Quantum math: garbage in, garbage out?

This post is basically a continuation of my previous one but – as you can see from its title – it is much more aggressive in its language, as I was inspired by a very thoughtful comment on my previous post. Another advantage is that it avoids all of the math. 🙂 It’s… Well… I admit it: it’s just a rant. 🙂 [Those who wouldn’t appreciate the casual style of what follows, can download my paper on it – but that’s much longer and also has a lot more math in it – so it’s a much harder read than this ‘rant’.]

My previous post was actually triggered by an attempt to re-read Feynman’s Lectures on Quantum Mechanics, but in reverse order this time: from the last chapter to the first. [In case you doubt, I did follow the correct logical order when working my way through them for the first time because… Well… There is no other way to get through them otherwise. 🙂 ] But then I was looking at Chapter 20. It’s a Lecture on quantum-mechanical operators – so that’s a topic which, in other textbooks, is usually tackled earlier on. When re-reading it, I realize why people quickly turn away from the topic of physics: it’s a lot of mathematical formulas which are supposed to reflect reality but, in practice, few – if any – of the mathematical concepts are actually being explained. Not in the first chapters of a textbook, not in its middle ones, and… Well… Nowhere, really. Why? Well… To be blunt: I think most physicists themselves don’t really understand what they’re talking about. In fact, as I have pointed out a couple of times already, Feynman himself admits so much:

“Atomic behavior appears peculiar and mysterious to everyone—both to the novice and to the experienced physicist. Even the experts do not understand it the way they would like to.”

So… Well… If you’d be in need of a rather spectacular acknowledgement of the shortcomings of physics as a science, here you have it: if you don’t understand what physicists are trying to tell you, don’t worry about it, because they don’t really understand it themselves. 🙂

Take the example of a physical state, which is represented by a state vector, which we can combine and re-combine using the properties of an abstract Hilbert space. Frankly, I think the word is very misleading, because it actually doesn’t describe an actual physical state. Why? Well… If we look at this so-called physical state from another angle, then we need to transform it using a complicated set of transformation matrices. You’ll say: that’s what we need to do when going from one reference frame to another in classical mechanics as well, isn’t it?

Well… No. In classical mechanics, we’ll describe the physics using geometric vectors in three dimensions and, therefore, the base of our reference frame doesn’t matter: because we’re using real vectors (such as the electric of magnetic field vectors E and B), our orientation vis-á-vis the object – the line of sight, so to speak – doesn’t matter.

In contrast, in quantum mechanics, it does: Schrödinger’s equation – and the wavefunction – has only two degrees of freedom, so to speak: its so-called real and its imaginary dimension. Worse, physicists refuse to give those two dimensions any geometric interpretation. Why? I don’t know. As I show in my previous posts, it would be easy enough, right? We know both dimensions must be perpendicular to each other, so we just need to decide if both of them are going to be perpendicular to our line of sight. That’s it. We’ve only got two possibilities here which – in my humble view – explain why the matter-wave is different from an electromagnetic wave.

I actually can’t quite believe the craziness when it comes to interpreting the wavefunction: we get everything we’d want to know about our particle through these operators (momentum, energy, position, and whatever else you’d need to know), but mainstream physicists still tell us that the wavefunction is, somehow, not representing anything real. It might be because of that weird 720° symmetry – which, as far as I am concerned, confirms that those state vectors are not the right approach: you can’t represent a complex, asymmetrical shape by a ‘flat’ mathematical object!

Huh? Yes. The wavefunction is a ‘flat’ concept: it has two dimensions only, unlike the real vectors physicists use to describe electromagnetic waves (which we may interpret as the wavefunction of the photon). Those have three dimensions, just like the mathematical space we project on events. Because the wavefunction is flat (think of a rotating disk), we have those cumbersome transformation matrices: each time we shift position vis-á-vis the object we’re looking at (das Ding an sich, as Kant would call it), we need to change our description of it. And our description of it – the wavefunction – is all we have, so that’s our reality. However, because that reality changes as per our line of sight, physicists keep saying the wavefunction (or das Ding an sich itself) is, somehow, not real.

Frankly, I do think physicists should take a basic philosophy course: you can’t describe what goes on in three-dimensional space if you’re going to use flat (two-dimensional) concepts, because the objects we’re trying to describe (e.g. non-symmetrical electron orbitals) aren’t flat. Let me quote one of Feynman’s famous lines on philosophers: “These philosophers are always with us, struggling in the periphery to try to tell us something, but they never really understand the subtleties and depth of the problem.” (Feynman’s Lectures, Vol. I, Chapter 16)

Now, I love Feynman’s Lectures but… Well… I’ve gone through them a couple of times now, so I do think I have an appreciation of the subtleties and depth of the problem now. And I tend to agree with some of the smarter philosophers: if you’re going to use ‘flat’ mathematical objects to describe three- or four-dimensional reality, then such approach will only get you where we are right now, and that’s a lot of mathematical mumbo-jumbo for the poor uninitiated. Consistent mumbo-jumbo, for sure, but mumbo-jumbo nevertheless. 🙂 So, yes, I do think we need to re-invent quantum math. 🙂 The description may look more complicated, but it would make more sense.

I mean… If physicists themselves have had continued discussions on the reality of the wavefunction for almost a hundred years now (Schrödinger published his equation in 1926), then… Well… Then the physicists have a problem. Not the philosophers. 🙂 As to how that new description might look like, see my papers on viXra.org. I firmly believe it can be done. This is just a hobby of mine, but… Well… That’s where my attention will go over the coming years. 🙂 Perhaps quaternions are the answer but… Well… I don’t think so either – for reasons I’ll explain later. 🙂

Post scriptum: There are many nice videos on Dirac’s belt trick or, more generally, on 720° symmetries, but this links to one I particularly like. It clearly shows that the 720° symmetry requires, in effect, a special relation between the observer and the object that is being observed. It is, effectively, like there is a leather belt between them or, in this case, we have an arm between the glass and the person who is holding the glass. So it’s not like we are walking around the object (think of the glass of water) and making a full turn around it, so as to get back to where we were. No. We are turning it around by 360°! That’s a very different thing than just looking at it, walking around it, and then looking at it again. That explains the 720° symmetry: we need to turn it around twice to get it back to its original state. So… Well… The description is more about us and what we do with the object than about the object itself. That’s why I think the quantum-mechanical description is defective.

8 thoughts on “Quantum math: garbage in, garbage out?

  1. In considering “line of sight” that acts as the third axis, this is considered the z-axis and the direction of propagation, as you probably know. However, I too have found the asymmetry of orbitals to be bizarre. When solving for rotational motion, we work around the z-axis again, which seems arbitrary unless we consider the z-axis again as the direction of propagation. However, I think this means we can conclude that the atom itself cannot STOP moving, otherwise the motion of the electron around the nucleus becomes erratic. (After all, the electron is or is acting along a wave in one direction.) Instead, the orbitals themselves exist in such asymmetric forms because the atom is traveling along the z-axis. If we work with this assumption, then this may actually make solving the three-particle-problem (proton-electron-electron, p-p-e, e-e-e, etc.) (slightly) easier. After all, the xyz-axis would not be arbitrary NOR could the quantum solutions of each electron around the proton be arbitrary because both electrons have to rotate with motion in respect to the z-axis. In some ways, this makes me think that the rotation of the electrons around the proton can again be treated as “two-dimensional” problems, though admittedly, we would need an integral (from calculus) to create the full “stack” of solutions. What do I mean? Consider this: Let each electron be rotating around the proton at a FIXED z-axis value. Then, the full “solution” is the “sum” of all of these solutions for each z-axis value of the electron. I make it sound simple, though admittedly, the math won’t be “easy” per se. However, a fixed z-axis value eliminates one degree of freedom.
    And I just thought of all this as I’m writing. Now I really want to try the math on this. (Of course, the math may not work out as I’d like, but we’ll see.) Thank you so much for your inspiration!

  2. I sat down this evening and punched out an equation for the potential energy to depend only on the distances of the electrons from the z-axis (which can be kept constant) and an angle theta that defines the offset angle (from the x-axis) of the primary free particle (electron) while keeping the secondary free particle (electron) static. I did this by forbidding motion of the electron along the z-axis. If I plugged my potential energy formula into the Schrödinger equation, I could use it to give me a 3D slice of the complete 5D solution (5-degrees of freedom because we can confine the secondary free particle to the x-axis, or more precisely, one plane of freedom (after all, it doesn’t matter where the x-axis is placed around the z-axis – only the relative distance between electrons matters for the xy-plane). (I misspoke in my previous comment by saying I needed an “integral from calculus”. What I was thinking was that my solutions were going to form slices of the whole picture.)
    Obviously, using only theta gives us one degree of freedom, which creates a… um… circle. The interesting part is the amplitudes at each point in the circle, which no doubt will not be the same as in a two-particle system. I would not expect the amplitude to be uniform, however, if the electron is truly acting as a wave (as in the two-particle system solution), then I would expect it to have a nice, complete oscillation around the center.
    Two have a complete 3D slice requires having two other degrees of freedom. I think the most useful thing to do (and in order to complete two “sets”, which I’ll mention in a moment) would be to allow the distances of the electrons from the z-axis change, but NOT the distance along the z-axis. The nice part about this is that, rather than using the complicated spherical coordinate curl operator, we only have to use the polar coordinate one (which is much more friendly). These three degrees of freedom (theta, e1 distance from z-axis, and e2 distance from z-axis) create a general solution (or a “set” of solutions) for a pair of z-axis values z1 and z2 (the distances of the electrons from the nucleus along the z-axis). The next general solution (set of solutions) is that of picking a fixed theta and distances from the z-axis value. Then we let the electron distances from the nucleus along the z-axis vary. In fact, some of these solutions are even symmetrical (as they should be)! There are only two cases: where both electrons are “above” (or both below) the nucleus and where each electron is on opposite sides of the nucleus.

    To get the true trajectory of the electron would require picking initial conditions and incrementing the electrons along their identified trajectories, as you would have to do for any sufficiently advanced problem. But that doesn’t give us nice pretty pictures. At least by obtaining slices, we can create graphs on the computer and combine them to create a more complete representation of these 5D orbitals.

    All of this treats the electrons as particles following harmonic oscillation. As I think about this, it seems to me that the wave function itself isn’t necessarily a “thing” so much as a law – that the particle, by its nature, must obey harmonic motion. This says something fundamental about the particle and the wave function because it means that the wave function is not some meaningless mathematical gimmick – it is the MOTION path of the electron, which is what it should be. At the same time, it doesn’t defy the interpretation that the squared value of the amplitude of the wave function is also the probability of finding the particle. However, rather than “God rolling the dice”, the electron is simply moving along that path. This makes quantum tunneling easier to understand: rather than the particle randomly “popping” out of its potential energy “well” (hole, ditch, whatever), it happens to be moving outside of it all along! Then, as it finds itself on the part of its path outside of the potential well, perhaps something collides with it or the nucleus moves away from it, wrecking its path just enough to where it finds itself escaping.

    And there you have it: Quantum tunneling from a deterministic perspective. It doesn’t seem like such a weird phenomenon to me now.

    Now let’s answer the question as to WHY the electron must obey the law of harmonic motion: The electron seems to be an oscillating wave of electromagnetic energy and mass, which is what we’re discussing here. In other words, the oscillations that form the wave function of the particle must “align” so that the wave doesn’t interfere with itself and self-destruct. The existence of the particle is the wave.

    That said, we can say that the particle does have both definite position AND velocity, despite the Heisenberg Uncertainty Principle (“HUP” for short). In fact, the HUP says something about the nature of the available SOLUTIONS but NOT about the nature of the particle. In other words, the HUP is misinterpreted, in my opinion. When we pick a position of the particle, the wave function equation spreads out over space. This isn’t wrong at all. In fact, this is what reality SHOULD say: it means that any SINGLE position in space can have ANY wave function. Hence, any and every wave is a valid solution! When we pick a velocity of the particle, the wave function extends to infinity again but allows us any point along that direction of propagation. Again, this is not wrong at all. It simply means that a particle of a particle velocity can be located anywhere along that wave. In fact, ALL positions along that wave are valid!

    Thus, my view of the electron now is that it is a point-wave that is obeying a its “space field”. Imagine if you were to create a 3D wave (2D is easier to imagine) extending over all of space and that this wave maintained complete uniformity throughout. That is, it looked no different anywhere. Let’s call this “wave-space”. Then pick a single point in wave-space. That’s a particle. Whenever that particle moves, it takes on the new wave amplitude of that position in wave-space. Interference with other particles makes it “pick” a new wave-space (not just a new wave-space position). Wave-space itself is a 3D electromagnetic field, wherein each point represents some amplitude of electricity and amplitude of magnetism.

    Naturally, then, this helps us make sense as to why photons, electrons, and protons interact, but not photons and, say, Higgs bosons (if they exist). Photons directly interfere with the very essence of the electron (a point in wave-space), causing it to move into a new wave-space. The resulting wave-space is still uniform (making the electron motion harmonic), but now the electron is traveling with a new velocity.

    Notably, I say “wave-space” as though it were its own “thing”, but I don’t mean that the space itself exists in its entirety alone and apart from the electron. The only “existing” part of wave-space is the point that the electron (or proton or photon) occupies. It’s simply a tool to represent the oscillation of the particle itself, which is an electromagnetic point oscillating like a wave as a moves through space.

    I’ll pause there for now. Thoughts?

    • Wow ! As mentioned in my other reply, we shouldn’t discuss by filling pages of comments. Let’s work on a little paper together for the Los Alamos e-print archives or… Well… If we can add enough academic references and all of the other garbage reviewers expect to see, perhaps for a more serious physics e-journal?

  3. Oh, and adding to that too: It should make sense now why an electron can interfere with itself in a single-slit diffraction experiment. The other particles forming the hole affect the wave-space that the electron can have, which ultimately affects its trajectory. In other words, the electron is “doomed” or “predestined” to follow along the harmonic path. The hole itself determines the wave-space, and the individual electron positions as they pass through the hole will determine where along that path they travel. And since there are multiple particles involved, the wave-space isn’t aligned like window blinds. Instead, as we see, the trajectories are spread out. Even still, an electron isn’t necessarily going to “fan out” (that is, just because it’s at the “top” of the slit doesn’t mean it’s going to be at the “top” of the detector screen). Instead, its final position may be determined by even the tiniest of digits in its angle as it enters the slit. The distance from the slit to the detector merely serves to amplify what “wave-space looks like” in an area the size of the slit. In other words, the single slit experiment acts like a kind of microphone, amplifying space. The smaller the slit, the smaller the sample of space. The larger the slit, the larger the sample of space, which, as it gets noisier, causes a loss of the diffraction pattern.

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