Like if you have word, the next logical word is word so you get word word word word...?
And if you wanted to group compare the similarity of documents, would you just learn an embedding then use cosine similarity or is there a better (and fairly quick) way?
@pss1 the easy way to approach this is to note that acceleration and a gravitational field are the same in many ways. So the tube accelerating sideways at an acceleration $a$ is equivalent to a tube standing on its end in a gravitational field with gravitational acceleration $a$.
In the case of a Young Slit Experiment, what exactly is a two-part wavefunction and what does it describe? Its ability to pass through both slits at the same, interfere with itself and then land as a single particle?
@JohnRennie ||the easy way to approach this is to note that acceleration and a gravitational field are the same in many ways. So the tube accelerating sideways at an acceleration $a$ is equivalent to a tube standing on its end in a gravitational field with gravitational acceleration $a$.||.......fluid is accelerating right?.......should I have to now consider the tube is rest and fluids flowing in the situations you mentioned.....??
"A photon between a vertical and horizontal angle is in a superposition state" What the duck...?
Here is what I really dont understand: "The interference pattern that we see indicates that the photons are in a superposition state: the wavefunction describing each photon has two parts, one for the photon passing through the left slit, and the other for the photon passing through the right slit."
@PM2Ring Thanks, I read it but it's still confusing :D "In this way, the particle can "move" smoothly from being entirely at A to being entirely at B", how? By being at both places at the same time and when measured, it's wavefunction (with position probabilities) breaks down?
So when that photon that's in a superposition state between vertical and horizontal angle has a wavefunction with probabilities of it landing on one on the other when measured?
How do we know that things are really not defined and when we look at them, then their "wavefunction breaks down..."?
I mean, it can be a vertical angle all the time, why should I think it's in a superposition of horizontal and vertical?
Schrodinger's cat can always be dead, why do we think it was both at the same time?
@NovaliumCompany Don't worry too much if it's confusing. This stuff has no classical counterpart, so it's not easy to make mental models of it. But eventually your mind will get used to it if you look at lots of examples of superposition. Feynman said this is the only hard thing in quantum physics. And it's often been said that if you haven't been confused by this then you haven't thought hard enough about it. ;)
But to try & answer your question. Let's say you have an ideal photon detector. If you send it a photon it will register it 100% of the time.
Now place a vertical polarizer in front of the detector. If you send in a vertically polarized photon, it will get detected with probability 1, if you send in a horizontally polarized photon, it will get detected with probability 0, that is, it won't get detected. Ok so far?
Great. Now we can represent the polarization of any photon as a kind of mixture of vertical & horizontal, like this V cos(theta) + H sin(theta), where V is the pure vertical state, H is the pure horizontal state, and theta is the photon's polarization angle, measured from the vertical axis, so theta = 0° for a vertical, theta = 90° for a horizontal. Ok?
So if theta is 45° then the photon has probability .5 of being detected. If we send 1000 identical 45° photons at the detector, roughly 500 will be detected, 500 won't register, and there is no pattern to which ones get detected.
@NovaliumCompany Don't worry too much for now about where the sin & cos come from. The important part of that is that cos^2 + sin^2 = 1
Macroscopically, if you send 45° light through a vertical detector, it dims the light. But at the quantum level, that dimming happens because each photon is either blocked or passed by the polarizer.
@NovaliumCompany Yes. And that equation I gave before tells us the probability if we know theta. So all 45° photons have the same state of sqrt(.5) V + sqrt(.5) H. There is nothing in their state that determines if they will actually get detected or not. There's just that probability.
Now here's the other tricky bit. ;) When we produce the photons with some polarization angle theta they don't "know" that they're going to be fired at a vertical polarizer. We might change our mind & rotate the polarizer on the detector to horizontal, or any angle in between. But we can still use that same equation to represent their state, and simple trig, to calculate the detection probability. Eg, if we make the detector horizontal, then sin^2(theta) tells us the detection probability.
So the photon doesn't "know" in advance whether or not it will be detected by a given polarizer setting, unless it matches the polarizer exactly, i.e., a vertically polarized photon will get detected with probability 1 by a vertical polarizer.
So the core ideas are 1) that all the different polarization angles can be represented by a combination of vertical and horizontal, and 2) the outcome of passing a polarised photon through a detector is probabilistic. There's no hidden information anywhere that can be used to determine in advance what the actual result will be for any given photon.
J. S. Bell came up with an ingenious but simple way to demonstrate that, using 2 different polarization angles. At least, the Bell inequalities show mathematically that there's no information hiding locally in the photon generator, the photon itself, or the polarizer or detector, that determines the outcome.
Much more can be said on this topic, but I think that's probably enough to digest for now.
And I need to take a break. Typing stuff on the phone is not friendly on the fingers. ;)
@NovaliumCompany I should mention that I've been using a simplified model of polarization. The more accurate model is a little more complex, because we need to work in 3D, not just a simple plane of directions, but the same basic principles apply.
Whats the meaning of the uncertainty principle in the case of electron "orbiting" and interfering with itself to form the harmonic waves? I mean, whats position and momentum (velocity) in that case?
Within the orthodox interpretation, at least, that picture of an electron "orbiting" is a classical one. It is meaningful to talk about what position/velocity we measure the electron as having; it is not meaningful to talk about what position/velocity it would have independent of that.
Ok but we know that the actual picture of an electron in the atom is in the form of harmonic oscillations right? How can that have velocity or position for them to be uncertain in the first place? And how does this picture of the electron collapse when we observe?
No, we do not have such a picture. At most, we have a picture which would be applicable in the classical limit.
That an electron experiences a harmonic potential does not imply that one can attribute to it some trajectory
What we do have is the Schrodinger equation as dictating how the wavefunction of that particle will evolve in time. The wavefunction dictates probability distributions for the observables of an electron (position/momentum), and the Schrodinger equation dictates how these distributions will evolve with time.
as to whether the electron "is" the probability cloud, or whether the probability cloud has a more complicated relationship with the electron's ontology, that is essentially still a complicated open question.
The measurement problem in quantum mechanics is the problem of how (or whether) wave function collapse occurs. The inability to observe such a collapse directly has given rise to different interpretations of quantum mechanics and poses a key set of questions that each interpretation must answer.
The wave function in quantum mechanics evolves deterministically according to the Schrödinger equation as a linear superposition of different states. However, actual measurements always find the physical system in a definite state. Any future evolution of the wave function is based on the state the system...
Hi all! I got a famliy of "vector fields" $f: \Bbb R^3 \to \Bbb R^3$ that satisy a certain physical property (that is not that relevant now) However they have all in common that theay are pointwise parallel to $\nabla\rho(\vec{r})$. for $\rho: \Bbb R \to \Bbb R$. parallel means now a multiple of any scalar including 0. The question: How to formulate that properly, shortly and concisely? I thought of $f$ is the gradient of an arbitrary scalar function $g$ of $\rho$. Like $f=\nabla g(\rho)$. But I am getting unsure ...
It is interpretation-dependent, to be clear. For instance, in the Bohm interpretation you do have meaningful trajectories. But those trajectories come with some very strange properties, e.g. nonlocality, first-order dynamics, etc.
If you know that $\vec f(\vec r) = h(\vec r) \nabla \rho(\vec r)$ it does not follow that there exists $\varphi(\vec r)$ such that $\vec f = \nabla \varphi$
As a simple counterexample, take $\rho(x,y) = x$ and $h(x,y) = y$
then $\vec f(x,y) = (y,0)$ has nonzero curl and is not equal to any gradient
If you have additional constraints on the $\vec f$'s which allow you to rule out that behaviour, then the story might change.
@NovaliumCompany It depends on what you mean by "what happens"
in a sense, we do. The wavefunction collapses.
but if you want a "deeper" explanation, then no, we don't.
We know the laws of quantum mechanics and we know how to apply them. But (I would argue, and this is a subjective statement) we don't really "understand" those laws to an extent that'd be sufficient to merit the term "understand".
@Rudi_Birnbaum yep. With what you've given, that's the furthest you can go.
Note that, in a philosophical sense, we're talking about metaphysics (in the sense that we're talking about the properties of a system apart from they could conceivably be measured)
People tend to use that word pejoratively but I don't
For me the point of QM and its interpretations is that it constrains the kind of metaphysics you can credibly adopt for the quantum world. But "constrains" is very different than "determines"
So there's a lot of freedom for different views on what's going on "beneath" QM, but they all have to be consistent with the QM formalism
(this is so long as we're not supposing that QM is merely an approximate description of something more fundamental. some people like that idea.)
But if (for a free particle) you measure position, then measure velocity, and then measure position again, you should not anticipate getting the same position back twice
if you choose to measure position, it will take a well-defined position immediately after you measure it (though it will then tend to spread as time goes on).
If you choose to measure velocity, then it will take a well-defined velocity, which it can retain or not depending on whether there is a nonzero force acting on it
@Semiclassical and that.
@AbhasKumarSinha This is not a "proof" because it does not start with hypotheses.
This is just a jumbled collection of formulae.
If it doesn't have text, it's not an argument, and if it's not an argument, it's not a proof.
Though, that said, the manipulations are roughly correct.
@AbhasKumarSinha you shouldn't need help with something as basic as a simple chain rule derivative if you want to read a book this high level tbh, why not learn simpler material first
You are not proving the last statement that's literally just the chain rule, you're supposed to apply this to the given example of $L$ which in this special case is $L = T - V$
You're saying that there exists $\vec B$ such that $$\nabla \cdot \left[ \left(\vec{B}\times \bigg (h(\vec{r})\nabla\rho(\vec{r})\bigg) \right) \rho(\vec{r}) \right] = 0?$$
@bolbteppa 1. Because I've been in the video, that Prof. also writes $-V$ instead of $-U$. 2. Your nick rhymes with that 3. You write more about Math SE than Physics SE (which I've heard about that from your former students on Quora) 4. You teach very well, (explain even a bit complicacy easily) means that you have some teaching experience. 5. You linked me a video long ago, which was from some different professor. which everyone would do, instead of giving theirs (specially who are indian)
In addition to that, You wrote your name a long ago (those who have not written that long name before can't type that fast!), this part gave me hints about it.
@AbhasKumarSinha maybe go through the introductory videos here then 'my' course here and then 'my' math course here and then 'my' quantum course here rather than the hardest books which casually use things that should be second nature to you
Bleh, I dunno. I'm personally sympathetic to the Bohm interpretation as providing a productive visualization, but there's real issues with viewing that as a "real" description of what's going on
It's a useful enough visualization for non-relativistic QM, but seems not to provide good lessons for how to understand relativistic QM or quantum fields
@AbhasKumarSinha I'm not sure what your goal is so not sure what advice to give but the videos above would be a good guide for extra study out of interest
At the same time, there's also a time-dependent probability current density $\vec{j}(x,t)$ (as dictated by the form of the Schrodinger equation)
together, those imply a time-dependent velocity field $\vec{v}=\vec{j}/\rho$. you can interpret those as generating streamlines of the probability density
That's just math, and you can take it as a convenient visualization (or not)
The basic image is that the wavefunction evolves in time, subject to the Schrodinger equation, and this same wavefunction determines what paths the particle can take
Hence why it's also referred to as the "pilot-wave" interpretation. The wavefunction evolves according to the Schrodinger equation, and it pilots the particle.
in the Bohm interpretation, one does insist that the particles have meaningful trajectories even when you're not observing them
this doesn't come without cost, though. for instance the Bohm interpretation requires a measure of nonlocality, which seems to conflict badly with the spirit of special relativity
The more common view is that this wave function collapse stuff is vaguely like saying if we flip a fair coin, until we flip it and observe which outcome we got we can only describe the situation of flipping a fair coin with a 'wave function' $\psi = \psi_{up} + \psi_{down}$ but once we measure it can only be one of $\psi_{up}$ or $\psi_{down}$, the idea this 'collapse' means anything is pretty much just misundertanding what QM is
and IMO it's only a productive visualization when there's only a few degrees of freedom involved. more than that and your configuration space has high enough dimension as to elude visualization
@bolbteppa eh, i think it's hard to boil it down to just that. i mean, if all you have is one type of measurement, then attributing classical values isn't such a big deal
it's once you have more than one measurement that you run into non-commuting observables
As an example: Suppose you ran a Bell-type experiment on entangled (singlet state) electrons where one observer only had access to one Stern-Gerlach device and the other could change their device freely
In that case you'll never be able to observe a violation of the Bell inequality.
At a minimum, the first person needs to have access to measurement settings $a,b$ and the second to measurement settings $b,c$. Once you've got that, then you can test the Bell inequality
The classicalness of this situation doesn't matter here, it's just a way of saying: 'before we measure there are certain possibilities, the wave function simply captures this all along with the probabilities of them happening in a linear combination, but once we measure only one of those outcomes actually happens'
Sure, but my point is that you could equally well account for that setup with a local hidden variables model.
It's only once you've got multiple measurement types that you can run into situations which LHV can't account for
(This is not to say one can't think of the up/down situation like that, to be clear. It's just that the necessity of doing so isn't apparent in that situation)
it ignores the incredible amount of experience that tells us how good our models currently are (in certain circumstances) and neglects more so, the spirit of science. Which is run with a model until you have something that contradicts it.
It may turn out that our models are wrong, and that we just keep going and going with better approximations. But possibly we arrive at a theory that is as good as it can be (not likely imo, but I doubt the future theories in 100 years won’t hold at there core principles from today). Even so, the theories we have now work ‘good enough’ in a lot of cases. For example. Classical dynamics explains an unbelievable amount of phenomena, and with additions of quantum theory explains even more.
My point is this, if you want to learn physics it’s also about understanding the stuff that works ‘good enough’ and knowing that as best as you can. Ultimately it’s going to be a part of the bigger puzzle.
@JakeRose Im not talking that physics will turn out to be wrong, Im saying that it may all be interpretation of our brain ans sensors. I mean, everything could break down to our brain, consciousness and interpretations...
The brain of a rat < the brain of a cat < the brain of a human < enchanced human brain
That’s more philosophy. Which physicists don’t take that well to. And fundamentally yes, everything is a measurement +interpretation of some sort. If you can’t measure it or at least measure something related to it then there’s little point even entertaining the idea
string theorist enters
But the point is, if you want to understand physics stop worrying about all that. Just learn!
@NovaliumCompany if every time you try and learn physics you stop learning because you talk about how it’s all gonna be trivial in 100 years then I’m afraid you won’t learn it very well
the way you enhance the brain is by studying and thinking.
The other some grad colleagues in a local forum were asking about a certain system of equations of unknown origin, and I was wondering if I would be allowed to ask about it in the Physics Exchange. My concern is that I'm not even sure if this system belongs in physics, and I was just hoping someo...
@JakeRose my own view is that the various interpretations of QM (I'm excluding modifications of QM, e.g. GRW) show that it's perfectly legit to have intuitions/concepts about what's going on with QM "behind" measurement
but any such metaphysics of quantum is at the very least constrained by the formalism.
So one shouldn't expect to be able to say much about what's going on in QM without having a rock-solid understanding of the bare formalism.
(one can of course critique such interpretations on grounds other than "is it compatible with the QM formalism". but if they can't do that much, they're not interpretations)