.

As a student of the history of science, I find the above quote to be of a very entertaining quality and especially applicable to this question and so many others that I run into on a daily basis. There’s an almost fractal effect to it that many would call “very meta.”

So I must ask what you mean by “derived from fundamental principles?” because it looks derived from fundamental principles to me.

IMO, One of the most central points of his paper was to call upon the fact that classical principles could be applied to the problem of quantum mechanics within a limit. He fairly dotes on the subject as though it is the very source of his reasoning.

In anticipation of an argument against any classical analogy, he goes on to state many of the problems with the view and continues:

At first sight it does not seem at all tempting, to work out in detail the Hamiltonian analogy as in real undulatory optics.

Then, in explaining that the problem lies in the size relationships in some cases where wavelength and the radii of orbits become similar. He goes on from there to state that it’s one of the most striking features of the new physics dealing with the quantum world that classical mechanics break down and that fact is in the forefront of all physicists’ minds at that time.

Taking into account this fact, which impresses its stamp upon all modern physical reasoning, is one not greatly tempted to investigate whether the non-applicability of ordinary mechanics to micro-mechanical problems is perhaps exactly the same kind as the non-applicability of geometrical optics to the phenomena of diffraction or interference and may, perhaps, be overcome in an exactly similar way?

He basically argues for the idea of scrapping a fully particulate nature to “particles” in favor of explaining all the strangeness in terms of wave behaviors. He points out that the Hamiltonian idea of a moving point can only be used in a situation where there is a large difference between the radii of curvature in a path and the wavelength. (the curvature is large compared to the wavelength)

Otherwise it must be studied as wave phenomena alone and the point particle concept discarded. IE for QM we must think of particles as waves which are spread out and non-localized.

However, this methodology is still classical in nature. The methodology of wave theory is well established at the time. Applying it to the problem at hand is the somewhat novel idea (of de Broglie) Schrodinger is expounding upon.

It is by turning a Hamiltonian system a bit on its head that he blends wave mechanics with more “ordinary” mechanics. Earlier in the paper he states:

Instead of thinking of the surfaces as fixed in space and letting the values of the constant wander from surface to surface, we may equally well think of a certain numerical value of W as attached to a certain individual surface and let the surfaces wander in such a way that each of them continually takes the place and exact form of the following one. Then the quantity u, given by Eq. (6) will denote the normal-velocity of any surface at any one of its points. Adopting this view we arrive at a picture which exactly coincides with the propagation of a stationary wave-system in an optically non-homogeneous (but isotropic) medium.

(Personally, it reminds me of the interchange of Lagrangian and Eulerian descriptions of flow.)

By using his system of surfaces as waves, he is able to deduce the necessity of a universal constant for determining amplitude. Then by plugging in Planck’s constant, he establishes the necessary relationship between energy and frequency in what he feels is an intuitive fashion.

This then leads to “wave packets” as a better description of particles.

The paper thus far is building up this comparison between “ordinary mechanics” and wave mechanics with the express purpose of showing how an orbit of a point particle which would not normally work when the radii of curvature is not large in comparison to wavelength, is actually approximately the same as a wave packet under the same influence of a center of gravity.

The important thing to note, however, is that the equations used are dispersive. There is a relationship between energy and propagation speed that leads to patterns of interference that give us the “virtual” phenomena of wave packets.

(This reasoning, as stated by Schrodinger, is from Louis de Broglie and therefore components of the derivation are wholly dependent upon de Broglie’s conceptualizations.)

The alignment of wave systems leads to “lumpy” effects that are second order and these effects are the base nature of a particle and therefore the reason the particle appears to be spread out over an area while also having the ability to localize.

As stated above this “motion as a whole” is governed by the laws of ordinary mechanics

It is important to note that this concept of superposition is in no way probabilistic but is so dependent upon subtle initial conditions and complex evolution as to require probability to reasonably handle mathematically. While there may not have been terminology at the time to describe it, this is known as deterministic chaos in recent times.

This leads us towards spherical harmonics.

Throughout the bulk of the rest of the paper he goes on describing how spherical harmonics provide an excellent physical understanding of quantum physics while also supplying the azimuthla and radial quantum numbers..

…by assuming [psi] to be a product of a function of the polar angles and a function of the radius r only.

…and he sums this discussion, somewhat in the middle of it, by saying:

So the whole of the wave-phenomenon, though mathematically spreading throughout all space, is essentially restricted to a small sphere of a few Angstroms diameter which may be called “the atom” according to undulatory mechanics. Any one of the above mentioned solutions (consisting of a product of spherical surface-harmonic and a function of r only) greatly resembles a fundamental vibration of an elastic sphere with a finite number of (1) spheres, (2) cones, (3) planes as “node surfaces.”

Which leads us to believe that the derivation may not have come directly from spherical harmonics but its agreement with it strengthened his probing into the subject strongly enough to provide a sort of retroactive continuity. This sort of back-propagation of confirmatory information can sometimes play a large role in the development of complex discoveries but the non-linear nature of the process sometimes doesn’t provide a simple satisfying narrative.

While he speaks on the boundary condition problem later in the paper, there is some apparently ad hoc reasoning calling upon requirements of the math alone that does little to satisfy, but he does appear to reference one of my favorite topics that goes by many names from “Huygens’ synchronization” to “parametric resonance.” I do not think these discussion topics, however, reveal the processes of derivation from first principles.

They too seem to be a refinement which comes after the fact.

So derivation from first principles looks entirely possible…

But does not seem to be the actual sequence of events for Schrodinger who apparently locked upon de Broglie’s thesis along with a strong belief that a mechanical reasoning could be found to explain how ordinary mechanics seemed to provide an approximation.

This led to expounding upon de Broglie’s work and hitting upon various conveniences and truths that proved the value of the direction he was taking.

The most striking of which was the analogy of Bessel functions and spherical harmonics leading to exactly the right numbers.

Spherical Harmonics:

Schrodinger Equation:

It is important to note that “Schrodinger’s cat” was a purposeful mockery of ideas which held that fuzzy probabilistic features underlay reality rather than simply complex mechanics that require probability to mathematically be dealt with.

One can even set up quite ridiculous cases. A cat is penned up in a steel chamber, along with the following device (which must be secured against direct interference by the cat): in a Geiger counter, there is a tiny bit of radioactive substance, so small, that perhaps in the course of the hour one of the atoms decays, but also, with equal probability, perhaps none; if it happens, the counter tube discharges and through a relay releases a hammer that shatters a small flask of hydrocyanic acid. If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has decayed. The first atomic decay would have poisoned it. The psi-function of the entire system would express this by having in it the living and dead cat (pardon the expression) mixed or smeared out in equal parts.

It is typical of these cases that an indeterminacy originally restricted to the atomic domain becomes transformed into macroscopic indeterminacy, which can then be resolved by direct observation. That prevents us from so naively accepting as valid a "blurred model" for representing reality. In itself, it would not embody anything unclear or contradictory. There is a difference between a shaky or out-of-focus photograph and a snapshot of clouds and fog banks.

(Emphasis added is my own)

Creation time: Oct 03, 2017 12:45 AM PDT

Excellent question in the context of a string!

The first thing to think of is the idea that a wave is just a tension or stress introduced in a smaller area or group of particles within a larger group of particles. This tension is out of balance or equilibrium so there is a natural seeking of least resistance that occurs. Imagine a crowd of germophobes in an enclosed arena who are very good at making their personal space maximized at all times. Stick another one into the arena and those next to him will adjust because the space around them is no longer equal. After they adjust, the next layer of people will no longer have equal space so they too will adjust and the wave then spreads out from the disturbance of the equilibrium.

“Wait, then what is the downstroke?” you might ask. There’s a combination of momentum and back-propagating adjustment involved. If the new guy in the analogy above ran in quickly then he’s going to have to be slowed down and back up because too much open space will be left behind him. Even if he did not go in quickly, if the reaction of the next person (or particle) is less than instant, then there will be a delay to deal with. So when the first person reaches his perfect middle, he’s messed up the next guy in the line and that guy has to adjust. Once he’s adjusted then the guy who just made him adjust has to adjust again and this goes back and forth in ever smaller parts.

An upstroke is people packed too tightly and a downstroke is them packed too loosely. What is too tight or too loose? Nobody knows until they’ve all adjusted repeatedly and the average is achieved.

When downstroke and upstroke meet the average of too much and too little averages out but there is still momentum to deal with so the disturbance continues to carry through after the apparent happy medium is reached for a moment.

“Hmm, that doesn’t go with a string very well at all…”

Yes, but it’s important to understand the distribution of stresses in a medium and the above analogy is for longitudinal waves but those on the string are transverse ones.

Now what we have is a partial rotation. Think about standing alongside the stretched string and pinching it between two fingers. If you then rotate your hand along the axis towards the ends of the string then the shape you have created looks like those waves on the string, don’t they? So when you have moved the end of the string up and down you’ve induced a rotation and that rotation represents a difference in the tension of the string in that area.

As you noticed in the above analogy, there’s the local effect of spacing or tension and then there’s the global effect of the average they have to eventually hit and each of these interact with and affect the other.

As one wave of rotation collides with the other, the local tension may be even greater via increased rotation (the string is actually stretched a little more in that little area) or it may be decreased down to the apparent average but the overall “average” is still not met and there is also still momentum of the particles as well so the wave continues.

So the temporary annihilation is a local effect that simply looks to us like the wave is removed but the stresses are simply hidden as they transform into other types of stress or motion. Additionally, the global disturbance is not removed because all particles have not reached equilibrium as a larger group.

Therefore the destruction of the wave is an illusion.

Creation time: May 22, 2018 03:53 PM PDT

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There are literally too many inventions based on wave mechanics to enumerate. Most of modern technology depends on our understandings of waves in general.

However, it’s crucial to understand that our grasp of wave mechanics is a general set of principles that are separate to those sets of information related directly to the difference between sound and light which is defined by special relativity.

So, what about just the difference between light and sound, what is that?

According to SR light propagates without a medium and is therefore unlike any wave in the universe. The entire idea of a wave is an action like a “car-ride.” or a “jog” yet we cannot have a care ride without a car and we cannot have a jog without a jogger. A wave, before Einstein, was understood as what occurs to an elastic medium when the forces are unequal and it therefore seeks equilibrium and over-corrects. Cars in traffic move in waves and so do crowds of people.

All these waves are mechanical waves and are therefore separate from electromagnetism according to Einstein.

So light, after special relativity, has the special property called “constancy” which allows it to basically shortcut through 3D reality using the 4th dimension which Einstein and Minkowski re-interpreted from Lorentz’s work. Lorentz created a system of math to track optical illusions and therefore had ambiguous data stored in a 4th dimension, mathematically.

Einstein didn’t change that math whatsoever, he just reinterpreted its meaning and therefore utterly changed our concept of light.

Now, even though it complies with literally every single other property and behavior of physical waves, we believe it has no “jogger” to jog and can cut through a new version of reality that Einstein described. Same math, different interpretation.

Most people do not remember this crucial division and historical fact that there was an aether theory that was mathematically equivalent to special relativity that special relativity was based upon. They do not understand the dividing line between mechanical and non-mechanical interpretations lies upon the concept of constancy.

Understanding this historical fact is CRITICAL to understanding this post.

So has this new understanding of reality and the difference between mechanical waves and electromagnetic ones yielded benefits? No!

The first thing people are going to try to bring up is the GPS system. It’s a good argument, not because it’s accurate, but because addressing it is difficult and technical. It’s also a red herring, so we’re multiple layers of division from the point but I have to address it since others will bring it up.

First, remember that Einstein was using the same math so the difference Einstein actually added (specifically constancy and only constancy) is very very subtle.

The GPS system how two major effects going on.

1. The differences created by relativistic effects is tiny and atomic clocks have errors.

2. Doppler effect is the largest effect.

So, even though there are relativistic effects that would occur under even Lorentz’s theory of mechanical electromagnetism, the GPS system has re-synchronization of clocks on a daily basis because the error level is greater than the relativistic corrections.

The second effect is often called relativistic doppler effect and is therefore somehow considered a relativistic effect, when in fact, in this context it is no different from the classical doppler effect found in mechanical waves. (though there are some differences in mediums that create longitudinal vs transverse waves)

This should not be considered a relativistic correction simply because we call it “relativistic doppler.” The difference actually added by relativity in this context is either nonexistent or negligible. (and the same would be true of the mechanical relativistic aether theory)

IE: GPS, by happenstance, is not dependent upon relativity in any way.

Secondary to this, it’s especially not dependent on constancy, even though it should be dependent upon certain relativistic effects if it weren’t for clock error. Those relativistic effects however, are NOT based in constancy and are not separable therefore from a mechanical wave theory. (no dimensional shortcutting magic mediumless waves needed)

“What about other technologies? I’ve heard there’s tons of inventions based on relativity!”

What exactly is relativity?

Unfortunately we call both Lorentz’s kinematics and Einstein’s electromagnetism “special relativity” and this leads to confusion since, time dilation, length contraction and the appearance of light speed constancy all came from Lorentz before Einstein. This is why it used to specifically be called Lorentz-Einstein relativity. Poincare, coined the phrase relativity and it is not just separate from constancy, Einstein later abandoned the idea of constancy during the development of general relativity.

"This is, in my opinion, not the limit of validity of the principle of relativity, but is that of the constancy of the velocity of light, and thus of our current theory of relativity." - Einstein (in reply to Abraham on the accusation that relativity fails because Einstein abandoned constancy)

Then there’s general relativity and it’s quite a different thing more related to gravity and we’re talking about light, specifically.

So, yes, there are technologies that use “relativity” but that’s glossing over the real issue.

The issue we’re addressing here is if our new understanding of light, constancy, which is codified in Minkowski’s convention, and the necessity of a new dimension and the behavior of light cutting through 3D reality, is something that has led to new technology. That answer is no

Since Poincare’s electromagnetism combined with Lorentz’s kinematics is an aether theory mathematically indistinguishable from relativity, the question only comes down to whether there is a mechanical medium or not. This is the one fundamental difference in our understanding. All the other principles of the theory that are used for invention don’t count as “special relativity” really.

The factor of change is not Einstein’s nor are most of the very revolutionary effects of “special relativity.” Those all belong to Larmor and Lorentz. That’s just history that nobody disputes.

It’s the new understanding of light and the new properties of the universe (4th dimension) required to uphold that new viewpoint that we are looking at.

That new “understanding” has gained us literally nothing. Not one single thing.

If you’d like to understand more of the historical context of this close association of classical aether theory with modern theory, how it shaped modern theory and the crossovers, then this is the paper for you:

[1804.01846] History of the NeoClassical Interpretation of Quantum and Relativistic Physics

Creation time: Aug 08, 2018 06:20 AM PDT

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Nobody else here is going to give you an affirmative, so I’m going to just to spite them. YES! (Hang on to your hat! The ride is going to be bumpy)

Imagine for a moment that Lorentz and Poincare were right in their interpretation of relativity. There is a preferred frame and constancy is an illusion caused by relativity.

This means that there is an aether and therefore if one understands the idea of particle formation that was developing through Lord Kelvin and JJ Thomson, then one understands that atoms are simply complex vortical knots in the substance.

If a particle can move through the substance like a vortex moves through its medium then there are some considerations of center of rotation that change as we move with respect to a flowing medium.

Where am I going with all this? The answer is wave-vortex duality.

Think about it for a moment. If you look at a simple sine wave pass by, the particles just make an up and down motion, (or just back and forth with longitudinal waves) but if you look at more complex waves, you start to get circles.

So if you are moving along with a point particle undergoing perturbation by waves then you’ll see a vortex in one place. If you’re moving at a significant speed with respect to it however then it’s just a wave.

Wave-vortex duality is the concept that waves pump vortices and vortices give off waves. They are somewhat interchangeable phenomena which can exchange energy.

So in the end, if there was an aether, then because of preferred frame mechanics, the interference patterns of some waves (especially counter-propagating ones) would make standing waves that were dependent upon your motion through the medium.

Wave-particle duality and wave-vortex duality would be the same thing.

Whether or not you say a standing wave vortex (a particle of matter) or just a wave (just electromagnetic phenomena) would be dependent upon your speed through that medium.

The speed of light, would, therefore act somewhat like the visible light spectrum in that it would represent an octave of phenomena in a range. That which was matter to you would be relative to its speed and that which was just waves would be those things beyond your “visual spectrum.”

It’s sort of like how we have an audio range. We can’s hear things of a given frequency, but if we moved fast enough with respect to the medium, then doppler effect would raise or lower the frequency to make it audible.

For this model I’m describing, your speed with respect to standing wave patterns that make up particles would be subject to doppler shift and therefore a transition point at each end of the spectrum would occur. As you moved, from your frame some matter would become more energetic and some would have less. Some matter would cross a transition point and some energy would. Just like a full range of sound would gain some audibility at one end and lose some at the other as you moved faster through the medium.

The hardest part to understand here is just the counter-propagation effects when it comes to creating standing waves. (which we see as vortex “particles”)

So yes, if Lorentz and Poincare were right, (and their theory actually is mathematically equivalent incidentally) the speed of light could possibly define a range of motions of the aether that determine whether or not something appears to be matter or energy to a moving observer.

This speed is your range of matter which is like an audible/visible spectrum.

The universe you experience would be determined by that range but just like insects see ultraviolet patterns on flowers that we do not, it doesn’t mean that it’s the only range that can be considered.

THIS IDEA, (even staying within its mythos) IS NOT WITHOUT POSSIBLY GIANT GAPING HOLES.

For instance, the speed of a wave in a medium is always the same because it is a property of that medium. Light wouldn’t even propagate in the forward direction once you reached light speed in a given direction. How would that work?

There’s still some miniscule possibility that there are enormously faster longitudinal waves in the universe but I’m not holding my breath.

Sure a universe with an aether would almost certainly have currents moving at massive speed differences so this would mean that things like stars would have their own speed bubbles around them of rotating aether like the heliopause, that represent local currents and therefore we’d have lots of weird border transitions to deal with.

Heck, even moving our particles (vortices) through a medium would involve a lot of weirdness with local currents and transition between vortical speeds and the surrounding media.

There’s a lot to consider to really determine if what I’ve described holds up to more scrutiny.

The point is that, some people just aren’t looking at the “more unlikely” propositions when they say no. Our idea of what is likely and what is not is just a form of bias that could easily be fundamentally flawed.

There are still many things that are not impossible, but simply seem very unlikely. (and something I’ve said above might actually be within the impossible range because I haven’t looked closely enough)

I can guarantee you that not one singe other person who reads this has EVER thought of this possibility. I’d wager it’s never been suggested in the history of mankind.

But the only way to find revolutionary new ideas is to explore the unlikely. Don’t lose your wonder!

If you’d like to learn more about the validity of aether theory in a modern context, read this paper:

[1804.01846] History of the NeoClassical Interpretation of Quantum and Relativistic Physics

Creation time: Aug 10, 2018 05:07 PM PDT

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Simpler yes, more broadly applicable no.

The most popular answer so far relays the consensus view that reality is just math with no need for mechanics. I regularly call this magical thinking, because that’s just what it is.

If we simply believe a description is an explanation, we are regressing back toward the caves. We are re-embracing faith about bottom level reality and it’s a very troubling trend that really started with Dirac as far as I can tell.

The map is NOT the territory.

Math is a tool of description that closely matches mechanics but it isn’t mechanics, it’s a description of it. Unfortunately there were decisions made out of pure desperation and “throwing our hands up” that resulted in reverting to faith.

“A widely accepted school of thought maintains that an objective picture of reality -in any traditional meaning of that term- cannot exist at all. Only the optimists among us (and I consider myself one of them) look upon this view as a philosophical extravagance born of despair in the face of a grave crisis. We hope that the fluctuations of concepts and opinions only indicate a violent process of transformtion which in the end will lead to something better thn the mess of formulas that to-day surrounds the subject.” - Schrödinger from "Our Conception of Matter." [reprinted in "What is Life and Other Scientific Essays (1956)]

That faith seems well placed in mathematics because it is so reliable, but unfortunately math is useless without the rational map between the symbols and that which is symbolized. We feel something like 2+2=4 is pure truth but it is not. It’s a symbol that is undefined. It’s a pointer without what is pointed to.

If it’s 2 apples and 2 apples then sure, we have 4 apples. What if, however, someone mistook oranges for additional apples, then the mathematical “truth” that we have 4 apples is a falsehood. This dependence of the link between the symbol and what is symbolized must NOT be ignored. This connective tissue between the two is absolutely crucial. A symbolic procedure must be proven to match the thing it symbolizes.

You can come to a rational conclusion and perform the operation correctly but if your mapping is wrong then you are wrong, period.

“This result is too beautiful to be false; it is more important to have beauty in one's equations than to have them fit experiment.” - Paul Dirac, The evolution of the Physicist's Picture of Nature Scientific American 208 (5) (1963)

“I learned to distrust all physical concepts as the basis for a theory. Instead one should put one's trust in a mathematical scheme, even if the scheme does not appear at first sight to be connected with physics. One should concentrate on getting interesting mathematics.” - Paul Dirac

Unfortunately that desperation led to a cultural shift that has directly damaged the progress of science. We now culturally believe that if something is reliable like mathematics that we’re not thinking magically but that’s patently untrue and it is the same mentality that has validated the schizophrenic ravings of numerologists for thousands of years now.

One can think quite magically about reality even with lots of perfect and useful quantification and mathematical rigor riding on top of the faith.

Consider the idea of spontaneous generation from long ago. If we were able to quantify exactly the meat-to-flies conversion ratio in precise mathematical terms, and accurately predict the speed of the meat conversion to flies based upon the weight of the meat, then we’ve contrived a precise system upon the top of what is basically a magical ideation about the nature of reality.

We can fool ourselves that our idea isn’t magical because of all the rigor constructed on top of it but the base is still magical thinking.

“Aren’t consistent results good enough?”