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Comment by lmm | Hacker News Reader

The Schroedinger Equation is the most true thing I've ever encountered. It is clear, simple, obvious in its correctness; the greatest piece of physics since Maxwell. Once you've understood it it becomes almost impossible to imagine that the universe could possibly work any other way. (Contrary to this article's claim, the equation describes the behaviour of Helium atoms perfectly well; it's a failure of our imagination or exertion that we can't find closed-form stable solutions, not a fact about the equation itself). So this kind of complaint seems utterly wrongheaded.

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.

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This is also the thrust of the original LessWrong Quantum Physics Sequence. https://www.lesswrong.com/posts/7FSwbFpDsca7uXpQ2/quantum-ex...

> 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.

> This is also the thrust of the original LessWrong Quantum Physics Sequence

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.

> Absolutely agree with his gripes in the first article about the traditional historically-oriented introduction to QM confusing students.

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.

Most of philosophy is for natural reasons at least partially a response to what previous people have asserted, so if you want a good understanding of the field of western philosophy you need the background.

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"

There are plenty of cases where newtonian or relativistic physics diverge from intuition as well, it's strange that we impart special significance to Quantum Mechanics.

> simple, obvious

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 long

Regarding your second to last point, have a look at https://en.wikipedia.org/wiki/Classical_limit

> 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?

In an equation like

  df/dt = k Δf

the Laplacian create a diffusion-like effect only if k is a real positive number. If k is a real negative number, it creates the opposite effect that concentrates whatever f represent and make it more spiky.

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 positive, the amplitude of the function decrease with time. And when ω is big, it decrease faster, so the small features disappear very faster than the smooth parts.

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

> - 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?

The 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.

> Did they really expect that the behaviour of the very small would be exactly like that of everyday macroscopic objects?

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.

There's a discontinuous boundary? That's news to me. It's pretty well known that classical mechanics is reproduced by quantum mechanics in the large scale. There's no discontinuity between the two. There's also a whole heap of people competing on trying to put ever and ever larger objects in superposition. Finally, decoherence has seen plenty of study and seems to be exactly what you're lamenting the lack of.

> There's a discontinuous boundary? That's news to me.

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.

> Quite a large number of physicists (and quantum computing people) seem to speak as if there is one.

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.

> 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.

> There's a discontinuous boundary?

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.

> 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.

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.

All of the "weird" phenomena are just everyday wave phenomena as far as I can see - e.g. the uncertainty principle applies just as much to plain old classical wave packets as it does to QM objects. And as far as I know there's no "peculiar discontinuity", just a gradual irrelevance - e.g. the uncertainty principle is exactly the same when applied to objects of any size, it's just that for large objects that uncertainty is too small to be relevant.

>I expect a theory predicting otherwise to bear the burden of demarcating exactly where the boundary between "macroscopic" and "weird not-macroscopic" lies

My feelings exactly, I really don't understand why such a relevant comment is being downvoted.

Please elaborate on why you find it to be clear, simple, obvious!

he is obviously a much smarter individual than us regular mortals

>Contrary to this article's claim, the equation describes the behaviour of Helium atoms perfectly well;

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".

If you go back in the past, contemporaries of newton were critical of his gravitational theory having an instantaneous action at a distance. Familiarity has just bled us of its strangeness over time.

> it's a failure of our imagination or exertion that we can't find closed-form stable solutions

You better not say that when any solitons are floating by...

Do you mean wave packet solutions? The Schrodinger equation is linear, so I don't think it can have solitons.

>The Schrodinger equation is linear

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 Schrodinger equation is linear, but the extension to Quantum Field Theory is not linear https://en.wikipedia.org/wiki/Quantum_field_theory so actual reality is even more weird.

Besides being a joke, I was a dispersive PDE guy and so have a perverted idea of what the Schroedinger equation is. It’s common in that field to study a nonlinear variant. A u|u|^2 term, for instance, admits sech type solutions.

What's natural about SE? From my limited understanding, SE tells us that all particles in the world are interconnected in a peculiar way by a "energy field" of some sort.

That would indeed by very unnatural. Fortunately that's nothing like what the Schrödinger equation says.

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.

You can actually skip 2, if you model yourself as part of the system as well (i.e. have an Ideal Observer who changes from “has XYZ knowledge of the system” to “observed ABC about the system” for all possible ABC, with respective amplitudes) – but then you're doing a lot of computation you're never going to use for prediction. (It's more elegant, though – even more so if you model yourself as a human, though this'll take more computational power than we have.)

I meant skip 3. Skipping 2 is like making a laptop without a processor.

And why is this "natural"? I find the collapse to be the strangest axiom, the rest I can handle.

Well one definition of "natural" could be that it "came from nature", and as the Schrödinger equation perfectly describes non-relativistic events in nature arguably it came from nature just as much as newtons laws of motion.

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.

Collapse isn't actually an axiom. It doesn't appear in the Schroedinger Equation at all.

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.

In standard quantum mechanics, collapse absolutely is an axiom. I really have a lot of sympathy for the Everettian no-collapse position, with effective collapse caused by interactions with the environment. Nevertheless, the measurement problem hasn't actually been solved. The Born probabilities haven't truly been derived, except under some truly restrictive assumptions.

> Nevertheless, the measurement problem hasn't actually been solved. The Born probabilities haven't truly been derived, except under some truly restrictive assumptions.

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.

It's a very simple equation and it's kind of the "square root of" the basic wave equation (d^2x/dt^2 = -k x). So it's the most obvious way to have an equation that can describe a space where waves are pervasive but there are also non-wave things, which is what reality is.

That's pretty vague. Something so obvious, correct, simple and clear should have a more straight forward explanation.

Well, what are you claiming is more obvious/correct/simple/clear? Certainly I'd hold that it's more so than any other major physics: relativity, Maxwell's equations, or even Newtonian mechanics are less clear IMO.

You might be thinking of Quantum Field Theory, which models particles as excitations in fields specific to those particles. It's completely compatible with the Schrödinger equation, but operates on a higher level of abstraction to allow easier understanding of systems of multiple particles.

Well, Maxwell is Lorentz invariant. Schroedinger equation is not.

There is an extension made by Dirac that is Lorentz invariant. It is even more weird, and to write the extension you need to consider positrons, so he predicted the existence of the positrons from this extended equation https://en.wikipedia.org/wiki/Dirac_equation

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