Is the Schrödinger Equation True?
scientificamerican.com
scientificamerican.com
People hold quantum mechanics to some ridiculous standard, far higher than any other physical theory. Did they really expect that the behaviour of the very small would be exactly like that of everyday macroscopic objects? Do they enjoy the mysticism of pretending that it's complex and hard to understand when it's really nothing of the sort? Frankly I think an honest approach to QM would present it as something banal and obvious.
> In the coming sequence on quantum mechanics, I am going to consistently speak as if quantum mechanics is perfectly normal; and when human intuitions depart from quantum mechanics, I am going to make fun of the intuitions for being weird and unusual.
Thanks for linking to this! Absolutely agree with his gripes in the first article about the traditional historically-oriented introduction to QM confusing students.
The rest of the articles (still reading them) appear to be pretty much the route taken by Feynman's _QED_ book (which, of course, is fantastic). It's also very interesting how his 1960's Feynman Lectures books seem to tell the student "QM is confusing and weird and hard and scary" roughly every ten pages or so, whereas by the 1980's with _QED_ he dropped the fearmongering language.
Off-topic but that's also my problem with any introduction to philosophy I've seen. This old Greek said something. This other old Greek thought something else. Some German philosopher said still something else.
Maybe that's what philosophy is all about, in which case I'm perfectly happy not to learn anything about it. But I have the feeling there really is something interesting to be found, if only the books would talk about that instead of painstakingly reconstructing the whole history of the field.
If your complaint was actually about it just being old greek guys and nobody from other places then I can just agree, it's really just western philosophy that's taught under the name "Philosophy"
That's a little bit of a stretch, don't you think?
I mean, here are questions about the SE I've so far never received simple, intuitive answers to:
- Why do we need complex numbers in the SE, what do they bring to the table and how does that tie in with physical intuition?
- How was the equation derived, intuitively?
- The laplacian in there feels like it's a "diffusion" term (as in the heat equation, things tend to spread and smoothen over time). Why isn't there ever a intuitive explanation in text books about this?
- [edit]: Many book trumpet that classical mechanics is just an approximation of QM, but unlike in special relativity where speed going to zero obviously gets you back to classical mechanics, I've never seen a clear example of a system where the quantum behavior "transitions" to classical behavior as a variable (e.g size) increases or decreases.
- the list is longThe complex numbers are the smallest algebraically closed field, they're the natural place for doing algebra.
> - How was the equation derived, intuitively?
I don't know if this how it was derived, but the way I see it it's pretty much the "square root" of the standard wave equation. So if you're trying to describe something where the solutions are mostly waves but you want to also include some other kinds of solutions, it's a very natural thing to try.
> - The laplacian in there feels like it's a "diffusion" term (as in the heat equation, things tend to spread and smoothen over time). Why isn't there ever a intuitive explanation in text books about this?
It's pure imaginary, so rather than a diffusion it's a rotation (phase change). So things tend to behave as waves with a frequency proportional to their energy.
> - [edit]: Many book trumpet that classical mechanics is just an approximation of QM, but unlike in special relativity where speed going to zero obviously gets you back to classical mechanics, I've never seen a clear example of a system where the quantum behavior "transitions" to classical behavior as a variable (e.g size) increases or decreases.
You tend to recover classical behaviour as energy and momentum go to infinity (so frequency goes to infinity and wavelength goes to zero). E.g. if you look at a double-slit interference pattern, as you increase the frequency/decrease the wavelength the classical bell curve becomes more and more visible.
In an equation like
df/dt = k Δf
When k is an imaginary number, it is not like diffusion o anti-diffusion (I don't know the technical name). The associated operator is unitary and the behavior is different.
Many books about the heat/diffusion equations and Schrodinger equation have this discussion. And the generalized versions where k is a functions that depends on x or t, or is a complex number instead of a real or complex number.
For an intuitive idea of what is happening, the idea is that you consider at time=0 the function
f(x,t=0)=A sin(ωx).
If k is real and negative, the amplitude of the function increase with time, and you have the opposite effect.
If k is imaginary, the function just rotate in the complex plane but does not increase or decrease the amplitude. So you get a wavy effect instead of a diffusion effect.
The initial equation is linear, i.e. there is no f^2 or something like that. With some hand waving you can extend the simple cases where f is sin or cos to more complex cases. The technical details are in https://en.wikipedia.org/wiki/Fourier_transform
I expect a theory predicting otherwise to bear the burden of demarcating exactly where the boundary between "macroscopic" and "weird not-macroscopic" lies (waving your hands and mumbling "thermodynamics" does not count). I expect that such a theory's proponents should be absolutely burning with interest in probing, studying and understanding what goes on at this peculiarly discontinuous boundary in Nature.
Those two things haven't happened, which is kind of sad.
Quantum Mechanics predicts the results of experiments better than anything else. If "predicts experimental results" is the only thing you care about, then Bohr was right, QM is complete, and there's nothing more to be written in that book.
I don't think he's claimed that the boundary was discontinuous, you've implied that.
However, there's no denying that QM and classical mechanics are two fairly different sets of equations.
I haven't come across a QM book so far that explicitly (i.e. with an example) demonstrates how one morphs into the other as a variable, say size, changes.
Special relativity is way better in that regard, when speed goes to zero, the equations converge to their classical analogues.
Ehrenfest's theorem would be an example where Newton's 2nd law is recovered as the wavefunction's spatial extent is reduced on the scale of the system being studied. More generally, you won't get an explanation that shows a morphing based on size because size is not the limiting factor. It's how isolated your system can be from the outside world. This is obviously much much harder as your system gets bigger. Decoherence theory (https://en.wikipedia.org/wiki/Quantum_decoherence) recovers classical probabilities as a system interacts with its environment.
I too find it unlikely that there is such a discontinuity. Quite a large number of physicists (and quantum computing people) seem to speak as if there is one.
> Finally, decoherence has seen plenty of study
I get the impression that decoherence is treated as merely a problem for technicians working on experimental apparatus, or an engineering issue to be dealt with on the road to quantum computation. Are there any results on fundamental limits to the degree of entanglement in a system? We've been trying to get that damn cat into superposition for ~85 years now, with no luck.
We've been setting records for bigger and bigger superpositions every few years - apparently we're up to 2000 atoms now. There's no reason to suspect there's a fundamental limit, any more than there's a fundamental problem that prevents nuclear fusion from working. That doesn't mean the engineering isn't devilishly hard, maybe even impractical (if it gets literally exponentially harder to maintain more and more particles in superposition), but it's not a fundamental physics issue.
The reason they seem to speak that way is because they are being a little sloppy for the sake of brevity. The physicists all understand that there is a blurry boundary.
There is an amusing instance where there was a similar confusion though: the soft diffraction limit was mistaken by many for a hard one. So super-resulution came as a bit of a shock.
My feelings exactly, I really don't understand why such a relevant comment is being downvoted.
Yes, I was disappointed to read this as well. I suspect what he meant is that we can't derive closed form solutions for the Schrödinger equation in the case of the Helium atom.
But the equation can perfectly be integrated numerically ... in these days of rather powerful computers, closed-form solutions to diffeqs aren't as important as they used to. As a matter of fact, for most problems in QM, I suspect there's just no other way.
One step further, when reading QM books, there seem to be a strong tendency to try and assign "physical meaning" to purely mathematical techniques used to find closed form solution to diffeqs (eg separation of variables). I am not convinced that it is a fruitful endeavor.
Another reason he may be saying this could also be because with the Schrödinger equation for the Helium atom, assuming it isn't separable along space variables, the solution ψ is a map from the configuration space, (i.e. ℝ6) to ℂ, something that's hard and annoying to visualize, whereas for the Hydrogen atom, you can draw pretty orbital probability density pictures in ℝ3, something everyone can "understand".
You better not say that when any solitons are floating by...
Is it though? Or have we just found a linear approximation to actual reality?
Also, the other day, I read in another HN thread that there are situations (Bose Einstein condensates) where the potential may depend on Psi thus making the equation non-linear (since the potential multiplies Psi).
The basics of quantum mechanics are:
1. Any system can be described in a linear "state space", whose basis vectors (roughly speaking) are each possible arrangement of the things being described (every thing's location). However, all possible states include (complex) linear combinations of these. This is usually described as the "wave function", you can plug in any value for coördinates of all your things and get a number back out.
2. States in a closed system evolve according to the (time-dependent) Schrödinger equation, -i h-bar d/dt (state) = E(state). This is sometimes phrased as "energy is the generator of time translation" (note that this phrasing holds true in the Hamiltonion formulation of classical mechanics too, but with different, related meanings). Actually calculating this can be difficult, and you might need to do various tricks such as decomposing in a convenient basis.
No all-encompasing energy field binding and penetrating us needed, though the equation is non-local in the sense that it could depend on the entire world. (In fact, it's quasi-local in practice; energy is usually just a function of where things are (local), and how different that coefficient is on infinitesimally adjacent states (i.e. spacial derivatives of the wavefunction). But this does let changes propagate at any speed, including faster than light. Basic QM is not Einsteinian relativistic. It is perfectly Galilean relativistic in a very nice way though.)
3. If you measure a system, you're going to get results that are probabilistic with the probabilities proportional to the (absolute value of) the square of this wavefunction. After that, the state is reset to whay you measured, and evolution will proceed according to 2.
It comes in when you treat the measurement apparatus as separate from the thing being measured. When you do that, collapse flows naturally from that assumption. It's a valid approach, and very useful, but many people find it philosophically awkward.
An approach that's philosophically easier, but less pragmatic, is to treat the masurement apparatus as part of the system. Now you've got one single enormous quantum system. The math of that is far too complex to actually run, and more importantly, too enormous to hold itself in an unstable equilibrium. It must fall into a more stable equilibrium.
The tricky part is when there's more than one stable state. The measurement apparatus exists in both of those states, and returns opposite results. The states are in local equilibria, and don't exchange significant amounts of state with each other.
You can call that "splitting universes", if you want, though I think that's needlessly messianic. Like the first one, it's perfectly valid. And pragmatically it acts just like collapse. There's a philosophical question about whether the other states "actually exist but can't be reached" or "don't exist at all since we can't prove it except by inference from the fact that this one exists and follows the Schroedinger Equation so closely". That's "philosophical" in the sense that there is no physical difference between the two, but just one of your mental model.
That's a lot of words; sorry about that. But the point is that you don't need a collapse axiom, and if you choose to add one, it's not strange.
True enough, but what else could you expect the experience of being in the Everettian universe to look like? I think it's fair to say that experimental results are at least compatible with reality following the Schroedinger equation without needing a collapse postulate.
Another measure of naturalness is "simplicity", and the Schrödinger equation is a fairly simple expression.
You might argue that it's results don't feel very intuitive, and therefore it is not natural. But then I why you think its intuitive that levers work.
This doesn't preclude there being a better model of course. Observations can get more sophisticated, revealing that your model needed adjustment. Or parsimony can drive and someone finds a way to describe multiple phenomena at once, eg electromagnetism.
As for why mathematics is great at describing nature, it's because there's loads of it, since it's an extensible system of logic. Someone somewhere is going to find a way of describing how asteroids go round, and being a kind of logic there's no way you would say it wasn't math, regardless of what exactly it was. Epicycles are math too, despite not being the exact right way to think about it.
Quarks are an example. Quarks were conceptualized as a mathematical simplification, and only later were recognized as real physical things - or at least as real as protons.
In QM we have physical quantities (an electron's mass) and unphysical quantities (an electron's phase). But there's open questions, like the wavefunction - can that be placed in correspondence with reality? We don't really know. This is the "ontic" vs "epistemic" debate.
Reminds me of tunnelling. That's one of those "it's weird but the equation should work on the other side of the barrier" situations, lo and behold stuff goes through the barrier.
For example in mechanics the coordinate systems being used aren't "real" in a physical sense because you are free to choose things as you wish. But the force you calculate is very much a physical fact, a property of the real world.
Why not? When you really get down to it, aren't both neutrons and neurons ultimately also models?
When most people think of a neutron, they think of a classical particle, yet we know experimentally that there is no such thing.
In QFT, particles can be interpreted to not exist, only fields -- particles are just a fuzzy (imperfect) abstraction of "field excitations". Particles may exist in the path-integral formulation of QM however. But it known among physicists that particles or fields don't define what is real -- what it matters is that the models predict observations. It is irrelevant which model is 'true' -- if they give equivalent results, both can be said to be [indistinguishably] correct (in their relevant domain). The examples go back a long way too, e.g. with the Hamiltonian and Lagrangian formulations of classical mechanics (wildly different but compatible mathematical formulations) -- so the 'model paradigm' has been known to physicists for a while.
I think in physics 'truth' is understood to be a (possibly non-unique) model that in principle would provide an exact prediction under ideal conditions, for a limited experiment. In our reality the ideal conditions are never going to be met. For example, we cannot ensure that a system has every atom in a certain position before performing an experiment (in fact the known quantum principles of physics themselves would contradict this); we might in theory know of a generalized state of a system (quantum superposition), but even then there are too many other assumptions (about experimental devices, about the individuals reading the experiments, about science as a whole) that go into the experimental paradigm.
I still find truth (as given) a useful concept, as a metaphysical principle that reality exists and has some working real mechanism (defined as being mathematically describable) -- we may never we able to write it down or discover it entirely, but it exists. Try to imagine an universe that is indescribable in principle (yields contradiction). There must be a rule.
It also sets the standard for our models.
https://chem.tufts.edu/answersinscience/relativityofwrong.ht...
This is not the case.
This is a simple example of how hard quantum mechanics is to think about.
Waves -- even in classical mechanics -- are not "countable", and are best represented in physical models with a continuum, such as field of real numbers.
They're fundamentally different. Quantum Mechanics glosses over this difference because when it was developed, this was too hard to deal with.
A lot of people who briefly studied QM at an undergraduate level assume that it "keeps going" and is applicable to all small-scale processes. In reality, it is not applicable to a wide range of phenomena, including explaining why particles come in countable units.
Similarly, the "infinite size" of electron clouds or the probability distributions of particles such as neutrons are a mathematical shortcut that was explicitly called out as such in the first QM papers. After a century, people just forgot and assumed that QM has a direct correspondence to reality, when it is a simplified abstraction designed to be analytically tractable.
Lastly, many people think that QM predicts that EM waves come in "countable" units called photons. This isn't true in general, and again, was called out as a simplification for the treatment of the emission and absorption of EM waves by atomic matter specifically. Atoms have electron orbitals that can take on only specific values, hence they can only absorb or emit EM waves with specific values. This is not a property shared with free particles or other fields that interact electromagnetically.
QM pedagogy needs to be rethought from the ground up, because nobody seems to be learning QM, but instead they're internalising a limited model they don't actually understand.
No, neutron observations in experiments are countable with integers. But the model that correctly predicts all those observations does not have "neutrons" that are countable with integers everywhere and at all times between observations.
> including explaining why particles come in countable units.
What observations of particles are you claiming that QM (which includes quantum field theory and the Standard Model of particle physics) cannot explain? I'm not aware of any.
> many people think that QM predicts that EM waves come in "countable" units called photons.
No, many people think that when you observe faint EM fields, those observations come in discrete units. When you shine very faint light through a two-slit apparatus onto a screen, you see individual dots on the screen as individual photons hit. The interference pattern that the wave model predicts only shows up over time in the pattern of the dots. Each individual photon observation is discrete. And that's what QM predicts. But QM does not say that there are always countable little photons everywhere and at all times between observations.
In the doubleslit experiment, can it be that the two slits set up magnetic or gravitational field that creates the interference pattern and photons are just a method to observe that pattern? Just like iron dust is used to see the magnetic lines that would exist regardless.
Physics equations don't have "if (...) { ... }" conditionals in them. The Universe doesn't seem to run on Boolean algebra!
The rules that govern neutrons either apply everywhere, or nowhere.
If the number of neutrons weren't so thoroughly tied to integer quantities, then deuterium could spontaneously convert to tritium and would decay at some non-zero rate.[1]
A mental model that I like to use (but certainly isn't mainstream) is to think of fermions as topological flaws, much like knots. In this model bosons are like wiggles in the rope.
I like this model because Fermions are very robust to outside interference except when interacting with other Fermions. This is much like knots in a rope. No amount of rope wiggling will ever undo a knot.[2] However, when two knots meet on a rope they obey a kind of "knot algebra" with strange and interesting rules. Preon and Rishon models are an attempt to formalise the mathematics of this, and work surprisingly well.
Continuous transformations of a topological flaw in a 3D volume of the "fabric" of the universe has Spin, which is hard to introduce otherwise. Topological flaws in crystals have even been used to model General Relativity!
Obviously, reality is not a 1-D rope with knots (tell the String Theorists that!). My model of this is a block universe where the parallel universes form a continuum. The knots are point-like only if you take a 3D spatial slice through the higher-dimensional block[3]. However, this point-like behaviour can never 100% manifest physically, because only interactions with other particles can ever be used by in-universe observers.[4] These particles in turn are a continuum across parallel universes. Thus any interaction is essentially the product of two continuous functions with each other, not infinitesimally small points bouncing off of each other.
I know this may seem a bit... "out there", but it's just a minor variation on MWI with elements of Rishon theory sprinkled on top.
[1] I'm obviously pulling this example out of a hat, but you get the idea. I'm not aware of any experiment that can demonstrate neutron number changing in any circumstance other than high energy collisions, or particle exchange between hadrons. A neutron floating about in space will remain a neutron. You won't get two neutrons suddenly turning up where you had one before. More importantly, you can certainly never have fractional neutrons. You can have a fractional expectation of finding a neutron, but you'll find a whole neutron.
[2] This begs the question of how photon-photon pair production can create an electron-positron pair out of nothing! This is like a sufficiently strong wave looping the string back on itself to produce a pair of opposite-handed knots. If separated you can call them two particles if you wish. If you bring them together they "annihilate" to form a rope that's locally highly curved. The curvature flattens, racing outwards as wiggles. These are the gamma rays produced by an positron-electron interaction. Similarly, long-wavelength wiggles move the whole knot around, but very short-wavelength wiggles can loop around the knot and through it. This could explain how some particles interact, but only above certain energy levels.
[3] A common mental model of MWI or parallel universes is something akin to pages of the book with slightly different content printed on each page. But this traps the physicist into thinking that spatial slices can only go parallel to the pages. In my model there's no preferred direction or "grain", so any slice is valid! This means that observers will disagree on essentially all measurements, including particle number. However, all observers will agree that particle numbers are quantised, and all observers agree that the particles follow the same rules. They just disagree about particle histories and particle futures.
[4] The refusal of some physicists to admit that everything obeys QM rules is a disease of the field. Either everything is QM or nothing is. QM experiments don't end at the bench top. The equipment, the physicist, everything is a part of the wave function. This insight is critical, and forms the basis of RQM and MWI, but not mainstream QM as taught in most universities.
But you'll get a proton, an electron and an anti-electron-neutrino. Possibly a photon as well. Remember, free neutrons only have a mean lifetime of less than 15min.
However, you'll still have three quarks either way.
Specifically, it comes from solving the wave equation given the boundary conditions, and those boundary conditions are fixed by the constraint that the square of the wave function (the probability of finding a particle somewhere = 1.
Similarly, atoms can be thought of as a "spherical box", the boundary in this case is a full revolution around the potential well.
The thing is: Not everything is a box, hence the abstract model that uses photons as a shorthand for the behaviour of the EM field in a box isn't fully general, and simply doesn't apply outside of those scenarios.
That doesn't stop the vast majority of people talking about photons (and bosonic particles in general) as if they're individually countable things zipping around in free space.
Neutrons can be counted like this. Photons can't.
Yes, they can. There are detectors that detect individual, free photons, just as there are detectors that detect individual, free neutrons. There are many, many experiments that have been done using them.
If you think you can disentangle the QM properties of the detector from the QM properties of the thing being detected, please write a paper explaining how and collect your well-deserved Nobel prize.
Which, if you are going to take this position, applies just as much to neutrons as to photons. The only evidence we have for either of those being "particles" is such detections. So either you should withdraw your claim that neutrons are particles, or you should admit that photons are. You can't pick and choose.
Composite fermion systems may act as bosons and be "squeezed." Bose-Einstein condensates are an example of this.
The wave-like properties of all fermions can be explained with any variant of the many-worlds interpretation (MWI) that you prefer.
Essentially in MWI each neutron in each universe is a point (actually three because of the quarks), but that point is not in the same place in all universes. The "closer" a parallel universe is, the closer the neutrons are to each other spatially. The universes aren't discrete, they form a continuum that is locally smooth, forming a differentiable manifold.
It's the distribution across the parallel universes that's continuous and wave-like, not the individual neutrons! The wave mechanics of QM is simply a description of the cross-universe distribution of neutrons interacting with the cross-universe distribution of other particles.
This was not the initial mental model developed in the early days of quantum mechanics. The founders of the theory flatly rejected MWI based on largely religious objections. It's absurd if you think about it, but the weird and complex mathematics of QM is simply a product of these people pretending parallel worlds don't exist.
It's a way of pretending functions can have multiple values in one place without actually admitting that this requires additional dimensions, otherwise they're not mathematical functions!
They also like to pretend that scalars have probability distributions... except sometimes when interacting with other probability distributions that magically stop being distributed at the same time.
In computer terms: This is like someone pretending that arrays are passed into a function using a CPU register that can have many values at the same time. It's just absurd.
Note that MWI as originally formulated by Everett does not work this way. Eg quoting from the Discussion section of his thesis:
> [I]t seems to us to be much easier to understand particle aspects from a wave picture (concentrated wave packets) than it is to understand wave aspects (diffraction, interference, etc.) from a particle picture.
or
> This view [ie Everett's view, as described in the thesis] also corresponds most closely with that held by Schrödinger.
The main point of MWI is not to go back to a more classical picture of point particles, but to remove wave-function collapse and Heisenberg cut from the description.
Quantum Mechanics is actually incredibly conservative, with theoreticians simply refusing to abandon notions shown to be incompatible with reality for over a century now.
Same with Spin. Okay, so particles aren't literally spinning spheres. We get it. Stop pretending they are and then Spin won't be so mysterious any more!
It’s just an evocative name?
Okay then, if it's impossible, don't print what it isn't. Put down in writing what it is. That is all.
However, unlike the operators for the usual sense of angular momentum, there are additional solutions that appear, which have the half-integer spins.
And, in a sense the spin contributes to the angular momentum.
I don’t see a problem with saying “it is in some ways kind of like if it were spinning, but it isn’t quite the same. Here is the math to describe it, and how that math is like and how it is unlike something spinning.” . It seems better than just saying “here is the math which describes it”.
I don’t think your classical CPU analogy is really applicable here.
The wave function is exactly that, many values describing the probability of the particle in space. The act of measurement is what sets the value of the wave function. Take the classic electron diffraction experiment. It’s not the measurement that defines the wave-like property there.
Take tunneling through a potential barrier, why can the particle tunnel? Tunneling is a consequence of the wave function defining the probability of particle in all space.
In the last few decades especially the "publish or perish" culture of academia in general (not just in the physics departments) has lead to a perverse incentive to make mundane topics seem more impressive by deliberately overcomplicating the mathematics.
Similarly, if you read through most (95-99%) of high-energy physics papers, they aren't even remotely connected to reality. They're advancing the state of the art for some obscure corner of mathematics for the solution to some simplified model which itself is an abstraction of an abstraction.
It is astonishingly difficult to find "down to earth" mathematics for just about anything related to theoretical physics.
Meanwhile, a lot of what physics "does" is actually very simple mathematically, it's just that that simplicity is intractable symbolically. (Nonlinear differential equations are notoriously difficult to solve algebraically, but relatively straightforward numerically.)
If you think about it, particle physics must be simple because while the Universe contains advanced maths degrees, it doesn't have one. A particle zipping around and bouncing off of things doesn't solve monstrously complicated Feynman Integrals moment-to-moment! The rules it follows can't possibly be that complex.
Sure, but could it be that Feynman Integrals are one of the least complex measurable predictions/calculation of those simpler, underlying rules, potentially acting in dimensions we can't measure?
This may or may not have been intentional by the author but you could also interpret the sentence as referring to whatever it is that does exist that we label neutrons as opposed to the models we currently use to define that thing.
How do free neutrons, etc. work into this?
https://www.youtube.com/watch?v=MBnnXbOM5S4 (3b1b on the Heisenberg uncertainty principle and how it arises)
Think of a free neutron as the wave envelope, rather than any of the bumps in the wave.
The argument is that models are only useful tools and should not be taken to be a full or only representation of the thing being studied.
I mean in some ways Mathematics is just like a completely made up world that is internally consistent so saying it is "truth" is the same as me saying "in a made up world where truth exists, truth exists"
Us mathematically, and computer inclined people like to think math has a sort of universal truth to it. But it only has truth within the bounds of it's own system, or it would break one of Godel's incompletness theorems, right?
As I write this, I'm kind of thinking that the "All models are wrong" George Box quote is almost a statement of Godel's theorem.
All models are wrong by definition. The goal of the modelling is to simplify things, makes them less complex then they are in the reality. So model is wrong. QED.
So to say "the model is wrong" is to say nothing. No new information transferred. While to say "the model is not valid" is to state that the model doesn't deliver it's promises. It is a non-null information about the model.
I mean, yes, it could be seen as an another way to say "all models are wrong, but some are useful". But I'd prefer not to say that, because I see "wrongness" to be a notion inapplicable to a model. It is ok though to apply it nevertheless to make a nice catchphrase.
> it only has truth within the bounds of it's own system, or it would break one of Godel's incompletness theorems, right?
I'd prefer not to say this either. I'm not so sure that there is no way to create The Mathematics, which would go beyond arithmetic so Godel theorems would be inapplicable to it. Godel theorems more then just a regular model, they are a model within a model modelling a model about a model. What do they say us about the reality I'm not sure. How to apply them to the reality without losing validity?
My favourite example for this is the electron. Does an electron how we describe it in quantum mechanics exist in nature? The answer is no. The electron is a model construct we have made up that describes natural phenomena. Nothing more.
This is where non-experts that didn't study physics get confused when they read that an electron is a wave but also a particle. All of these concepts are just physical models that again do not exist in nature. But they describe different aspects of nature pretty well. And so we assign them a meaning beyond what the mathematical or physical model contains, a part of reality.
I don't think it's meaningful to say that the electron described by QM doesn't exist in nature... something exists in nature and all we can ever do is describe it with a model.
Indeed, having completed a graduate degree in physics, by the time you've been introduced to a different model of the electron for the third or fourth time, it begins to sink in.
Yep. I immediately started thinking about how this “real”/“not real” dichotomy applies to programming. We write programs that confidently declare not only that they’re defining facts for the purpose of the computation, but transferring them and interoperating with other such stated facts. But they’re all abstractions, they’re all convenient fictions we use to describe, model and hopefully send electricity through minerals to achieve the result we defined.
Even stepping a bit away from the physics involved, just considering the humans who do lithography and are closest to reality, their mental model and work is still an approximation of that reality that’s effective at their level of abstraction.
And this doesn’t just apply to human-mechanics interaction. It’s something my messed up brain thinks about with human-human interactions all the time. We have mental models of how the electrical impulses travel through our fellow animated bags of meat, but they’re not just approximations they’re often very flawed. But we still interact and we hopefully don’t become overly obsessed with the underlying details we don’t and probably can’t fully grasp.
Well technically I think the answer would he "We don't know" instead. We don't have anything better than the standard model - the perceived clash between wave and particle is resolved by a more general theoryz for example.
And to be fair, with regards to the standard model, we know the standard model is not fully correct and we still don't have anything better.
Most scientists, I think, use some definitions along the lines of "Reality is whatever appears in our best models/explanations of our observations of the world".
If all observations of electrons we have ever made obey some law, we say that's the real behaviour of electrons. If we find out it doesn't always work, we realize reality is something more complex, and at some point hopefully we find out what it is.
Starting with the laws of physics, no, we cannot in practice calculate a consciousness. But we also can't inspect a lottery machine and predict the balls which come out either. Nobody thinks the latter is is because lottery machines also contain magic, yet so many people think that of consciousness.
Could consciousness involve extra-physical explanation? Sure. But there there is an equal amount of evidence that lottery machines do too (i.e., none).
When people talk about magic, they are referring to the strangeness of subjectivity, and what it is at a very fundamental level. I think a better analogy is being a Greek philosopher and trying to understand what matter is. No tool at the time can measure things in a way that would meaningfully lead them to an answer, nor did they have any theories where matter could be related to something they could study. So when we are confronted with the same situation, I think it's perfectly reasonable to assume that we, like them, are missing a massive piece of the puzzle. Calling it magic feels kind of right when it's something so large and consequential.
One of my favorite books is "The Man Who Mistook His Wife For A Hat" by Oliver Sacks [1]. It contains a number of amazing cases of neurology where someone has suffered a brain injury and the facade of our apparently sophisticated consciousness is exposed a bit. For example, a guy who suffered a stroke and became blind, but was unable to accept it, so he lied to himself and rationalized all the resulting outcomes. The rest of his brain was 100% normal, was an articulate, intelligent person in all other regards. But he would confabulate extraordinary excuses to maintain his belief system. That story just drives home how much we as humans are capable of rationalizing.
I don't recall if it was that book or another that describes Capgras delusion [2] where a person sincerely believes that someone they know intimately, say a parent or spouse, has been replaced by a doppleganger, but are otherwise fully rational. They can even state that it sounds crazy, but nevertheless it is true. Or a form of it where they feel this way about themselves, leading them to believe they are dead yet still animate, is the Cotard Delusion [3]. What causes it? When one looks at their mother, an array of neurological connections to associated feelings and memories are activated; apparently in these cases, that doesn't happen in the normal way, and rather than concluding that something went wrong in their heads, they experience as this person in front of my looks identically to my mother but must be someone else and that is why those feelings are not arising.
The point of these two small examples is that our experience of consciousness and "me-ness" is a self-consistent facade, not nearly as deep as we feel it is, as site-specific injuries can result in profound flaws in that system.
[1] https://www.amazon.com/Man-Who-Mistook-His-Wife/dp/149151407...
Why thank you, Mr. Science Communicator. Such a tremendous insight that all I needed was a mindset to find, eh, quantum field theory?
(Apologies for the sarcasm, but - really?)
(Also: no disrespect intended to (other) science communicators.)
She is cold, methodical, and notably critical of the search for mathematical beauty. She is pretty clear about the fact that reality is what we observe, and equations are just a tool we use to make models.
Perhaps it doesn’t. Perhaps language is just a tool we, humans, use for thinking and representing our knowledge.
Mathematics is a language...
Mathematical objects need not exist anywhere else but in the Mathematician’s head.
Since we are the only species who have Mathematics, I will continue insisting Maths is invented, till we can compare notes with other life forms.
even if the golden ratio were as prevalent as it popularly thought, it still wouldn't really situate math (the map) in a causal role; rather it would suggest a common physical factor which the math models.
in general, it arguably doesn't seem terribly surprising that lots of things in the world have share simple patterns, even without causal connections, simply by virtue of that there are far fewer simple patterns to go around than complicated ones, sort of a generalized 'law of small numbers'
I know I can improve my communication skills, but it’s extremely frustrating to have such a hard time expressing a thought without having drastically different intentions bolted onto it. Maybe you’re right that it’s not terribly surprising. But I didn’t even kind of suggest that “math” has a “causal role”. I made another comment higher up relating how the way I think is another set of approximations and abstractions of reality. My point was far from saying that there’s some “truth” in math, quite the opposite.
Addressing your response, I’m sure there’s a bit of detectable woo even after clarification, but it’s just wonder. I don’t think there’s a magical energy core connecting all life. I just think that it’s a fascinating thing about physical reality that a mathematical constant as defined by humans is something that finds expression in a lot of seemingly unrelated places.
It’s entirely possible that the ratio is just common enough in growth patterns, or just a pattern than emerges when the right set of otherwise common constraints come together.
It’s just neat to me, that’s all.
You are somehow insisting that the mathematical expression (1 + sqrt(5))/2 is in some ways similar to a snail shell.
Don’t you find it even a little strange that just about anyone not schooled in algebra/arithmetic would doubt your claim?
I am simply pointing out that where both of us observe/experience the pattern, your mind ascribes to it significance by relating it to the Mathematical construct called "Golden ratio".
My mind doesn't make that relation. I call that pattern (and other patterns like it) a "snail shell pattern".
We index the experience differently.
From your perspective a Mathematical expression describes the pattern. From my perspective an English expression does.
Either way, we index the pattern in our respective minds.
You index (understand?) it as a "Golden ratio".
I index (understand?) it as a "snail shell".
Your understanding it Mathematically, I do not. Which is why I said that you assign Mathematical meaning to the pattern.
This is essentially what Linguistic relativism is all about: https://en.wikipedia.org/wiki/Linguistic_relativity
> I index (understand?) it as a "snail shell".
> Your understanding it Mathematically, I do not. Which is why I said that you assign Mathematical meaning to the pattern.
Huh? I spoke of snail shells. If your characterization were accurate we probably wouldn’t even know we’re talking about the same thing.
You connected the dots between the two...
The fun begins when it comes to the deeper extrapolations of what can be readily observed. It seems possible that an alien civilization could be much more knowledgeable than us, yet never formulated the equivalents of our QM equations.
If that’s the case, the property you’re describing belongs to abstract thought and language generally, of which mathematics is a subset.
Much of what you point out is also what Quine identified in his paper "The two Dogmas of Empiricism" which is trivially the question "Is logic/mathematics empirical?". In my world-view inductive types (computations) are empirical.
But since you seem to be optimising for "pinning things precisely", general consensus/agreement may not be of any empirical value ;)
To take a specific example, I'm curious what post-structuralism actually means as applied to the structure I was trying to refer to - there are certain symmetry groups that seem to apply everywhere we look, but the actual structure is itself isn't a symbolic concept in the same way the standard model is.
There is certainly an element of wheel re-invention in first principles thinking, but acquiring maker's knowledge seems like a more rewarding endeavour than acquiring only user's knowledge.
This process of creating new languages/jargon has a name in Computer Science: Meta-linguistic abstraction [1], and so the value of Philosophy (to me) is in the first paragraph of the Wikipedia article:
>metalinguistic abstraction is the process of solving complex problems by creating a new language or vocabulary to better understand the problem space.
And so this I find it fascinating when you say that...
>there are certain symmetry groups that seem to apply everywhere we look
Because your meta-language for talking about your experiences is the language of symmetry groups.
Post-structuralism (and all Philosophy, really) is merely an attempt to escape the reductionist nature of that sort of language-centric thinking - it's an attempt to drag us out of the cave of Logocentrism [2]. Mathematics is only up to isomorphism - there's nothing in the symbolic denotation without the human connotation, so if structuralism is about symmetry (equalities) , post-structuralism is about asymmetry (inequalities).
[1] https://en.wikipedia.org/wiki/Metalinguistic_abstraction
I don't believe that most modern physicists dispute that the equations we use are anything more than models. For example, you won't go to a seminar on microrheology and have a physicists point out in disgust that the presented derivation is "wrong" because it used Newton's second law without the relativistic component. While this is perhaps an extreme example, it is the case that we often have to ignore certain effects/make certain assumptions to make equations solvable and interpretable. If we believed that our physics equations represented some "absolute truth," then we would never allow ourselves to make these sort of assumptions in the formulas we derive.
"a friend showed Pauli the paper of a young physicist which he suspected was not of great value but on which he wanted Pauli's views. Pauli remarked sadly, 'It is not even wrong'."
An equation can be perfectly descriptive without explaining anything, and just as easily used in a patently incorrect or at least unverifiable explanation. A new physics learner may be uncomfortable with this notion, as everything up to that point was taught as cause and effect, no difference between explanation and description. As in the example, the description could take a different form. God could use matrices. Or you could say that god prefers to minimize the lagrangian. Or you could simulate with finite element analysis and say that it is an emergent phenomenon of many points of mass all following a set of rules. Questions for each explanation abound. I think at the bottom of all of this there has to be an admission that the universe works in this particular manner because it wouldn’t exist for us any other way, and we do exist, so it must.
Newton's equations are approachable, closer to human experience and comprehension. Feynman's diagrams, perhaps a step in the right direction.
True? Science is not math; it has no proofs. A 'theory' is the expression of a model that's stood the test of time - but one piece of evidence can force it to be re-built. Science forces us to search for and adapt to the new. Feynman said, "When someone says ‘science teaches such and such’, he is using the word incorrectly. Science doesn’t teach it; experience teaches it."
A couple of the earliest things I recall proving in physics classes: 1. acceleration due to gravity on earth g is the same ~9.8 m/s² for all objects whose mass is negligible compared to the mass of the earth. 2. if it weren't for atmospheric drag, thrown projectiles fall at velocities much smaller than the escape velocity would trace a parabolic curve.
In what sense were these not proofs?
Scientific theories are used to support a wide range of phenomenon, but are later superceded by better theorems that can encompass more phenomena. Experimental evidence can provide support for the theory, can disprove a theory, but cannot prove the theory.
Unrelated to mathematical proofs, but there are counter examples for each of your statements.
1.
The acceleration due to gravity on earth is not constant, it varies depending on location (easy to experimentally refute that it's not 9.8 everywhere). For example, being on mount everest versus at sea level in the Arctic would give different rates of acceleration.
Adding a tilde in front of 9.8 m/s^2 doesn't help make the statement look like a proof.
2.
Given that acceleration due to gravity is not constant, then the trajectory will no longer be a parabolic curve.
I'm not interested in nitpicky arguments. Good bye.
Well, this article seems to be misleading drivel, to a first approximation. Why are people supposed to waste their time reading this?
Well. Anecdotally, when there's open house at the physics department, the guys that will tell you how quantum mechanics and/or relativity are wrong tend to have a background in engineering.
Also note that not everyone subscribes to what the article calls the 'Gospel of Physics': Quite a few of us are well-aware that the map is not the territory...
I don't think the author is being dishonest. But that admission is extremely striking to me.
[edit: Can some admin give me guidance on how to make the point I'm clearly trying to make? I'm not sure how to make this point without being discourteous.]
But if their problem is not being able to stomach the "just-because" explanations that every QM resigns to using to explain wtf complex conjugation has to do with reality, then I completely, totally sympathize. Learning QM consists of learning to silence your objections to the fact that none of the things you're being asked to believe make any sense.
It gets somewhat more scrutable when you deal with relativistic QM, where, at least, the terms in Schrodinger equation start to make intuitive sense (it's the first couple terms in the Taylor expansion of E psi = sqrt(p^2 + m^2) psi = m sqrt(p^2/m^2 + 1) psi = (m + 1/2 p^2/m)psi ), but then spinors enter the picture and all intuition goes out the window again.
Yes, but the reason you're learning those things is that that is the way reality actually works. That's what happens when you actually do experiments. Richard Feynman once remarked: "quantum mechanics was not wished upon us by theorists". We have QM because physicists were dragged to it kicking and screaming by actual experiments that showed them that reality does not work the way classical physics describes. And the reason all those difficult mathematical tools have to be taught is that that's the only way physicists know how to build models that make correct predictions about what happens when you do those experiments.
In other words, yes, you can say QM doesn't make sense, but that's because reality doesn't make sense--at least not if "make sense" means "works the way my human intuitions, which never evolved to deal with quantum phenomena, say it should". Yes, reality doesn't work the way our human intuitions say it should. That means we have to retrain our intuitions--which is what a lot of learning QM is really about.
My teacher told me, that I will really know physics when I will be able to derive my own equations. I extended this definition to "I will really know physics when I will be able to create the physical model of a physical thing, to be able to watch it with my own eyes".
But it's not just complex conjugates, but eigenvectors as well. Also spectacular: "the square root of a negative number, which by definition does not exist"
Uh. That's a complete lack of connection to anything beyond 10th grade math. Maybe not the ideal foundation for criticizing QM.
And yes, I get the complaint that QM comes with a lot of "if you just hold your nose and accept this", and intuition doesn't really work for it - but that's hardly a valid criticism by itself, outside of saying "well, this is hard to understand".
And the idea that a mathematical model isn't perfect and as a result of that can't be true really just betrays a complete misunderstanding how models work.
This piece boils down to "I don't understand it, and so it can't be correct".
Having studied both engineering and physics, I can tell you that there is nothing observable in the universe that requires complex arithmetic/algebra/theory. We use it because it makes the math a lot more convenient.
BTW, the use of complex math for real world applications goes way beyond QM, and predates it significantly. Engineers use it all the time: AC Circuit theory is full of it. As is electromagnetics. As is power (motors, etc). As is anything involving Fourier or Laplace Transforms (i.e. almost all engineering). You can come up with purely real mathematics to describe all of the above, but it would be tedious.
But can you call a theory true if no one understands it? A century after inventing quantum mechanics, physicists still squabble over what, exactly, it tells us about reality.
I think his question is a little loaded - it's a cliché that "nobody understands it", and I'm sure there are physicists who would bridle a little at that being true.
His following claim (disputes over what it tells us) - while true - is not quite the same thing, so the argument is flawed in that regard, but not because of a problem learning maths.
I understand non-relativistic quantum mechanics, i.e. the Schrodinger equation, so close to perfectly as makes no difference. So do thousands of other physicists.
We don't all understand it in the same way. The atomic scale is removed from the human scale by an order of magnitude of orders of magnitude, they do things differently there, so humans can't appeal to their experience to say that one self-consistent description of atomic phenomena is real and another isn't. But the way I understand quantum mechanics is self-consistent, so is the way any the next physicist understands it, and we'll all agree what result it predicts for any experiment you can imagine. If you want more than that, science can't help you.
not an admin, but seems like you made your point
I wonder if there would be less objection by the author if he was exposed to geometric algebra, to help with difficulties encountered by sqrt(-1) philosophical issues.