Wave-Function Collapse

Quantum physics is weird. How weird? “Too weird for words,” as we used to say, and there is a literal truth to words being inadequate in this case. There is no way to look at the quantum world that doesn’t break one’s mind a little. No one truly understands it (other than through the math). It’s like trying to see inside your own head.

Since we’re clueless we make up stories to fit the facts. Some stories advise that we just keep our heads down and do the math. (Which works very well but leaves us thirsty.) Other stories seek to quench that thirst, but every story seems to stumble somewhere.

One of quantum’s biggest and oldest stumbling blocks is wave-function collapse.

My relationship with quantum physics predates quarks. I was already an avid physics fan when The Tao of Physics and The Dancing Wu Li Masters came out. (For old time’s sake, both books still sit on my shelf along with the newer ones that obsolete them.)

I’ve watched the field evolve. String theory grew from an early gravity theory into a career-consuming institutional juggernaut. What started a sensible physical theory ended a monstrous mathematical swamp.

Through it all, the wave-function collapse remains an enduring mystery.

Understanding why requires looking into wave-functions and what we mean by “collapse.” (I’ll end with a story I think might explain these things.)

§ §

The wave-function of a quantum system is described by the Schrödinger equation, which is a central aspect of quantum mechanics.

That equation is sometimes written like this:

It’s a complex-valued equation, meaning it uses complex number math. The star of the show is Ψ (psi, typically pronounced “sigh” although some say “sea”). It’s the complex vector representing the current state of the quantum system.

The co-star, Ĥ (“H-hat”), is the Hamiltonian operator that sums the kinetic and potential energies of the system in question.

What the whole thing describes is the quantum system defined by H-hat. The vector, Psi, is the system state at any given time, t.

Solving the equation gives the probability of finding the system in the specified state (value of Psi). Note that using the equation at all requires defining the Hamiltonian for the system in question.

It’s worth emphasizing that the Schrödinger equation describes a quantum system’s wave-function (the wave-like nature and evolution of that system). The actual ontological nature of a system’s wave-function is unknown.

§ §

A quantum system has a property called phase, which is the source of quantum interference.

The wave-function sees (quantum) reality as fundamentally wave-like, and phase is essentially a point on a wave. Think of it as a single hand on an analog clock marked, not by hours, but by degrees around the circle (noon is 0°, six o’clock is 180°).

Wave phases.

Waves (mathematically speaking) go ’round and ’round like clocks. Phase is what o’clock it is (expressed in degrees).

When a quantum system has more than one path from point A to point B, the clock ticks separately along each path. If the paths are the same length, the clocks match when they meet. If the paths have different lengths, the clocks show different times.

Quantum interference involves the two clocks. If they show the same time, they combine. If they show the opposite time (say 3 o’clock and 9 o’clock) they cancel. Between same and opposite, they average out (between 0 and 2×).

(This is adding and cancelling is what generates the pattern in the two-slit experiment.)

§

What’s important here about quantum phase is that it’s very fragile. (A key difficulty in designing quantum computers involves protecting the phase of quantum computing bits.)

A quantum system with detectable phase is said to be coherent. (Related to, for instance, coherent laser light, a situation in which all the clocks show the same time.)

But the phase of a target system, S, is easily lost to surrounding quantum systems. It often requires a vacuum and near absolute zero temperatures to prevent decoherence of a system under study.

What happens, if not prevented, is that the wave-function of S entangles with the wave-functions of surrounding systems, each of which has its own wave-function and phase.

Combining two different system phases.

When two phases combine, the phase of the resulting wave is their average. That means the resulting phase is shifted from both (unless the two were exactly in phase). So the phase of S is smeared out among all the surrounding systems it interacts with.

In turn, those surrounding systems affect S. Their phase information combines with S, and the result is like trying to hear one voice in a room with hundreds of people talking.

The bottom line is that S decoheres and its phase information and quantum interference behavior is lost. The surrounding systems also change to reflect the interaction with S.

This interaction is called a measurement.

It’s directly implicated in the apparent “collapse” of the wave-function.

§ §

With that background, let’s consider what happens in a specific quantum system — a single photon in a two-slit experiment.

For definiteness, let’s say we have a laser capable of emitting single photons. It shines through two thin slits in a metal wall to our detector — a piece of film capable of responding to individual photons (more on that in a bit).

The basic two-slit experiment.

A photon is a simple enough system that we can define a Hamiltonian for it. If we use it and solve the Schrödinger equation for every point between the laser and the film, we get a wave pattern. It looks very much as if someone dropped a rock where the photon leaves the laser. The ripples spread out from the source.

Crucially, after they pass through the two slits, they meet and interfere.

The peaks are places the photon is most likely to be found if we look in that spot. In particular, the peaks and troughs along the piece of film are places where the two paths reinforce and cancel out. (Forming the famous interference pattern.)

Ultimately, the photon’s flight must end, either in the various walls of the experiment or in the film (it can’t just vanish; it has to end up somewhere).

If it ends up in the walls, it generates a tiny, tiny bit of heat as the wall atom absorbs it and jumps to a higher energy level. That atom immediately dissipates the energy to nearby atoms. Eventually the disturbance is lost as the system reaches equilibrium. The energy distributes and dilutes to nothing.

If the photon hits the film, it again increases the energy of an absorbing atom, but here the material is primed to record the photon. (There may even be a cascade effect that alters the state of multiple atoms.) The energy of the photon causes a chemical change in the film that later chemical processing amplifies and fixes.

At this point, I want to mention something very important: Note that in all cases, the photon, which came into existence in the laser, ends that existence when absorbed by an atom. This will be important later.

§ §

This brings us back to the great mystery, the idea of wave-function collapse.

The problem is that the Schrödinger equation describes the linear evolution of a quantum system. Further, it’s fundamentally a wave description. The abrupt change from this smooth wave behavior to localized point behavior represents a discontinuity physicists haven’t truly explained.

Worse, there are implications of non-locality. (Experiments testing Bell’s Inequality seem to establish quantum physics is non-local under almost any reasonable interpretation. It seems something we’re stuck with.)

But the show-stopper is trying to explain, exactly, mathematically, what happens when a coherent wave-function appears to collapse.

The story solutions range from ignoring it to specifying infinite worlds. I find both too extreme. Next time I’ll tell you the story that I think might account for it.

Stay coherent, my friends!

About Wyrd Smythe

The canonical fool on the hill watching the sunset and the rotation of the planet and thinking what he imagines are large thoughts. View all posts by Wyrd Smythe

15 responses to “Wave-Function Collapse

  • Christina Schmidt, MA

    Sir, I don’t pretend understand a word of it but I appreciate your clear efforts to educate nonetheless. Hope you (and your skilled fingers) have a great weekend! 😀

    • Wyrd Smythe

      Well, thank you — geeks gotta geek! (I have a lot of teachers and preachers in my family tree, so it’s apparently in my DNA to try to explain things. (Or maybe its just basic man-splaining.🤷🏼‍♂️))

  • SelfAwarePatterns

    Nice description Wyrd! A couple of observations.

    I think you alluded to this (and so understand it), but just to be clear, decoherence and the wave function collapse are not the same thing. As you noted, decoherence explains the appearance of the collapse, but if we’re accepting decoherence, then we’ve dispensed with the collapse as a fundamental event. (We could choose to retain it as an emergent event, but just saying “decoherence” seems clearer.)

    Second, while for all practical purposes information in decoherence is lost to the environment, technically it’s preserved in the same manner as when only two particles entangle. The difference is there’s currently no way to model the staggering number of interactions involved in entanglement with the environment.

    I’m looking forward to seeing how you try to cut the Gordian knot!

    • Wyrd Smythe

      You’ve implied I don’t know the difference between decoherence and w-f collapse repeatedly, but it’s never been true.

      And of course it’s believed that quantum information is never lost. I think that one should be looked into, though. It causes problems when it comes to black holes, and it seems obvious information can be created (so why not lost).

      • SelfAwarePatterns

        I can only call it based on what you write Wyrd. Here you gave me enough to give you the benefit of the doubt.

        As I understand it, black holes may be an exception to information conservation, or not, depending on which theories you favor.

      • Wyrd Smythe

        The “benefit of the doubt.” Wow. Um,… thanks?

        Based on numerous past exchanges, it’s not clear to me you always understand what I write, so I’m not sure how to process these benefits of your doubts. I can say that having my opinions dismissed as “summary” and, when I protest, implying I am a liar is a bit much. Here you’re continuing to suggest I don’t know what I was talking about. It’s getting a little old.

        Maybe you should find a blogger whose opinions you have more regard for.

      • SelfAwarePatterns

        I definitely didn’t mean to imply you were a liar. Honestly, I’m sorry I asked about the old statements. I should have known better. I wish it could be undone, but it can’t, so I can only say: I’m sorry.

        On “summary”, I shouldn’t have used it, but in my defense, the language at the end of your previous comment stung, and I’m human.

        We have a lot of common interests, and I do have a lot of regard for you. I’d hate for our conversations and friendship to end. But there needs to be space for respectful disagreement.

      • Wyrd Smythe

        Apologies do carry weight with me. I think the history here may demand some kind of change; I dislike how our debates can get derailed. I fear my challenges have polarized things; I sometimes wonder if one defends ideas one doesn’t fully support rather than agree with me. Maybe that’s on me.

        I apologize for stinging you. That was definitely provocative phrasing on my part. (I assume you mean the crack about it being “increasingly inexplicable” that anyone thought MWI was more than a math fantasy.) At that point I was frustrated by our previous discussions about MWI and the one about the surface of a torus (which left me feeling you didn’t believe me).

        (There is also that the more I look into MWI, the more I don’t see the attraction. Adding decoherence to the mix seems to take away the parsimony argument. If decoherence explains collapse, who needs many worlds? Worse, decoherence seems to explain why the quantum state is not amplified into the detector. Plus, in Everett’s paper, I seem to have found the big assumption he makes in discussing the wave-function of large systems. I wish I understood the attraction.)

        I quite agree about respectful disagreement (although it would be pretty funny to respectfully disagree about it). What frustrates me is ignoring questions or points or even walking out on a discussion. On MWI, IIRC, I was questioning why the energy and probability problems didn’t bother you when they seem like showstoppers to me. I was questioning why infinite actual worlds is so easy to swallow but those issues can be waved away. All I wanted was a cogent explanation for a belief, but that seems where a number of our discussions go off the rails. (It’s what often derailed our computationalism discussions. My wanting to know why.) Again, maybe that’s on me.

        Maybe it has to do with my weird learning curve, but I need to know the content and logic of someone’s belief in order to reach respectful disagreement. It’s mostly needing to know someone’s thinking is well-grounded enough to take seriously. (I’ll say this: With many people I don’t bother because I know it isn’t. They do not lead “well examined” lives.)

      • SelfAwarePatterns

        Thanks Wyrd. I’m grateful for your forgiveness.

        It does seem like I need to think about how I handle these discussions. And I definitely take responsibility for some things leading to frustration. In particular, it looks like the way I end discussions can be problematic, so I think I should lay my cards on the table.

        Incidentally, this isn’t specific to you. It’s the rules I generally follow for internet discussions, but we have so many discussions it’s probably more obvious for you.

        I generally walk away from a discussion for two broad reasons. First, if it seems like we’ve both presented our arguments, and I notice that we’re beginning to repeat ourselves (which can be hard to notice if it’s spread out over days), or if I’ve just reached a point where my only response is to repeat past points, I’ll usually move on. My experience is that after this point is when discussions start to become progressively more heated. I used to announce this, but it seemed like the announcement itself was often interpreted poorly, so I took a cue from a lot of other internet friends and started just moving on.

        The other reason is that either the other person seems to be getting upset, or I’m getting upset. In that scenario, my experience is that continued participation rarely makes things better. And announcements here almost always make things worse. Sometimes I’ll let something like this sit for a day or two to give myself a chance to cool off and revisit it, but often it scrolls out of view and I forget to go back. (Incidentally, I really should have taken a time out before responding to that initial comment. Again, sorry, my bad.)

        Speaking of scrolling out of view, sometimes I just get distracted, or need to do research, and it ends up slipping through the cracks. It’s been particularly bad over the last couple of months with work periodically erupting into panic periods of intense activity.

        So, I’m open to suggestions. I do think there are things we disagree on where we’re never going to agree with each other’s reasons. (I find that true for everyone I know.) There’s definitely value in discussing those reasons, but if we loop too much on them, I suspect the frustration will only climb. Maybe in those scenarios you should just downgrade your assessment of how well examined my life really is. 🙂

      • Wyrd Smythe

        I do think online debates are a challenge. The lack of emotional vocal cues makes the words matter more than they should in what is really a casual conversation. We all abbreviate ideas or figure the other person knows what we mean.

        I’ve been online since the 1980s and I’m not sure I’m entirely happy with any prolonged debate I’ve ever had (and there have been a lot). The problem might be that it’s a debate at all. A successful dialectic almost requires formal rules and a moderator. There needs to be something to prevent what you very rightly identify as the problem of looping. Moderators are great for that, but not practical in comment discussions.

        In general, what seems to, at least sometimes, work is trying to stick to what I think and why I think it and then asking questions to prompt the same from them. In my mind, the first step is mapping the territory; seeing where we agree or disagree. But some conversations don’t seem to make it that far. Sometimes the questions are off-putting or challenging, tension builds, and things go awry.

        When things get as far as seeing the map, it’s then sometimes possible to dig into the differences. It can require some care. I think it’s important to agree with and validate the other person as much as possible. (I’d like to believe that if you go back over our conversations, you’ll find a lot of my replies do contain agreement.)

        I think Richard Brown is good at the dialectic in his interviews. It’s interesting to watch his detailed interviews with people with contrary views. For one, he seems genuinely interested in what they think and why, which I identify with. And he’s good at really delineating the points of disagreement without raising hackles. That’s a skill I wish I was better at. As I said, sometimes my questions seem to derail things. (I sometimes wonder if I’m on the autism spectrum somewhere. There are some social skills that seem beyond me; a piece missing.)

        Regarding exiting a debate, I’ve known you long enough that I do know your stated policy. The etiquette of online communication is still forming, and I know many are fine with conversations just ending. I’m old enough for that to feel vaguely weird, especially when it seems mid-conversation.

        Maybe the misinterpretation you sometimes get comes from mentioning the looping. I can see that being taken wrong. It has a vague taint (unintended I’m sure) of blaming the other person. One of the great challenges we have as intelligent articulate people is, as they say on Archer, phrasing. I’ve been working on myself for years to neutralize my phrasing because as an intelligent articulate person, phrasing is a weapon.

        Here’s an exercise that might raise your eyebrows. Compare two, say, book review posts, one where you really agreed with the author and one where you didn’t. Pay careful attention to the subtext of your phrasing and choice of words. We all load our speech for effectiveness, and intelligent articulate people do it best. Those of us that read a lot, especially. It can be surprising to see how clearly we telegraph our opinions.

        As to exiting a debate, maybe the trick is — somehow — not letting it be a debate to begin with. I’m at a point in life where conflict of any kind makes me unhappy, so I try to approach debate as an exploration not a challenge. It’s a work in progress, obviously. I’d like to be more like Brown, but I’m not sure I have it in me.

      • SelfAwarePatterns

        I’m with you on the challenges on online debates. Often a statement someone might make jovially looks angry and bitter in writing. We can sometimes soften it up with emoticons, but I’m not sure they successfully counteract the visceral reaction we feel, particularly since the emoticon usually comes at the end, after the punch.

        I agree with mapping the territory. There’s a lot to be said for making our case and asking questions. Although with questions, we have to be careful, since it’s very easy to fall into the rhythm of a prosecuting attorney.

        I agree on Brown. He does seem very careful to be tolerant of different viewpoints. I do occasionally hear him express strong views, but I’m listening not watching, so I may be missing some crucial body language. But there’s a lot to be said for operating under the assumption that people are going to be in different places. But I also agree, it can be very hard to consistently stick to that. (Brown himself on Twitter can sometimes go on the attack with the best of them, so he’s human too! Which actually makes me feel better. It’s a strategy, not an inborn trait.)

        On withdrawing from the debate, maybe I need to find an explicit way to do it without mentioning the reason, although I don’t want to be dishonest about it. I’ll have to give it some thought.

        But I agree. The best option is to try to avoid having it be a debate. I definitely need to give some thought on how I can do my part to avoid it.

      • Wyrd Smythe

        True about emoticons and other attempts to communicate tone. I try those and to write a little humorously to try to send the message this isn’t that serious. But you never can tell how someone will react. They might see it all as sarcasm. Communication is tricky. (I have a whole back history of ex-girlfriends and one ex-wife to attest to that.)

        Good point about prosecuting attorney. I very much suspect that’s a big part of what gets people (perhaps a bit justifiably) unhappy with me. Pointed questions intended to open uncomfortable boxes. I do get that. The way my mind works, it figures, metaphorically speaking, innocent people have nothing to fear from questions. If one’s opinions are grounded in rational thought, questions about it shouldn’t be a problem. I think that’s fundamentally the “Spock” in me who just doesn’t quite get how human brains work, the possible autism I mentioned.

        (It’s weird. I know, intellectually, exactly what the “parable of Spock” is, but putting it into action is as difficult for me as comforting someone by saying, “It’ll all be okay,” when I’m dubious it will be okay. That’s the human thing to do, but my mind sees it as lying. I was raised to be as honest as possible, and I took that to heart, so it’s hard.)

        Wonder if Brown’s teaching experience might be where he gained those strategies. (I agree that’s what they are. I do wonder about natural predisposition in people, but nature-nurture is a tangle I won’t second guess.) I have no doubt he’s not perfect at it. It might be, too, that Twitter makes it a little too easy to fire off a first thought one might modify given another minute to think about it. Or not. 🙂

        On exiting a debate, what about summarizing the basic points, pointing out the disagreement (which is fine), and then asking if there are other points in question (and hope they say no). It sort of bounces the ball back while suggesting the points so far are closed. If they persist with old points, I’d just acknowledge, re-summarize, and keep asking about new points. Who knows, that might even turn the conversation in a new direction.

        But of course,… advice from me about dealing with other people needs to be taken with a whole salt shaker.

        Thinking about Brown’s tactics, maybe the trick is being super upfront about disagreeing when it happens. Saying clearly, I disagree, here’s a bit of basic why, moving on then… The idea being to just let it be. Maybe if a discussion hits enough connection points it can move back to looking that the disagreement spots? There is the dishonesty thing you mentioned as well as it being difficult to put one’s ego aside enough (that sense of, if I explain this just one more time, then they’ll get it; it’s hard to let go of the sense one is right).

        I dunno. As I said, it seems a challenge just figuring out the map sometimes. It’d be so much easier if there were virtual reality bars we could hang out in.

        The current social climate doesn’t help. I’ve thought the general social edginess was on the rise before the virus. This stay-at-home business now really has us all jumping out of our skins. I think for my own peace of mind, I’m going to try to avoid debate of any kind. It just feels too much like arguing right now.

      • SelfAwarePatterns

        On Brown and his teaching experience, I have noticed that teaching philosophers, particularly philosophers still teaching undergraduates, have much more diplomatic skills than a lot of people. When you have to deal with naive and uninformed opinions from students all the time, it probably gives you muscles that make it much easier to handle other philosophers or the general public.

        (It’s not universal though. I’ve seen some philosophers be far from diplomatic, although they tend to be in senior positions and probably only now have to deal with graduate students.)

        On summarizing, I’ll think about it, but I may be too lazy for all the work involved. (Sorry!)

        Yeah, I definitely feel the, “If I only explain this one more time, they’ll get it,” impulse, but I think I’m pretty decent at letting it go. (Not always, but usually.) The trick for me seems to be exiting in such a way where the other person doesn’t feel like they were robbed of that chance.

        One thing the last 24 hours has reminded me of. Friendships are more important than persuasion. It’s something I apparently have to be reminded of from time to time.

      • Wyrd Smythe

        Indeed.

        I think pursuing science (or worse, math) can trap one into being overly focused on seeking correct answers — grasping for absolute truths. I haven’t thought of this in years, but it applies: I got my ears seriously pinned back by a high school girlfriend when I corrected her spelling in a love letter. An early painful lesson in that being correct isn’t always the right choice. (What often puts me at odds with the world is how often being correct isn’t the right choice. It’s hard to swallow in some cases. Not that one; she was completely right. 😮 )

  • Wave-Function Story | Logos con carne

    […] Last time I started with wave-functions of quantum systems and the Schrödinger equation that describes them. The wave-like nature of quantum systems allows them to be merged (superposed) into combined quantum system so long as the coherence (the phase information) remains intact. […]

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