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What is a wavefunction? Answered

In the Physics topic on the EPR paradox, NachoMahma asked about wavefunctions and "collapse."

Let's put aside the whole "collapse" issue -- not all physicists agree that it is a sensible concept. NM's comment has a link to the Measurement Problem, and I'm not a good enough theorist or philsopher to contribute to that argument.

What is the wavefunction?
  • "Is wavefunction only a convenient way to say it's located somewhere close to here, but we're not sure exactly where until we measure it?"
  • "At any particular point in time/space the object is in a definite spot with a definite set of properties, but we can only make a reasonable guess?"

No. The wavefunction, spread out over all of space (I'm speaking non-relativistically here, but the formal interpretation applies to spacetime), is the fundamental "thing" in QM. "Objects" are wavefunctions. If the wavefunction is localized (non-zero for a small contiguous set of coordinates, zero everywhere else) then treating it like a particle makes sense. Otherwise, it doesn't; the thing behaves like a wave, showing diffraction, interference, and lots of other effects. My preference, when I talk about these things, is to just call them "quanta." They are not particles, they are not waves; they are their own kind of entity with well defined, if really hard to understand, behaviour.

How do I get to that point? Well, quantum mechanics is one example of a "field theory" (electromagnetism is the most familiar classical field theory). The equations we write down (the Schrödinger equation non-relativisitically, the relativistic Dirac and Klein-Gordon equations) to describe how quanta behave are coupled partial differential equations (PDEs), which relate the values (and derivatives) of the field at every point in space to their evolution in time.

A PDE which relates the time and spatial properties of a function is either a wave equation (if the solutions are sines and cosines) or a diffusion equation (if the solutions are exponentials). The Schrödinger equation is a wave equation, and we call the solutions wavefunctions. Electromagnetism also has a wave equation, which is how we get radio, light, etc.

The difference is that the functions in EM are "real valued:" the value of the field at each point in space/time is a regular floating-point number (the "phase" in EM is determined by the relative values of the field and nearby points). The wavefunction is a '''complex valued''' field -- at each point in space/time, the field has both an amplitude and a phase (or equivalently a real and an imaginary component). This means that wavefunctions can interfere in ways more complex than simply "adding" or "subtracting", which can have quite interesting consequences.

You get probabilities by taking the square (norm) of the wavefunction. This procedure gives you a real value, a probability, at each coordinate. When you make a measurement, those probabilities determine which coordinate value you see as the "location" of the quantum. The actual result is random, but that isn't because "we're not sure exactly." The quantum objective does not have a single coordinate location until we make the measurement.

How that happens, whether by "collapse," "decoherence," "many worlds splitting" or something else, is a subject of intense philosophical and experimental argument.


. It still sounds to me like it's a case of "we don't know what it is or how to describe it (at least so as Nacho can understand)." S's cat knows if he's alive (where he is), even if we don't. . Maybe I'm just stuck in a non-quantum world, but it seems to me (ie, I'm assuming) that, at any point in time, everything has a definite place (position, speed, direction, &c;). Just because we can't figure it out in advance, doesn't mean it's not there. . Please excuse me if I'm being dense. They didn't teach QM when I went to school.

You are "stuck in a non-quantum world" :-) What you're describing is essentially a hidden variables theory for QM. One concrete example is Bohmian mechanics.

People who want a deterministic theory in place of QM usually also want their theory to be local. That is, interactions between particles (entities) only happen at points (or small regions) in spacetime where both particles are located --- no "action at a distance."

(An aside: long-distance interactions, like EM or gravity, are also considered local because their effects can be accurately accounted for by considering their field's value only at the point where the particle being acted on is located.)

The problem is that in our Universe, you cannot have both a local hidden-variables theory and make predictions for quantum experiments which match reality. That's the essence of Bell's Theorem, which is where this whole discussion started :-)

In other words, if you try to construct a mathematically consistent mechanics where "at any point in time, everything has a definite place," then you will be able to use that mechanics to calculate the outcomes of definite, concrete experiments (such as a spin-spin correlation measurement). If you actually run the experiments, you'll discover that your calculations are simply wrong. Nature gives you results that match what canonical QM predicts, not what your "hidden variables" theory predicts.

So, do you give up locality (allow spacelike separated entities to interact instantaneously)? Or do you give up hidden-variables? Or both, and just "accept" that the bizarre mathematics of QM actually describes nature?

As Adrian Monk put it, "QM is weird."

. Bohm sounds to me like he knew what he was talking about. Wish I could understand the math. :( . I'm going to have to do some more studying/ruminating. Looks like this (non-)local thing is going to give me problems. Seems to me that non-locality is just an undiscovered "thing." . If non-locality turns out to have a speed limit, wouldn't that make it local - as with G and EM?

Bohm's theory predates Bell's Theorem. Indeed, Bohmian mechanics continues to be analyzed and evaluated by very good theoretical physicists because it seems like a natural way to avoid the philosophical confusions inherent in conventional quantum mechanics. I don't think (and this is my personal opinion, not a statement of physics) that it's correct.

You suggested, "If non-locality turns out to have a speed limit, wouldn't that make it local - as with G and EM?" Well, if that "speed limit" is not c, then I think it would violate special relativity. The recent experiment in Geneva was specifically testing whether such a speed limit exists. Their result is that if there is one, it must be greater than (about) 1000c, and possible an order of magnitude higher.

. OK. Let me chew the cud for a while. I'm sure I'll have more questions. . Thanks for being so patient. You must feel like one of your grad students teaching an Intro To QM class to Freshmen.

P.S. You are not being dense. The interpretation of quantum mechanics has been a source of fundamental confusion and philosophical angst for the last 80 years or so. Some of the finest minds of the last few generations have struggled with it unsuccessfully.

Re your msg in my EPR topic: > If you've played with diffraction gratings, interference fringes, or even standing waves in water, I think that's enough to give you a vague mental picture of wavefunctions. For some mental imagery on how they can "also behave like particles," I recommend reading up on solitons. . I dunno. It still seems to me that this whole particle/wave business just means we haven't figured something out. . I've worked with 3-phase theory (I was an industrial electrician in a past life), so I have a vague understanding of wave(form)s. Maybe that's my problem. Is there a one-to-one correspondence between electrical waves and quantum waves?

Ah! That actually helps. The math of wavefunctions is precisely the full AC arithmetic (with resistors, conductors, indcutors, coax, etc.). At each point in space(time), the value of the field is a complex number, with amplitude and phase, written as psi = A exp(-i phi).

You ask, "Is there a one-to-one correspondence between electrical waves and quantum waves?" No, but there is an analogy.

Electrical (electromagnetic) waves are real valued -- that is, at each point in space, the field has a magnitude (a floating point number) and a direction (a vector with three real-number components). The "phase" is a relative property of the EM wave as a whole -- where the magnitude is a maximum, we can define the phase to be zero, and where the magnitude is zero, we define the phase to be (+/-)pi/2.

With a QM wavefunction, each point of the field itself has a phase, and so the whole constructive/destructive interference business gets much more complicated. You can use EM waves, or water waves, to carray around some nice mental images, but they aren't identical to the reality.

I'm sorry this sounds so vague and obscure.