  Did I say anything about a little billiard ball going one way or the other and possibly bouncing off a mirror?  That's not how reality works.  Reality is about complex amplitudes flowing between configurations, and the laws of the flow are stable.

But if you insist on seeing a more complicated situation that billiard-ball ways of thinking can't handle, here's a more complicated experiment:

In Figure 2, B and C are full mirrors, and A and D are half-mirrors.  The line from D to E is dashed for reasons that will become apparent, but amplitude is flowing from D to E under exactly the same laws.

Now let's apply the rules we learned before:

At the beginning of time "A photon heading toward A" has amplitude (-1 + 0i).

We proceed to compute the amplitude for the configurations "A photon going from A to B" and "A photon going from A to C".

"A photon going from A to B" = i * "A photon heading toward A" = (0 + -i)

Similarly,

"A photon going from A to C" = 1 * "A photon heading toward A" = (-1 + 0i)

The full mirrors behave (as one would expect) like half of a half-silvered mirror—a full mirror just bends things by right angles and multiplies them by i.  (To state this slightly more precisely:  For a full mirror, the amplitude that flows, from the configuration of a photon heading in, to the configuration of a photon heading out at a right angle, is multiplied by a factor of i.)

So:

"A photon going from B to D" = i * "A photon going from A to B" = (1 + 0i)
"A photon going from C to D" = i * "A photon going from A to C" = (0 + -i)

"B to D" and "C to D" are two different configurations—we don't simply write "A photon at D"—because the photons are arriving at two different angles in these two different configurations.  And what D does to a photon, depends on the angle at which the photon arrives.

Again, the rule (speaking loosely) is that when a half-silvered mirror bends light at a right angle, the amplitude that flows from the photon-going-in configuration to the photon-going-out configuration, is the amplitude of the photon-going-in configuration multiplied by i.  And when two configurations are related by a half-silvered mirror letting light straight through, the amplitude that flows from the photon-going-in configuration is multiplied by 1.

So:

• From the configuration "A photon going from B to D", with original amplitude (1 + 0i)
• Amplitude of (1 + 0i) * i = (0 + i) flows to "A photon going from D to E"
• Amplitude of (1 + 0i) * 1 = (1 + 0i) flows to "A photon going from D to F".
• From the configuration "A photon going from C to D", with original amplitude (0 + -i)
• Amplitude of (0 + -i) * i = (1 + 0i) flows to "A photon going from D to F"
• Amplitude of (0 + -i) * 1 = (0 + -i) flows to "A photon going from D to E".

Therefore:

• The total amplitude flowing to configuration "A photon going from D to E" is (0 + i) + (0 + -i) = (0 + 0i) = 0.
• The total amplitude flowing to configuration "A photon going from D to F" is (1 + 0i) + (1 + 0i) = (2 + 0i).

(You may want to try working this out yourself on pen and paper if you lost track at any point.)

But the upshot, from that super-high-level "experimental" perspective that we think of as normal life, is that we see no photons detected at E.  Every photon seems to end up at F.  The ratio of squared moduli between "D to E" and "D to F" is 0 to 4.  That's why the line from D to E is dashed, in this figure.

This is not something it is possible to explain by thinking of half-silvered mirrors deflecting little incoming billiard balls half the time.  You've got to think in terms of amplitude flows.

If half-silvered mirrors deflected a little billiard ball half the time, in this setup, the little ball would end up at Detector 1 around half the time and Detector 2 around half the time.  Which it doesn't.  So don't think that.

You may say, "But wait a minute!  I can think of another hypothesis that accounts for this result.  What if, when a half-silvered mirror reflects a photon, it does something to the photon that ensures it doesn't get reflected next time?  And when it lets a photon go through straight, it does something to the photon so it gets reflected next time."

Now really, there's no need to go making the rules so complicated.  Occam's Razor, remember.  Just stick with simple, normal amplitude flows between configurations.

But if you want another experiment that disproves your new alternative hypothesis, it's this one:

Here, we've left the whole experimental setup the same, and just put a little blocking object between B and D.  This ensures that the amplitude of "A photon going from B to D" is 0.

Once you eliminate the amplitude contributions from that configuration, you end up with totals of (1 + 0i) in "A photon going from D to F", and (0 + -i) in "A photon going from D to E".

The squared moduli of (1 + 0i) and (0 + -i) are both 1, so the magic measuring tool should tell us that the ratio of squared moduli is 1.  Way back up at the level where physicists exist, we should find that Detector 1 goes off half the time, and Detector 2 half the time.

The same thing happens if we put the block between C and D.  The amplitudes are different, but the ratio of the squared moduli is still 1, so Detector 1 goes off half the time and Detector 2 goes off half the time.

This cannot possibly happen with a little billiard ball that either does or doesn't get reflected by the half-silvered mirrors.

Because complex numbers can have opposite directions, like 1 and -1, or i and -i, amplitude flows can cancel each other out.  Amplitude flowing from configuration X into configuration Y can be canceled out by an equal and opposite amplitude flowing from configuration Z into configuration Y.  In fact, that's exactly what happens in this experiment.

In probability theory, when something can either happen one way or another, X or ~X, then P(Z) = P(Z|X)P(X) + P(Z|~X)P(~X).  And all probabilities are positive.  So if you establish that the probability of Z happening given X is 1/2, and the probability of X happening is 1/3, then the total probability of Z happening is at least 1/6 no matter what goes on in the case of ~X.  There's no such thing as negative probability, less-than-impossible credence, or (0 + i) credibility, so degrees of belief can't cancel each other out like amplitudes do.

Not to mention that probability is in the mind to begin with; and we are talking about the territory, the program-that-is-reality, not talking about human cognition or states of partial knowledge.

By the same token, configurations are not propositions, not statements, not ways the world could be.  Configurations are not semantic constructs.  Adjectives like probable and possible do not apply to them; they are not beliefs or sentences or possible worlds.  They are not true or false but simply real.

In the experiment of Figure 2, at right, do not be tempted to think anything like:  "The photon goes to either B or C, but it could have gone the other way, and this possibility interferes with its ability to go to E..."

It makes no sense to think of something that "could have happened but didn't" exerting an effect on the world.  We can imagine things that could have happened but didn't—like thinking, "Gosh, that car almost hit me"—and our imagination can have an effect on our future behavior.  But the event of imagination is a real event, that actually happens, and that is what has the effect.  It's your imagination of the unreal event—your very real imagination, implemented within a quite physical brain—that affects your behavior.

To think that the actual event of a car hitting you—this event which could have happened to you, but in fact didn't—is directly exerting a causal effect on your behavior, is mixing up the map with the territory.

What affects the world is real.  (If things can affect the world without being "real", it's hard to see what the word "real" means.)  Configurations and amplitude flows are causes, and they have visible effects; they are real.  Configurations are not possible worlds and amplitudes are not degrees of belief, any more than your chair is a possible world or the sky is a degree of belief.

So what is a configuration, then?

Well, you'll be getting a clearer idea of that in future posts.

But to give you a quick idea of how the real picture differs from the simplified version we saw today...

Our experimental setup only dealt with one moving particle, a single photon.  Real configurations are about multiple particles.  Tomorrow's post will deal with the case of more than one particle, and that should give you a much clearer idea of what a configuration is.

Each configuration we talked about, should have described a joint position of all the particles in the mirrors and detectors, not just the position of one photon bopping around.

In fact, the really real configurations are over joint positions of all the particles in the universe, including the particles making up the experimenters.  You can see why I'm saving the notion of experimental results for later posts.

In the real world, amplitude is a continuous distribution over a continuous space of configurations.  Today's "configurations" were blocky and digital, and so were our "amplitude flows".  It was as if we were talking about a photon teleporting from one place to another.

We'll get atoms and molecules and humans and all that stuff, out of a differentiable amplitude distribution in a continuous configuration space, later.

If none of that made sense, don't worry.  It will be cleared up in future posts.  Just wanted to give you some idea of where this was heading.

Part of The Quantum Physics Sequence

Next post: "Joint Configurations"

Previous post: "Quantum Explanations"

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Comment author: 10 April 2008 08:07:09AM 3 points [-]

Eliezer, in case you plan to discuss Bell's-inequality-type experiments in future posts, I suggest that you use the GHZ state (not the EPR pair) to show how local realism is ruled out in QM. The GHZ state is a much cleaner result, and is not obscurred by the statistics inherent in Bell's inequality.

Comment author: 10 April 2008 09:08:35AM 3 points [-]

I think some of my readers may be overestimating the degree to which I intend to explain quantum mechanics, here. I'm not doing a textbook. I'm trying to get (reasonably smart nonphysicist) readers to the point where they're no longer confused, and the remaining difficulties are mere matters of math.

Still a useful suggestion though, thanks.

Comment author: 10 April 2008 09:44:56AM 0 points [-]

A configuration can store a single complex value - "complex" as in the complex numbers (a + bi).

Any complex number? I.e. you're invoking an uncountable infinity for explaining the lowest known layer of physics? How does that fit in with being an infinite-set atheist - assuming you still hold that position?
I'm speaking as a nonphysicist reader, so I may well be missing something awfully obvious here. Any clarification would be appreciated.

Comment author: 10 April 2008 10:36:58AM -1 points [-]

Eliezer (and Robin) this series is very interesting and all, but.... aren't you writing this on the wrong blog?

I used to like this blog better when it was all about overcoming bias

Comment author: 10 April 2008 11:05:19AM 4 points [-]

For a rather silly reason, I wrote something about:

... explaining the lowest known layer of physics ...

Please ignore the "lowest known layer" part. I accidentally committed a mind projection fallacy while writing that comment.

Comment author: 10 April 2008 12:11:00PM 2 points [-]

botogol:

Eliezer (and Robin) this series is very interesting and all, but.... aren't you writing this on the wrong blog?

I have the impression Eliezer writes blog entries in much the same way I read Wikipedia: Slowly working from A to B in a grandiose excess of detours... =)

Comment author: 10 April 2008 02:49:02PM 1 point [-]

Any complex number? I.e. you're invoking an uncountable infinity for explaining the lowest known layer of physics? How does that fit in with being an infinite-set atheist - assuming you still hold that position?

In case you didn't notice, he's talking about a complex number, not all the complex numbers.

Comment author: 10 April 2008 03:45:17PM 2 points [-]

So... I'm confused. You say:

"...the half-silvered mirror rule is "Multiply by 1 when the photon goes straight, and multiply by i when the photon turns at a right angle."

We appear to have defined everything needful, except the word "when".

Accepting that we are just performing 'operations' on 'configurations', what decides which operation will be performed? Is it the configuration of the incoming photon? Is it some magical (i.e.quantum) property of a half of a silver?

Comment author: 13 September 2010 08:21:21AM 6 points [-]

It was intended to be clear that all operations are performed and propagated throughout the entire system, I think.

Comment author: 28 February 2013 03:33:23AM 2 points [-]

It's not clear to everyone; I didn't get it until reading your comment.

Comment author: 27 November 2016 04:19:24PM *  0 points [-]

I was confused about it too, but understood that what is meant is, that the 'half-silvered mirror rule' is a rule that does two things at once, namely x: (x*1, x*i), so it's a multi-valued operation.

Comment author: 10 April 2008 05:52:06PM 8 points [-]

Eliezer, I realise there's still a way to go, but I just wanted to let you know that this is already much more useful than any conversion I've had about QM with anyone in the past. Thank you.

Eadwacer, I might be wrong, but I'd assumed both operations are always performed.

Comment author: 10 April 2008 05:58:49PM 0 points [-]

as far as uncountable complex states... well, the actual complex values don't matter so much as the relative phases. Maybe best to think about it almost as a geometrical principle, of sorts.

Here I'm just speculating, but maybe relative phase (that is, angle when representing the complex value in polar notation) can only be shifted by rational amounts? That is, the relative phase between thingie 1 and thingie 2 is x*2Pi, where x must be rational?

I'm not saying this _is_ the way it is, but I can certainly see, based on my, as of yet, limited knowledge, that it could be that way.

Comment author: 10 April 2008 06:18:47PM 0 points [-]

I guess, Eliezer, that I would be concerned about convincing everyone that the universe runs along like a computer, computing amplitudes locally (which seems to be the gist of your discussion). To do so would certainly make people *feel* like QM isn't confusing; it would just be wave mechanics. But this would give people a false confidence, I think, and is not how the universe appears to operate.

But this is the first post, so I'll try to confine my criticism until you've wrapped up your discussion.

Comment author: 10 April 2008 06:33:50PM 3 points [-]

Psy-Kosh, when QM is formulated rigorously (something that is rarely done, and only by mathematical physicists) the amplitudes must be able to take on any number in the complex plane, not just the rationals.

Sebastian Hagen, I believe Eliezer is explaining to us the best model physicists have for the way the world works on the (sorta) lowest level we understand, not his personal beliefs on the nature of reality. This model must include the irrationals, to be self-consistent. This does not prevent the universe from being discretized (no uncountable sets) on a more fundamental level from QM.

Comment author: 10 April 2008 06:36:43PM 1 point [-]

aren't you writing this on the wrong blog?

As far as I know Robin doesn't actually have a separate economics blog and he seems to drop any economics topic that interests him into this one, so neither Eliezer nor Robin always stick closely to the "bias" theme. Does it really matter?

Comment author: 10 April 2008 07:04:24PM 0 points [-]

Jess: You mean that physics, as we understand it, absolutely requires that there will exist complex phase differences such that when divided by 2*Pi, the result will be irrational?

Oh well then, bang goes that idea. :)

Comment author: 10 April 2008 10:58:53PM 3 points [-]

I was a little disturbed when you offered up the experiment that allowed us to reject the hypothesis about a half-mirror changing each time it reflects a photon or lets one through. How do we know there aren't other experiments that could discredit the amplitude hypothesis? I'm sure there's a good answer, but don't expect me to take too much on faith.

I also thought it was odd that you called configurations real, when they just seem to be a mathematical construct that describes the behavior of photons bouncing off of mirrors. Couldn't some other construct just as easily explain what's going on (in an equivalent fashion)? It's sort of like saying that y''+5y'+ y = 0 is as real as a spring bouncing up and down, when it's actually only a model for describing what the spring is doing. Perhaps I misunderstood what you meant by "real."

Of course these are only questions, not criticisms. This is the best explanation I've ever heard--please keep it up!

Comment author: 10 April 2008 11:21:46PM 6 points [-]

Eadwacer: Accepting that we are just performing 'operations' on 'configurations', what decides which operation will be performed? Is it the configuration of the incoming photon? Is it some magical (i.e.quantum) property of a half of a silver?

Amplitude flows to both end configurations, every time. That is the law of the amplitude flows. It is not one or the other.

Sebastian: Any complex number? I.e. you're invoking an uncountable infinity for explaining the lowest known layer of physics? How does that fit in with being an infinite-set atheist - assuming you still hold that position?

Infinite set atheism is not part of standard physics. Probably the best hope for reconciling infinite set atheism with a continuous universe is some equivalent of holographic theory that bounds the total information, meaning, you could always describe a wavefunction with some finite number of bits. I doubt I'm going to go into that. 'sides, my infinite set atheism could simply be wrong.

Jeremy: I was a little disturbed when you offered up the experiment that allowed us to reject the hypothesis about a half-mirror changing each time it reflects a photon or lets one through. How do we know there aren't other experiments that could discredit the amplitude hypothesis? I'm sure there's a good answer, but don't expect me to take too much on faith.

Obviously, real physics is based on a hell of a lot more experiments narrowing things down than just the ones I'm describing here. This is not the evidence. This is just here to help you interpret what the real evidence is evidence for. To see how the theory was nailed down historically by replicable experiments, you would have to read real physics books. This series is not about presenting the full evidence for QM as a hypothesis, it is about rendering the hypothesis itself non-confusing. This is my ambitious goal.

Comment author: 10 April 2011 07:00:56PM 2 points [-]

I'm sorry; I'm still a bit confused by this. "Amplitude flows to both end configurations every time," so when a single photon is fired (as in figure 1), I agree that the amplitudes of A->1 and A->2 are both 1. Does that mean both detectors click? (I was under the impression that only one detector would click.)

Comment author: 19 April 2011 10:04:35PM 5 points [-]

"The detector" is not a machine for "measuring the amplitude". In fact, we have no way of "measuring the amplitude". The only tool we have "measures the ratios of the squares of the amplitudes", and that tool is this: "run the simulation a bunch of times and compute the ratio of detections at 1 to detections at 2".

Comment author: 10 April 2008 11:31:47PM 1 point [-]

Psy-Kosh: I have never heard of anyone ever successfully formulating quantum (or classical) mechanics without the full spectrum of real numbers. You can't even have simple things, like right triangles with non-integer side length, without irrational numbers to "fill in the gaps". Any finite-set formulation of QM would look very different from what we understand now.

Comment author: 10 April 2008 11:34:09PM 0 points [-]

Psy-Kosh: I have never heard of anyone ever successfully formulating quantum (or classical) mechanics without the full spectrum of real numbers. You can't even have simple things, like right triangles with non-integer side length, without irrational numbers to "fill in the gaps". Any finite-set formulation of QM would look very different from what we understand now.

Comment author: 11 April 2008 01:22:11AM 1 point [-]

Eliezer,

I've found that Jaynes's infinite set atheism is a little too extreme. It forces you to take the slow route every time when you want to explore stochastic process priors like Gaussian process and Dirichlet process priors. I reserve infinite set atheism for observables -- no infinite sets of observations allowed.

Comment author: 11 April 2008 01:28:09AM 0 points [-]

Jess: Basically, my (extremely vague) notion was that since there's a "planck time" below which little (as far as we know) can be meaningfully said, effectively all quantum operations/changes over time/whatever are integer number of some "planck" versions of themselves, or combinations theirof.

Soooooo..... maybe.... possibly... there may be some sort of "quantum of phase shift"... But I concede that it was just speculation on my part based on vague notions.

Oh well, thanks. :) (would be slick if it actually did work out that way, sounds like, from you, that may not be much of an option)

Comment author: 11 April 2008 02:52:02AM 2 points [-]

Just so everyone's on the same page, continuum atheism doesn't entail disbelief in all irrationals. The hypoteneuse of a right triangle, pi and e are all in the countable set of computable reals.

Comment author: 11 April 2008 05:05:08PM 2 points [-]

If a photon hits two full mirrors at right angles, then its amplitude is changed by i*i = -1. Does it matter whether the second mirror turns the photon back towards its source, or causes the photon to continue in the direction it was going originally? Do you get -1 in both cases?

Comment author: 13 April 2008 03:13:13AM 1 point [-]
Comment author: 19 April 2008 01:32:32AM 1 point [-]

Okay, what happens in this situation: Take figure 2. The arrow coming in from the left? Replace it with figure 1, with its mirror relabeled E and detector 2 removed (replaced with figure 2). And lengthen the distance to detector 1 so that it's equal to the total distance to detector 2 in figure 2. And I guess call the detector 1 in figure 2 "X" for "we know you won't be getting any amplitude". Now what? Here's what I get...

A photon is coming toward E (-1,0)

A photon is coming from E to 1 (0,-1) A photon is coming from E to A (-1,0)

A photon is coming from E to 1 (0,-1) A photon is coming from A to B (0,-1) A photon is coming from A to C (-1,0)

A photon is coming from E to 1 (0,-1) A photon is coming from B to D (1,0) A photon is coming from C to D (0,-1)

A photon is coming from E to 1 (0,-1) A photon is coming from D to X (0,1)+(0,-1) = (0,0) A photon is coming from D to 2 (1,0)+(1,0) = (2,0)

From this I conclude that detector 1 will register a hit 1/5 of the time and detector 2 will register a hit 4/5 of the time. Is that correct?

Comment author: 22 April 2008 04:46:39PM 1 point [-]

Here's what I was missing: the magnitudes of the amplitudes needs to decrease when changing from one possible state to more than one. In drawing-on-2d terms, a small amount of dark pencil must change to a large amount of _lighter_ pencil, not a large amount of equally dark pencil. So here's what actually occurs (I think):

A photon is coming toward E (-1,0)

A photon is coming from E to 1 (0,-1/sqrt(2)) A photon is coming from E to A (-1/sqrt(2),0)

A photon is coming from E to 1 (0,-1/sqrt(2)) A photon is coming from A to B (0,-1/2) A photon is coming from A to C (-1/2,0)

A photon is coming from E to 1 (0,-1/sqrt(2)) A photon is coming from B to D (1/2,0) A photon is coming from C to D (0,-1/2)

A photon is coming from E to 1 (0,-1/sqrt(2)) A photon is coming from D to X (0,1/2sqrt(2))+(0,-1/2sqrt(2)) = (0,0) A photon is coming from D to 2 (1/2sqrt(2),0)+(1/2sqrt(2),0) = (1/sqrt(2),0)

Detector 1 hits 1/2 of the time and detector 2 hits 1/2 of the time.

Comment author: [deleted] 15 September 2013 09:42:31AM *  0 points [-]

What I was about to say. It really doesn't matter yet, but it's better to get the reader used to unitarity straight away. (Though I wouldn't explicitly mention unitarity this early -- I'd just replace the rule with "Multiply by 1/sqrt(2) when the photon goes straight, and multiply by i/sqrt(2) when the photon turns at a right angle" and everything that follows from that. If the maths gets too complicated with all those denominators, just make the initial amplitude -sqrt(2) rather than -1.)

Comment author: 04 June 2008 08:16:48AM 0 points [-]

The prediction for what happens when you block the B to D path is wrong. We have three final configurations, not two as in the above.

* From the configuration "A photon going from B to D", with original amplitude (1 + 0i) o Amplitude of (1 + 0i) * 1 = (1 + 0i) flows to "Block with absorbed photon". * From the configuration "A photon going from C to D", with original amplitude (0 + -i) o Amplitude of (0 + -i) * i = (1 + 0i) flows to "A photon going from D to F" o Amplitude of (0 + -i) * 1 = (0 + -i) flows to "A photon going from D to E".

Applying the magical detector, we get a 1/3 probability for each of these outcomes; detector 1 detects, detector 2 detects, neither detects (block absorbs photon).

Does this work?

Comment author: 13 June 2008 02:13:10AM 1 point [-]

"So... I'm confused. You say:

"...the half-silvered mirror rule is "Multiply by 1 when the photon goes straight, and multiply by i when the photon turns at a right angle."

We appear to have defined everything needful, except the word "when".

Accepting that we are just performing 'operations' on 'configurations', what decides which operation will be performed? Is it the configuration of the incoming photon? Is it some magical (i.e.quantum) property of a half of a silver?" ~ Hendrik Boom

I feel the same way as above.

Comment author: 30 December 2009 08:44:47PM 6 points [-]

My understanding is fuzzy, but the sense I get is that the word "when" just attaches to the possible configurations rather than being a condition. A rephrasing of the idea might be:

"Multiply by 1 for the configuration in which the photon goes straight, and multiply by i for the configuration in which the photon turns at a right angle."

Which results in two configurations every time, rather than different configurations depending on what the photon does.

Comment author: 28 February 2013 03:42:51AM 0 points [-]

'I think the problem some of us are having is reading "They would send a photon toward the half-silvered mirror ..." precludes the possibility of there being two results. I didn't think to not picture a little (billiard) ball being propelled towards a "half-silvered mirror" and it ending up at one (and only one) of the detectors.

Comment author: 13 June 2008 06:13:34PM 1 point [-]

So from what I understand:

A photon is merely our way of interpreting an amplitude wave in 3d-space. Such a 3d amplitude can be described by an x,y vector or (to simplify things), a x+yi complex number. (The complex number multiplied by i is really just a way of getting the same result as an x,y vector, due to the properties of complex numbers.)

Correct me if I am wrong so far, because I am about to get a bit fuzzy.

From what I see, the half-silvered mirror sends the amplitude wave in both directions, and is capable of reversing the phase of the amplitude -- thus, it is possible for amplitudes to intersect and cancel out. When an amplitude gets to a detector, we can see its magnitude, and count that as 'a certain number of photons'. This is why 'origin of the photon' is inconsequential (as discussed in the next article', because a 'photon' is a factor of our reading the amplitudes.

That's what I got from the article, is that a correct conclusion?

I think that I must be missing something here, because I have a few questions.

How does the half silvered mirror split the amplitude? Is the 'wave' split into two sub-waves with half the amplitude, and we just send 'photons' in bulk so it looks like 'half of the photons' got to one detector and 'half' to the other, when really each wave is split and gets to both?

Which seems like just treating photons as waves... Furthermore, as far as my understanding goes, waves are a result of particle transmission. What I'm getting at is, what is 'causing' these amplitudes, or how can an 'amplitude' be measured (and what medium is the wave measured in)?

Comment author: 18 July 2008 12:26:28AM 0 points [-]

Hi, I'm the kind of guy I think this article was meant to target - I did not have an understanding of QM, but did start with enough base knowledge to follow the article without tripping over language or math in it.

I must say that I fried my brain trying to decipher what you're trying to say. From one paragraph to the next, there's a constant feeling a big hidden mental leap has been made. All of a sudden, one is left lost between notions that were introduced, but never explained.

For example. In Figure 3, from prior knowledge, I would suppose if you counted individual photons, they'd ALL end up in detectors 1 and 2 (none would be absorbed by obstacle) - this goes to explain what in fact happens when we think a particle "knows" where it's going to end up. Please correct me if I'm wrong here.

It should also be equally possible to explain the logic behind the quantum eraser - my intuition tells me that for some reason the information that "evades" us would, given better understanding, simply be a configuration that was not possible, and it should be clear what the link between the hidden information and the way configurations work is.

Comment author: 29 March 2012 06:23:05PM 0 points [-]

For example. In Figure 3, from prior knowledge, I would suppose if you counted individual photons, they'd ALL end up in detectors 1 and 2 (none would be absorbed by obstacle) - this goes to explain what in fact happens when we think a particle "knows" where it's going to end up. Please correct me if I'm wrong here.

You are. If you were to put a detector 3 there instead of an absorber, it would go off half the time, and detectors 1 and 2 would each go off a quarter of the time.

Comment author: 29 June 2013 05:05:02PM 1 point [-]

Are you implying that the presence of a detector instead of an obstacle changes what the other detectors detect, or not?

The text is unclear here:

Detector 1 goes off half the time and Detector 2 goes off half the time.

Does "half the time" mean "half the time that any detector goes off", or "half the time you shoot a photon"? I would expect that, with the obstacle in place, half the time you shoot a photon no detector would go off, because the first mirror would deflect it into an obstacle. Seeing no detector go off is distinct and observable, so I don't see any way it could be eliminated as a possibility like the other case described here where two possible timelines that lead to the same world interfere and cancel out. So I would assume Eliezer means "half the time that any detector goes off". If so, I'd like to see the text updated to be more clear about this.

Comment author: 01 November 2009 08:31:45PM 1 point [-]

I have a question similar to Nate's. How does a half-silvered mirror work? More specifically, what is it about light or about half-silvered mirrors that means there are two paths for a photon out of a half-silvered mirror (compared to a full mirror, for example)? My guess at the moment is that the answer might start "light doesn't actually travel in straight lines..."...

Comment author: 01 November 2009 08:40:30PM 0 points [-]
Comment author: 01 November 2009 09:01:24PM *  2 points [-]

You can't explain yourself? I followed your link. It looks like part of why half-silvered mirrors "work" for the purpose of seeing someone without them seeing you is that one side is kept brightly lit while the spying side is kept dark. I think "beam-splitter" is possibly a more accurate term for my question, which I looked up and found

Another design is the use of a half-silvered mirror. This is a plate of glass with a thin coating of aluminum (usually deposited from aluminum vapor) with the thickness of the aluminum coating such that part, typically half, of light incident at a 45 degree angle is transmitted, and the remainder reflected.

(Wikipedia) Of course, this doesn't actually explain anything - why should there be a thickness of aluminum such that part of the light is reflected while the remainder is transmitted?

Would a beam-splitter still work if the silvered and non-silvered parts were much larger (i.e. a chunky block pattern)? If you fired a single photon at that would it still make sense to calculate amplitude as you do in this post (considering the two outward paths and multiplying one by i, the other by 1)? Perhaps the distance between a silvered part and a non-silvered part needs to be close to the wavelength of the photon?

Comment author: 17 December 2010 10:18:06PM 0 points [-]

If the cross-section of the photon was spread out so that it hit both silvered and non-silvered parts, some would reflect, yes. But it wouldn't reflect quite like a mirror - diffraction effects would make things wonky, so people use half-silvered mirrors, which are nice.

How do they work, you ask? Did you ever take a course on wave mechanics where you calculated reflection and transmission coefficients? It's exactly like that, except now the probability is essentially what's "waving." (if you haven't, see here)

Comment author: 08 April 2013 03:20:54AM *  0 points [-]

To answer your question as to how a half-silvered mirror works, first it might be a good idea to discuss how a full mirror works.

Classically speaking, the silver in the mirror has electrons that can freely move around. The electromagnetic fields of the incoming light accelerate the charged electrons in the silver, inducing electric current.

The currents flowing in the silver create their own electric fields, which by Lenz's law, cancel out the electric field inside the silver, and in doing so, send an oppositely-shaped wave back out into the void (the reflected wave).

Because silver is not a perfect conductor of electricity, the topmost layer of silver does not completely cancel these fields, and so the light can actually penetrate a small distance into the metal (typically nanometers) before it's finally converted into electric current.

If the silver coating is very thin, thinner than the penetration depth, then the component of the light wave that has penetrated through the metal will escape out the other side and keep going. That is, the resistance of the thin silver is high enough that the induced current doesn't completely cancel out the electric fields of the photon.

This classical explanation is also the same as the quantum one.

They also make beamsplitters that are like you describe -- I think they call them "Polka dot beamsplitters". I don't remember what they're used for. They would work the same way, but if you have a focused laser beam, the beam spot would be so small that it would either hit a full-mirrored section or a transparent section, and not both. You would need to use a lens on both sides of the beam splitter to spread the beam out to encompass the whole beamsplitter, and then gather it back. I think as long as the polka dots are not on the same scale as the wavelength, it wouldn't cause a problem.

Comment author: 06 November 2010 02:39:14AM *  1 point [-]

I thought that was a really good, logical, simple explanation. Looking forward to reading the next episode.

Thanks!

Comment author: 06 November 2010 04:32:10AM 9 points [-]

The post you're replying to is from April 2008; the next part is Joint Configurations and you can follow along by selecting "Article Navigation > by author" and clicking the right arrow, or follow the whole thing in a more organised way by following the Quantum Physics Sequence.

Comment author: 17 December 2010 10:08:17PM *  0 points [-]

This is very cool. I know that's just in my head, but now I just want a half-silvered mirror to test this with my kids.

Comment author: 03 February 2011 04:01:28PM *  0 points [-]

An xkcd take on this material

I collapsed laughing. Your results may vary.

Comment author: 03 February 2011 10:40:19PM 1 point [-]

It's funnny - but your brackets don't match.

Comment author: 03 February 2011 10:51:21PM 0 points [-]

Fixed. Thx.

Comment author: 10 February 2011 11:18:29AM 1 point [-]

"we send some photons toward the half-silvered mirror, one at a time, and count up how many photons arrive at Detector 1 versus Detector 2 over a few thousand trials. The ratio of these values is the ratio of the squared moduli of the amplitudes. But the reason for this is not something we are going to consider yet."

OK, but I'd still like to see a little link or something here that takes me straight to the next article where this is properly dealt with, since this seems to be the biggest gap in understanding that the current article leaves open: over and over you tell us that there are no actual probabilities involved in the phenomena at the level of the territory, yet in my quote you have exactly a probabilistic description, with multiple trials that arbitrarily yield one of the two possible results, in stark conflict with the rest of your explanation which tells us the same thing is actually happening each time (the amplitudes are always the same).

The tiniest extra hint would do a world of good here (or at the end of the article): is it that quantum impurities always stray unpredictably into our experimental setups in the real world and actually change the amplitudes involved? Or what?

Comment author: 06 May 2011 02:43:00PM 0 points [-]

"Adjectives like probable and possible do not apply to them; they are not beliefs or sentences or possible worlds. They are not true or false but simply real."

Based on all the "i"s in the equations I think you meant to say "complex" =p

Comment author: 08 June 2011 08:00:57PM *  1 point [-]

Is it possible in reality to fire a single photon?
(Post modified)

Comment author: 08 June 2011 08:25:05PM *  1 point [-]

Is it possible in reality to fire a single photon?

Yep. Chad Orzel blogged a recent example of such an experiment.

Also, welcome to LessWrong!

Comment author: 11 July 2011 10:14:56AM 0 points [-]

after the computer program above calculates the amplitude (the same every time we run the program), can we incorporate in the program additional steps to simulate our magical measurement tool (the detector)?

Comment author: 12 July 2011 11:11:11PM 3 points [-]

Would it be possible to actually set up this experiment at home (i.e. without an expensive physics lab)? Any particular pointers would be wonderful, even if it's just giving a common name that this setup uses. The sequence seems wonderful, but I'd prefer not to take it on faith if I can take it on empirically-demonstrated-it-myself instead :)

Comment author: 17 August 2011 11:23:06AM *  1 point [-]

start from figure 2, turn the half mirror at D round so it faces the other way, now E will light up instead of F. Since his explanation doesn't allow for that we've just proved his explanation is wrong.

Anyone know if that's right? EDIT: seems clear to me both detectors must light up if you do this. EDIT2: it turns out that by "turn around" he means through 180 degrees, which should surely mean no change.

Comment author: 17 August 2011 12:07:49PM 1 point [-]

Can you clarify the question? Do you mean turning the mirror by a quarter-turn from its current orientation, so it's diagonal in the other direction? I compute that if you do that it should work out with 1/4 chance of E, 1/4 chance of F, and 1/2 chance of neither (if it's reflected back towards B or C it will never reach either). Exactly like the classical case, actually...

Comment author: 17 August 2011 12:43:21PM 1 point [-]

Edited to clarify - I agree re quarter turn, but it turns out he means half turn. I think our thought-experiment half-silvered mirrors are unchanged by a half turn.

Comment author: 17 April 2012 07:22:18AM *  1 point [-]

It turns out he was referring to this error; see How accurate is the quantum physics sequence?

Comment author: [deleted] 10 September 2011 12:35:20PM *  3 points [-]

What affects the world is real. (If things can affect the world without being "real", it's hard to see what the word "real" means.) Configurations and amplitude flows are causes, and they have visible effects; they are real.

<nitpick>Only ratios between amplitudes are “real” in that sense, because if you multiply the amplitude of everything by, say, exp(2πi/3), nothing actually changes.</nitpick>

Comment author: 12 December 2011 05:15:03AM 1 point [-]

Now, for complicated reasons that we aren't going to go into today - considerations that belong on a higher level of organization than fundamental quantum mechanics, the same way that atoms are more complicated than quarks - there's no simple measuring instrument that can directly tell us the exact amplitudes of each configuration. We can't directly see the program state.

I'm not sure if you cover this in further articles... but it is worth saying:

The amplitudes of each state are not unique... there are more than one (in fact, there are infinitely many) different configurations that get you the same observable probability density, each differing by a phase factor.

Comment author: 16 December 2011 11:46:34PM 3 points [-]

I... Er... What. Where did the whole 'amplitude' thing come from? I mean, it looks a lot like they are vectors in the complex plane, but why are they two dimensional? Why not three? Or one? I just don't get the idea of what amplitude is supposed to describe.

Comment author: 13 April 2012 05:07:42PM 0 points [-]

For that matter, amplitude of a wave...but what is waving? Where's the realism?

Comment author: 12 March 2012 03:35:34AM *  0 points [-]

Eliezer, regarding the Fig.1 experiment above you're saying "The half-silvered mirror obeys the same rule every time." "This same result occurs—the same amplitudes stored in the same configurations—every time you run the program (every time you do the experiment)." OK, mathematical result is the same. However, physical results at detectors 1 & 2 are not the same: click at either of them is not predictable. There is symmetry in math vs asymmetry of physical result for any individual photon. Is there any "quantum explanation" for such physical dissimilarity?

Comment author: 12 March 2012 03:52:03PM *  0 points [-]

Well, from the perspective of everyday life—way, way, way above the quantum level and a lot more complicated—the magical measuring tool is that we send some photons toward the half-silvered mirror, one at a time, and count up how many photons arrive at Detector 1 versus Detector 2 over a few thousand trials. The ratio of these values is the ratio of the squared moduli of the amplitudes.

The ratio of "photon at detector 1" and "photon at detector 2" (averaged over enough trials) is 1.

If none of that made sense, don't worry. It will be cleared up in future posts.

Edit: This was actually written as a response to one of these comments.

Comment author: 12 March 2012 03:16:58PM 0 points [-]

In regards to the first experiment (Fig.1) "the little two-dimensional arrow for the configuration "Detector 1 gets a photon" has the same squared length as for "Detector 2 gets a photon"." This mathematical equality should have resulted in each photon arriving at detectors 1 & 2 simultaneously. But this never happens. Could anybody explain to me reason for such a discrepancy between math and reality?

Comment author: 13 March 2012 03:18:48PM *  0 points [-]

Eliezer is saying that when the ratio of the squared moduli is 1, than Detector 1 goes off half the time and Detector 2 goes off half the time. But why it should be necessarily interpreted this way? Is this another QM rule? What prevents, in this case, an alternative interpretation: a photon must split in half and arrive at both detectors at the same time?

Comment author: 04 April 2015 03:17:06PM 0 points [-]

Clarification: an amplitude is the value of a configuration?

so { a photon going from A to B = (-1 + 0i) } is a configuration and { (-1 + 0i) } is an amplitude?

Comment author: 07 December 2015 12:44:31PM *  0 points [-]

Thanks for this explanation. I've tried to read it some time ago but have not really coped with it. Now after reading again it was interesting for me to check if this is explained on some other internet sources and how exactly. So I checked one of the first top search results and here is what I saw: http://physics.stackexchange.com/questions/91695/double-slit-expirement-fundamentals-half-silvered-mirror-version

There some guy asked for an explanation of this experiment and answers are all about optical refraction and phase shifting, which honestly speaking would not clarify the matters for me. Now this is all ok about theory and numbers. But I tried to google some videos which will show this in action and found that actually it is not really easy (or not possible at all) to set up all these pieces and see it in action. For example, in this video https://www.youtube.com/watch?v=M6y_igUpyCg guys tries to set it up and does not succeed. Is that about he is usint a laser which in fact transmitts a bit different photons every time? Could you recommend some video where this experiment can be seen in action?

Comment author: 19 May 2016 01:23:10AM 0 points [-]

Great post, Eliezer! I have one question, though, and maybe some of the folks here can answer it as well: why do we multiply amplitude in Figure 2 by i if it either turns "left" or "right" at 90 degrees? In the complex plane, we multiply a vector by i if we want to rotate it 90 degrees clockwise, and by -i if we want to rotate it 90 degrees counterclockwise...

Comment author: 08 December 2016 01:19:36PM *  0 points [-]

So the final program state is:

Configuration "A photon going toward A": (-1 + 0i)

Configuration "A photon going from A toward 1": (0 + -i)

Configuration "A photon going from A toward 2": (-1 + 0i)

Why does the bolded configuration still exist in the same way? Shouldn't it go back to zero once the photon has reached A, since the rest of the post seems to imply a timely order of things?

Comment author: 19 December 2016 08:06:52PM 0 points [-]

silver, check out rule #3 for calculating probability amplitudes:

https://en.wikipedia.org/wiki/Probability_amplitude#The_laws_of_calculating_probabilities_of_events

It basically states that in order to calculate the amplitude of a photon going to 1 is a product of the amplitude of it going to A and then from A to 1. So we do need to remember what the amplitude was for a photon going toward A.