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Is the Schrödinger Equation True?

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Re: Is the Schrödinger Equation True?

#151

Post by jfengel »

bollu wrote: post #130
wnoise wrote: post #97

Earlier quoted context omitted.

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 de…

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

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.

Re: Is the Schrödinger Equation True?

#152

Post by lumost »

FeepingCreature wrote: post #67
lmm wrote: post #65

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 imaginati…

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.

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.

Re: Is the Schrödinger Equation True?

#153

Post by disabledalpha »

ur-whale wrote: post #138
lmm wrote: post #65

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 imaginati…

> 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 equatio…

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

Re: Is the Schrödinger Equation True?

#154

Post by jl2718 »

frongpik wrote: post #87

I've seen a clever mathematical trick that derives the Schrodinger equation out of GR. As you know, there's this invariant interval thing: ds2 = c2 dt2 - dx2 - dy2 - dz2. The trick was to add a small oscillation multiplier to ds2, so it would become ds2 exp(a cos wt) = ... Mixed with lorentz transforms, this yields the static SE and mixed with GR tensors stuff this yields the time dependent SE. I can't judge if this…

This seems to be a more reasonable approach - derive a relation from a law of conservation. Starting from the differential form is fraught with indeterminism. Assuming this works, I wonder about maxwell’s equations. I assume they would have to be consistent with a conserved electromagnetic quantity added into this. Gravity?

Re: Is the Schrödinger Equation True?

#155

Post by gus_massa »

ur-whale wrote: post #138
lmm wrote: post #65

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 imaginati…

> 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 equatio…

> 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

Re: Is the Schrödinger Equation True?

#156

Post by tasty_freeze »

BobbyJo wrote: post #104
tasty_freeze wrote: post #102

> The “laws” of physics, Wigner adds, have little or nothing to say about biology, and especially about consciousness, the most baffling of all biological phenomena. 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, y…

I dislike this comparison. The reason we can't explain consciousness isn't because it complex or because there are too many variables for us to track and predict, as in the case of the lottery machine, but because we haven't decided what we mean when we say 'conciousness'. Pretty much any component of consciousness one may think of can be explained in isolation as a simple material system. Self-reflection, continuity…

I agree everyone has a different definition for consciousness. But I'd argue that you could take any one of them and my argument would still apply.

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...

[2] https://en.wikipedia.org/wiki/Capgras_delusion

[3] https://en.wikipedia.org/wiki/Cotard_delusion

Re: Is the Schrödinger Equation True?

#157

Post by jl2718 »

Toss a ball in the air and watch it spin. You ask why it spins and I tell you that it is because god is multiplying its state by a quarternion. Is that true?

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.

Re: Is the Schrödinger Equation True?

#158

Post by eyelidlessness »

ukj wrote: post #125
eyelidlessness wrote: post #114

Earlier quoted context omitted.

It sounds like we understand it the same way, with different words.

You could say that. But then it begs the question "What does it mean to understand?" 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: ht…

> 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.

Huh? I spoke of snail shells. If your characterization were accurate we probably wouldn’t even know we’re talking about the same thing.

Re: Is the Schrödinger Equation True?

#159

Post by ukj »

eyelidlessness wrote: post #158
ukj wrote: post #125

Earlier quoted context omitted.

You could say that. But then it begs the question "What does it mean to understand?" 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: ht…

> 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. Huh? I spoke of snail shells. If your characterization were accurate we probably wouldn’t even know we’re talking about the same thing.

These were your words: "a snail’s shell expresses it (the golden ratio)"

You connected the dots between the two...

Re: Is the Schrödinger Equation True?

#160

Post by BobbyJo »

tasty_freeze wrote: post #156
BobbyJo wrote: post #104

Earlier quoted context omitted.

I dislike this comparison. The reason we can't explain consciousness isn't because it complex or because there are too many variables for us to track and predict, as in the case of the lottery machine, but because we haven't decided what we mean when we say 'conciousness'. Pretty much any component of consciousness one may think of can be explained in isolation as a simple material system. Self-reflection, continuity…

I agree everyone has a different definition for consciousness. But I'd argue that you could take any one of them and my argument would still apply. 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 exampl…

Whether or not it is a facade is inconsequential, and not necessary for my point. Whether or not the performer on stage is a magician or an illusionist, the trick is of a caliber that defies our explanations and theories. Either way, it is enormously important for us to get backstage to see how it (it as in subjectivity) is done.
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