9:16 Small correction: probability is the area under the square of Psi, not the square of the area (i.e. square Psi first, then find the area) Nice video, though!
@@LookingGlassUniverse but how does ψ2 represent probability of finding the e - and how is that different from radial probability distribution function (4πr2ψ2 )
Nicely explained. Just one request, not just to you, but to all physicists: Please stop saying that a particle _is_ in multiple places at once and make it clear, that it is about _probabilities_ of being in any of those places. It took me (not a physicist) a while to learn about that distinction, and although that oversimplification might be helpful to spark interest in quantum mechanics, it's still misleading. It's those oversimplifications that often prevent real understanding of a topic.
@@LookingGlassUniverse Thank you for your answer! Please don't take my input as an attempt to correct you, that's not my place. I'm a layman and I'm well aware of Dunning-Kruger. But I'm trying to understand. I watched a bunch of videos (and will continue doing so), and one that helped me a lot understanding the superposition was FloatHeadPhysics' video about Schrödinger's cat. My understanding from that video is, that a superposition is something, that can not really be expressed in terms of classical physics. Saying that a particle "is" in a place (or multiple places at once) is an expression rather associated with classical physics, and thus misleading when talking about quantum physics. I had other moments where I stumbled about simplifications that prevented me for a long time getting real understanding, so I might be a bit sensitive in these regards. For example I understood the GRT when I understood the idea of bent spacetime, and that gravity is not an actual, but rather a fictional force (similar to the centrifugal force). It was Vsauce's video "Which way is down" that gave me that heureka moment. So when I'm nitpicking here it's really just about trying to understand it right, and I very much welcome your input on this. Besides: It's really amazing that people like you share their knowledge in the way you do, it's highly appreciated! Edit: The keypoint in FloatHeadPhysics' video is around 10:38.
@imagiro1 I saw that video too lol Honestly to say a particle is in all places at once before measurement isn’t necessarily incorrect, and in fact there is no way to prove this as incorrect because upon measurement a definite state is formed, eliminating the prior superposition. I’ve read that a safer interpretation of position and momentum is that particles actually don’t have any position or momentum prior to measurement, and that measurement itself creates position and momentum states. Who knows with this stuff, you know? Gotta love Mahesh though
@@austinlincoln3414 Yeah, I like the idea of a particle not actually having any of these properties until measured, but I didn't dare to say it like this - and I don't know if it is valid to say it like this. But as I said before, I think it makes a difference for someone who tries to capture the idea of superposition, to think of a particle actually being in multiple places at once or perceiving superposition as a new state, that can't be expressed through classic physics. To me it was quite helpful. However, of course I can be completely wrong - would not be the first time :) Btw, you might also like the video "Visualization of Quantum Physics" by udiprod. It was the first video that gave me what I needed to actually have ideas about quantum mechanics that turned out to be the basic idea behind quantum field theory, plus some other insights. Yet again, I'm grateful to everybody who shares such a knowledge, the more the better. And Mithuna does a great job. I hope she will still be involved in this conversation.
@imagiro1 Thanks for the recommendation! I’ll have to check it out. I wish I understood quantum mechanics more or could do the maths, but I haven’t gotten to that point yet. I read some of this book called introduction to quantum mechanics that was pretty good. That was where I heard about particles having no states until measurement. Also is your pfp from 2001? That’s dope
The wave function reminds me of the state of fan blades that spin so quickly that the position of the blade is blurred, where it appears to occupy everywhere on its path simultaneously and therefore simply appears to be in a state of superposition. Not that it is of course but rather it is our inability to keep up with its precise location at any time.
Sadly, simple explanations such as it oscillating between states quickly does not work. This can be shown in Bell's theorem, as it would violate the speed of light limit. The only way to get around particles existing literally in multiple states at once without violating the cosmic speed limit is to just treat the outcome of any physical interaction at all (even between two particles) as both random and relative to a chosen frame of reference, which is just relational quantum mechanics.
The analogy is good for visualizing the "blurriness" in position, and because you chose motion as the source of blurriness, you'll notice that it's also a good analogy for vilualizing the "blurriness" in momentum (the faster they spin, the harder it is to know how fast the blades spin). BUT! That is exactly why it's not a good analogy for quantum mechanics! In QM, it should be easier to see how fast the blades spin if they are blurry, and if the blades are perfectly clear, it should be impossible to tell how fast they spin.
@@Heulerado I am still struggling to understand the meaning of these words, like collapsing a wave function, and the concepts. i think the crystal only made it worse. so far, the way i see it is, "i have two measurements i can read, but i can only read one thing at a time. i can tell you the position at some exact moment in time, or i can tell you the rate over a given time span, doing either of which i will, weirdly, call "collapsing the wave function". and these confusing words and poor examples are why people think photons are watching to see if you are looking at them, and changing their behavior if they see you spying on them.
Very helpful and informative, thank you so much. Its refreshing to have this type of material presented in a clear and common-sense manner. Thanks again.
I think you're aware of this, and you were just trying to avoid the use of complex numbers for the sake of simplicity, but I think it's important to note that at 15:39, the Fourier transform of a delta peak is not a (co)sine, but a complex exponential. Conceptually this is important, because a cosine-shaped wave function suggests that the electron cannot be at certain positions, whereas a complex exponential suggests it can be anywhere. Also, the statement that 'it's going to be traveling at the speed given by that momentum' is a bit dubious. The phase velocity of a single momentum plane wave is given by c=E/p (follows from Einstein's E=hf, de Broglie's p=h/lambda, and c=lambda*f). It is the group velocity v=dE/dp of a collection of plane waves ('wave packet') that will travel with a velocity v=p/m.
Maybe the impulse p's velocity used in the Uncertainty Principle is *ACTUALLY* v/√(1 - v^2 / c^2) to avoid the spike becoming a Dirac delta [extended]function by complying with special relativity's universal speed limit of v < or = c.
Maybe the impulse p's velocity used in the Uncertainty Principle is ACTUALLY v/√(1 - v^2 / c^2) to avoid the spike becoming a Dirac delta [extended]function by complying with special relativity's universal speed limit of v < or = c.
You have clarified so much for me. I was struggling for a while with these concepts. Thank you for making this. I'm excited to check out the other videos!
That was a terrific video. It's the first time as a lay person to get that kind of detail about what a wave function is. I do have a 4-year degree in computer science and electrical engineering from a long time ago, but even now I could roughly understand what's going on. It makes me wonder why so much hand waving is being done instead of just giving some details like this. I understand that plenty of people have no college-level math so showing them these things in a mathematical way would be useless. But you also managed to present intuitive description. So I guess in summary I wonder why a video like this hasn't been done in the past, or somehow I missed it despite watching hundreds of videos related to this.
When you hear someone saying "here's an 'anology' for a wave function..." and see a piece of salt-like crystal at her hand, your expectations makes a top until....well the anology completes with the Fourier transform :-) Don't get me wrong. I am an electrical engineer all too familiar with F. transform and even took a lecture (40 years ago) in quantum mechanics where the uncertainity principle was derived in an amazing and supring way using the Fourier transform as far as I recall, I still couldn't help smile your mundane(!) anology :)
Wonderful explanation! 1:37 Your analogy with the glass cube leads me to believe that you would enjoy my visualization of the fractional fourier transform (can be found on my channel)
Well, technically it is the area underneath the function defined by the product of the conjugate of the wavefunction and the original wavefunction; and/or the area of the function defined by the square of the magnitude of the wavefunction. The wavefunctions take a position in space and time and returns a complex number, a linear combination of 1 and sqrt(-1)
Thank you a billion times! Your videos are so interesting! It would be great if you were able to walk us through the application process where the equations are actually used to create a product.
I wonder if it would be fruitful to explore the wavefunction in other bases besides position and momentum. Perhaps a density basis, or energy density basis. Or energy-frequency dual basis.
Great presentation with easy to follow mathematics aiding clear communication. This is probably the best explanation of the Wave function I've even seen online. The course youre offering looks very enticing after watching your teaching skills. Though the link says the product is not available. Is it still going to be offered?
Ooh dear, a c r i s t a l ... I can see certain people banging their tables and shouting "I knew it has to do with cristals!!!"🙄...Sorry just couldn't resist. 😉I love this series! Public education at it's best! AND, the "dead" declared social media alive and kicking;-) Wonderful❤
the crystal analogy is better than you let on: we do see projections onto a space axis or a momentum axis, and they are 90 degree rotations of the projection direction. If you don't believe me, see "fractional Fourier transform".
The electron is only in a position to a reference point, right? as our earth and sun are travelling at great speeds, how does this relate to the electrons existence?
Why do crystals and chairs and dogs and trees appear to us as objects that persist through time? I understand that particles locally coalesce when measured but what keeps the particles in the chairs and trees coalesced and consistent through time? Is it a constant interaction with the environment or do they coalesce once and then persist without experiencing a delta in energy that destroys the persistence? Or is there some information encoded in the wavefunction, if it is universal, that causes objects to persist?
Just because an object makes waves or an interference pattern, doesn't mean that the state of it has a wave-function. A wave-function has nothing to do with a wave at all. The wave-function is not meant to describe the physical trajectory of a particle, but to encode the probabilities of their state. The interference pattern, a measurable phenomenon, strongly suggests a wave-like behaviour, but it has absolutely nothing to do with a wave-function, and it isn't proof that light moves in a wave-like trajectory either. It has behaviour like a wave, but it does not necessarily move exactly like a wave. A wave-function does not describe reality at all. It is a mathematical tool to calculate the probability amplitude of a particle's state, not a physical wave. Quantum mechanics allows us to calculate the likelihood of where a particle might be, but it does not claim to describe the actual path that the particle takes or where it exactly is. You are only calculating where it might be, where it probably is. This is often misinterpreted and overstate to the general public by physicists, particularly when it comes to the double-slit experiment, saying things like "particles decides how to behave" and other such nonsense. No, they don't decide how to behave. They just behave like they do. Figure out how they do it instead of making a particle seem like it is reliant on cognitive abilities to behave. You should be very sceptic when someone invokes quantum physics. It means that they are talking about something that might be true, but there is also a probability that they don't know what they are talking about at all. People do this with AI as well, but that is another subject. Furthermore, there is no collapse of the wave-function, simply because there is no physical wave-function. When it is said that a particle is in a superposition, it's only so because the probability of that particle can be in either of two states, and the probability says that it can be in both states at once too, but it never is in two states at the same time in reality. Only information can be in two states at the same time. Superposition is a a combination of outcomes, that is all. When measured, the particle is in one state, and one state only. By that I mean states that are not opposed to one another, and contradictory, so that it becomes a paradox. If you use quantum physics to calculate something that is exactly the same result as in nature, then that means you can also find the result of that very same thing with classical physics. Just because you use quantum mechanics to describe an object, does not make quantum physics real. Just because it has unparalleled accuracy doesn't mean the calculation you made is what reality is like. It just makes the probability of what you found to be true, and as such, you can find that to be true with classical physics as well. What you did was to use probability as a tool to find the actual real object. Good. Now that you found it, you can find it again by observation and measurements, unless it's a photon, because nothing else in the universe moves as fast as it, so only another photon can track or observe another photon.
When you say that the infinite spike is not a real function do you mean to say that the function must be continuous? If so, is that function necessarily nonzero everywhere? does that mean that there is a small but nonzero chance of finding an electron miles away from the testing station? Is this what quantum tunneling is?
The wave function does not have to be non-zero everywhere. In the particle in a box scenario the wave function is zero outside and on the sides of the box.
I like your channel a lot but I believe you have a subtle misconception about what the wave function is. The wave function isn't a/the state in a Hilbert Space it is a representation of the state in a chosen coordinate space (may be position or momentum or something else). It is inherently tied to a coordinate system. Yes it contains all the information of the state but it isn't the state. The same way the coordinates of a vector represent an abstract vector. This also allows the wave function to be a Dirac Delta since it is only defined in the spectral decomposition of the state and is always integrated over. en.m.wikipedia.org/wiki/Quantum_state
I agree that physicists use the word "state" rather than "wavefunction" for the obvious reason that it's, uh, not a function. When we do use "wavefunction", we usually give it its full name in whatever basis it may be ("the wavefunction in the momentum basis", or "the position wavefunction"), unless it's obvious from context, etc. But I think it's valid (or at least instructive, if taught like she has in this video) to generalize it and talk about searching "THE wavefunction", which we can project into the various bases. That would be describing the state vector |ψ>, like the video says. The big problem is that it's, uh, not a function. Physicists are not new at giving things the exact wrong names, but I think we shouldn't be encouraging it. So a big disclaimer would've been nice.
@@Heulerado Thx for the reply yes some physicists mix the terminology. I believe this has the historical reason that and most of the original equation for QM also work for the representation of the state. Hence the distinction is not necessary in this context. But i have a particular issue with the following scene at 12:31. Where she clearly states the opposite of what is correct. Which will confuse people if they try to really understand it.
As I understand it, MWI proponents like Looking Glass Universe (which I do not support MWI personally) argue that the wave function is only relative to a chosen coordinate space because ψ is just a limited perspective of a much bigger Ψ referred to as the universal wave function. If you change your coordinate system you change the perspective that you are looking at Ψ from so that changes ψ. Hence, for them, it is still describing the ontological state of the system, but just from a particular limited perspective, kind of like describing one side of a box at a time due to your perspective because you cannot see the full box at once.
@@amihartz Yes, measurement basis effects the output probability distributions and measuring in the sub-hilbertspace as you are suggesting is described by the Generalized Measurement via partial traces (en.wikipedia.org/wiki/POVM) in open quantum systems. As far as i know one needs an open quantum system for the MWI interpretation to work and so far I don''t see how these interpretations are at odds. AFAIK one can write down a version of MWI using still the abstract vector notation, no problem. Can you point in the direction where this is a problem?
Where physicists frequently drop the ball in their explanations of how a particle can be in a superposition of quantum states is in failing to make clear that the multiple "places" they're referring to are not physical locations in 3D space. These are coordinate positions in the abstract, multi-dimensional realm known as Configuration Space, where the quantum wave-function is defined. Moreover, the "weights" associated with these coordinate positions are not merely negative, they are complex-valued probability amplitudes, a completely non-physical aspect of the quantum wave-function. Where such a superpostion becomes physically observable is at the point of measurement, where it is probabilistically projected into a particular location in 3D space in accordance with Born's rule (which calculates the conjugate square of the superposition's probability amplitudes to produce a real-valued probability of observing the particle in that location). In short, subatomic particles do not occupy multiple locations in 3D space; they manifest as a superpostion of locations in the abstract, multi-dimensional realm of Configuration Space.
When you say the electron could be anywhere within the box, and not discretized to multiples of a tiny Planck length, you are suggesting we've measured that and know it is true by measurements. But we can't measure a position that accurately. Isn't it true that in fact space MIGHT be discretized to the Planck length?
12:32 The left hand side is NOT the wave function. It’s the state vector (or the ket vector). It’s not even a function! The function \psi(x) on the right hand side is the position basis wave function. It’s called position basis wave function because the right hand side is the base decomposition of the state vector by the position basis.
I've never understood why people often call the state vector the "wave function" when it's not literally a function. Although, maybe it is because my learning is through quantum computing and not through traditional physics education. I always associated the state vector with |ψ⟩ and the wave function with ψ(x)=⟨x|ψ⟩.
Whether you call it a vector or a function is irrelevant. It's an element of a Hilbert space and the only meaning of that is really just unitarity, which physically is completely trivial.
@@schmetterling4477 State vectors and wave functions are different. State vector is written |\psi> and position basis wave function is written . They are completely different mathematical objects.
@@vonneumann6161 Are we discussing the difference between a physical vector and its coordinate representation now? That is completely irrelevant in this case since there are no physical vectors in quantum mechanics. All of this plays out in a completely abstract vector space. Unlike physical vectors none of it exists in reality.
The wave function \psi is complex valued. Probabilities are defined by the square |\psi|^2. Obviously by passing from \psi to |\psi|^2 we have wiped out a lot of information (namely the phase). If this were the whole story, QM would work with ordinary real valued probability densities instead. "Superpositions" of probability densities can also easily be defined: every convex combination of probability densities is another probability density. You need to explain why this is not enough for QM, i.e. what the significance of the phase is. In particular in view of the fact that \psi and a*\psi describe the same state , for every scalar a with |a|=1.
There are no probabilities in quantum mechanics. That's just a misnomer. What we are estimating are event frequencies. For those we need the Born rule because they depend on the spectral absorption function of the measurement system. The |psi|^2 case is just a lazy way to write the Born rule for a unity measurement projection operator. There is no actual measurement system that implements such an operator. It's a convenient fantasy for theorists, no more, no less.
If we design a certain quantum cellular automaton and then install a quantum particle in it and a device that measures it, how complex does this device have to be for the wave function of this particle to collapse (how minimal can an observer be whose components are also quantum)?
No one really knows where exactly this threshold between classical and quantum is. And why. And what exactly is the collapse and how it happens. See "measurement problem".
@@thedeemon The transition from quantum to classical is given by repeated irreversible energy transfers. We have known this very well since the late 1920s. That you do not know this is merely a function of your intellectual laziness. All of this can be found quite easily in the physics library.
@@schmetterling4477 Oh c'mon, multiple books (even quite recent) have been written on this subject of measurement problem still not being solved. Update your physics library.
When something collapses, like a building, it goes from having large extended structure to a pile of rubble much smaller than its original size. When a wave function "collapses" it's reduced to one of the basis states. So if just measured where the particle is, the answer you'll get is "at x=1" and the state it's now in is the wave function for a particle at x=1. This will in general be a much simpler state than before you measured it. So it's collapsed in the sense that it's squished down to a spike at just one place on the x axis.
The wave function has been measured/quantized/communicated to have practically settled into a quantum state called an eigenstate (one of the countably many dictated by the measuring apparatus and the object being measured) which can be made by a subsequent measurement to produce a specific rational number called its eigenvalue. Every eigenvalue isn't a point but an interval because the number 1 = 1/3 × 3 = 0.33... × 3 = 0.99... is between 0.9 and 1.1 = 2 - 0.9, between 0.99 and 1.01 = 2 - 0.99 and so on. There is *ALWAYS* a gap between the two bounds. All other rational numbers can obtained by scaling the number one with a nonzero rational number.
As one produces the eigenvalues from eigenstate and the eigenvalues always have a fuzziness due to the limited resolution of the measuring apparatus, there is an inherent uncertainty in the eigenstate. A wave function which has "collapsed" means that it is practically in an eigenstate with a fuzziness not measurable by the apparatus but can be consequential via positive feedbacks in subsequent measurements (i.e. the Butterfly Effect in Chaos Theory).
A classical measurement had occurred meaning there is now a new distribution of where the particle could be based on each infinitesimal of the basis or whatever and how each infinitesimal would react classically. At least that’s what Landau told me in his cool special books
I've always been curious how an (e.g.) electrons wavefunction relates (or doesn't) to it's associated electromagnetic waves. Ditto for a photon. Are these entirely unrelated? Which feels unlikely, but I've never seen it addressed.
Amazing question. The EM field is pretty intimately related to “photons”- vibrations of the field are what leads to light. This topic is worthy of a whole course in itself!
@@salerio61 My point wasn't the level of education, but the level og educational representation. Two very different things. No one expects this level of academic expertise to be taught in elementary, middle, or high schools. But when a presenter has strong academic qualities and the ability to visualize in a way that gives students a better chance of learning a subject, regardless of level, a good foundation is laid for a broad understanding of the subject among students.
@@davidzuccaro1906 The uncertainty principle has nothing to do with quantum mechanics to begin with. It's just a general property of Fourier transformation-like operators and applies just as well to water waves. In quantum mechanics these quantities aren't describing the position and momentum of one system to begin with. They are describing distributions of positions and momenta of a quantum mechanical ensemble... which is an abstract that nature doesn't know anything about. For an actual individual measurement you can get any combination of position and momentum that you want.
Can you do a piece on the nature of WF collapse and what constitutes an observation (like why doesn’t it collapse sometimes and not others - for example recording the result on a computer but deleting the information before checking it? Turning on and off photon detectors so they are keeping a record/not keeping a record? Removing them? Adding detectorsbut leaving them unplugged etc Does this maintain interference?). This is not really clear in all the QP videos I’ve seen. I feel this is a very misunderstood aspect of quantum mechanics.
It would be interesting since she believes in the Many-Worlds (multiverse interpretation) and I have never been able to visualize what it even _means_ to make an observation in terms of Many-Worlds as it is not clear to me where the observables even are in such an interpretation. In relational quantum mechanics, it is very simple, an observation is just an interaction between any two physical systems _from the perspective of one of those systems,_ i.e. the realization of the properties of the system does not occur for physical systems not part of the interaction, meaning that what properties of a system have been realized or not depends upon reference frame.
Wave function collapse is a meme on the internet. It's not a thing in physics. You won't even find that term used in well written textbooks like Sakurai. It has no discernible scientific function. I haven't even seen a precise definition of what it is supposed to mean.
@@schmetterling4477to me it means how observations (single outcome) don’t tally with what QM seems to say - which is the universe is a wave function yet we don’t see superpositions.
@@jamesmather7896 You won't find a single quantum mechanics textbook that says that the universe is a wave function. Wheeler liked to talk about "the wave function of the universe", but that is a horrible idea that leads to nowhere. The universe does not have a wave function. A wave function is a mathematical abstract that allows us to estimate experimental event frequencies by APPROXIMATING an actual (serially performed) quantum experiment with an imaginary (non-existent!) ensemble of identical copies of the same experiment. Many theorists don't seem to know this (or they just don't care about this level of detail) and then they are talking nonsense like Wheeler did. Laymen do, unfortunately, have a tendency to amplify the nonsense in their minds while completely suppressing the actual science facts. :-)
@ then how do you explain single outcomes in observations instead of us observing the superposition of the wave function? There is no doubt that there is some addendum to Quantum Physics as that is what we observe - call it what you will: decoherence / WF collapse / measurement / interaction etc.
Great explanation, but I have to nitpick about the math. A few times you say a function that’s zero everywhere but infinite at one point “isn't even a real function.” No, it’s a perfectly fine function mathematically! Maybe what you mean is that it’s not continuous, or its range isn’t finite?
No, that's not a function but a distribution. The mathematicians had this figured out in the 1920s, which is why von Neumann could write his book about QM in 1932 without needing any new math.
It was very nicely described and I get it but what IS the wave function though? Is it just a mathematical model that works or is it representing something physical in reality?
She believes in a class of interpretations called MWI which argues that the wave function is not only a literal physical entity but there is only actually one of them in existence, and in fact it is the _only_ thing in existence, i.e. the entire universe is just one giant infinite-dimensional "universal wave function" and the little ones we perceive are just due to us only perceiving a part of it at a time from a particular perspective.
No one knows. It's something different physicists take different positions on. Of the various interpretations of quantum mechanics, some say it's real and others say it's not; but even the ones that say it's real are a bit vague on where/how the wve function is, since the function evolved in a many-dimensional configuration space, not 3d physical space. Given that the wave function can interfere with itself (as in the double slit) it seems awfully likely that it is physically real to me. But we are a long ways off from understanding how that can be or how it works.
@@erinm9445 I see. It's interesting though because you could train some neural network to make correct physics predictions but would you call that science? Probably not right because it would not tell you anything about the physical world.
When you say that, actually the electron is spread out, isn't that an interpretation of the math, a hypothesis, and not necessarily what is actually the case? Don't we not really know what is actually the case, we only know what the outcomes are that we measure? And we know they're well predicted by the wave function equation, so we know that whatever is the actual case, it must fit with that wave equation. Isn't it even possible that until measured, the electron doesn't exist in physical space at all?
यह अपारदर्शी और पारदर्शी ऊर्जा है जो केवल ईस आवृत्ति में होती है जैसे अवकाश या वायु या प्रकाश लेकिन दर्शीय ऊर्जा जैसे तरल और पदार्थ में क्या होता है? अवस्था के अनुसार आवृति मे बदलाव आता है और इसकी डिजाइन और पैटर्न भी बदल जाते हैं। 0=(-0/-0)
So the position vector is a vector in a basically uncountably infinate dimensional space!?! The physics gods had me in mind when they made this place. Realtivity and its differential geometry and quantum with its mind bending abstraction that somehow logical if you think about it hard enough. both my favorite things.
Basically, yes, but the dimensionality is really irrelevant. All important properties are following from the unitarity condition which specifies that we aren't losing any members of the quantum mechanical ensemble during its evolution.
No. It's a mathematical abstract. It is ontologically equivalent to a probability distribution. There are lots and lots of such abstracts in the mathematical description of nature. Real numbers, for instance, are also just abstracts. Nature doesn't implement them.
@@schmetterling4477Thank you very much for your answer, but I don't see what are your arguments. Yes, you should read the article in detail, the important finding is on page 8. The incompleteness of QM, it is known as a reference Penrose is explaining this in many UA-cam videos; the scandal of QM was written by Nico van Kampen in 2008 etc. prof UNNIKRISHNAN described the foundational problems of QM in section 1, also related to experiments, in section 3 he is showing how this action wave formulation resolves the foundational problems of QM. In Prof Unnikkrishnan own words "The only way that the action principle can work is if the action is manifest in dynamics as a periodic entity in both quadratures, capable of interference. THEREFORE Hamilton mechanics (equation) that does not use this periodicity is incomplete. The equation in the article is the correct one. Then there is a single action mechanics- there is no division into classical and quantum mechanics ( since the QM measurement problem was also resolved in this way, there is no need for classical apparatus either, unlike in the current theory). QM is not wrong but it is right about ONLY statistical results. And its foundational premise is flawed." In any case if you don't have solid arguments based on experiments, please read in detail the article and references, don't let you trap, if not good luck with your views of reality.
@@schmetterling4477Thank you very much for your answer. The important finding is on page 8. The incompleteness of QM, it is known as a reference what Penrose is explaining in other youtube videos; another one: the scandal of QM was written by Nico van Kampen in 2008 etc. prof UNNIKRISHNAN described the foundational problems of QM in section 1, also related to experiments, in section 3 he is showing how this action wave formulation resolves the foundational problems of QM. In Prof Unnikkrishnan own words "The only way that the action principle can work is if the action is manifest in dynamics as a periodic entity in both quadratures, capable of interference. THEREFORE Hamilton mechanics (equation) that does not use this periodicity is incomplete. The equation in the articleis the correct one. Then there is a single action mechanics- there is no division into classical and quantum mechanics ( since the QM measurement problem was also resolved in this way, there is no need for classical apparatus either, unlike in the current theory). QM is not wrong but it is right about ONLY statistical results. And its foundational premise is flawed." . so take care and good luck !!!!
To be honest she has such beautiful face, voice and especially eyes that it is difficult to concentrate on the information provided. Still she explains so clearly that even I understood everything. Another problem is to remember))
@@ArnMH81 Dude, she starts you off with an animation of the double slit. The double slit has absolutely NOTHING to do with quantum mechanics. It's a simple classical wave experiment from 1801. You can do it at home with water waves in your bathtub. If you are willing to think for more than a minute about it, then you will also notice that Planck's constant does not show up in the double slit at all. How in the world is an experiment that doesn't contain Planck's constant OR at the very least two-quantum correlations like entangled quanta supposed to be about quantum mechanics? ;-)
@ she has another video when she performs double slit at home. I actually did it myself after her video. Anyway I guess the point is not to make sure that double slit experiment is not some lie. It has been done millions of times. But in terms of accessibility of the information provided she is very good. Ok I don’t deny, her voice and looks influence my opinion. I am bias if you need truth )). Still I think she explains quite simple. For example the math for wave functions ant operators- some of them I didn’t want to go deep into because I had that feeling of insurmountability. After her explanation I realized that I was able to get it. And I studied further.
@@ArnMH81 The double slit is a demonstration of classical wave behavior. As such it is mildly useful. Beyond that it teaches nothing. In the context of QM it teaches us nothing and it is not being used in the professional teaching of QM for a number of reasons. For one thing it is not even unitary. Why ask a beginner who is just about to learn about wave functions to toss them into the bin right away because the sub-trivial example doesn't work with them? Very, very poor teaching, indeed.
@schmetterling4477 no I mean a discreet quantum universe can't really exist. The inverse extrapolation of Gödel is that his theorem is a complete description of everything, a TOE
It's misleading to say the wavefunction is the "state" of a system. It's actually a representation of our knowledge of what would be observed if the system were measured. (See the Born rule.) Our knowledge of the system state is usually limited & incomplete, so the predictions of measurement results are usually probabilistic, with the probabilities summing to 1. "Superposition" is a fancy way of saying there are at least two possible results if the system were measured. In other words, each of the probabilities is less than 1.
States in QM are abstract vectors in Hilbert spaces, and projecting a state vector onto the position basis gives the wave function. The wave function contains all the information of the state in the form of complex coefficients, so it's perfectly fine to call the wave function the state.
>simplelife1021 : The claim that the known information is the _complete_ information is an unproved metaphysical claim promoted by Bohr, which was disbelieved by Einstein, Schrodinger, Bohm, Bell et al. Also, a projection can lose information. Consider a photon that passed through a horizontally-oriented polarizing filter. Projecting its state to a vertically-oriented basis wavefunction (in order to predict the probabilities of the possible results of vertically-oriented measurements) suggests the photon is in a superposition of up & down polarizations, which loses the info about its actual horizontal polarization prior to the measurement.
@simplelife1021 : The claim that the known information is _complete_ is a metaphysical claim promoted by Bohr, disbelieved by Einstein, Schrodinger, Bohm, Bell et al. A projection can lose information. Consider a photon that passed through a horizontally-oriented filter. Projecting to a non-horizontal basis loses the info about its horizontal polarization.
The wave function have nothing to say about the elementary particles contained information and behavior. If something is hidden behind complex abstract definitions or math formulas, is wrong. The truth is always easily understandable but not always easy to live.
For it is written Vs Life we're living Alphabetic effects above neck => It's this way it's that way Mathematics is rooted in much the same way Doing my best to stimulate content Keep up the good work
This video shows a natural quantized effect using a version Euler’s Contain Column Theory? This video of an engineering test models the quantum physics problem called “Particle in a Box”. The wave function is used to predict the location of quantum particles, atomic structure and molecules. ua-cam.com/video/wrBsqiE0vG4/v-deo.htmlsi=J5jCMxq3pRe_P-gG Very stable trusses and truss core panels have been fabricated using the process shown in the video. The video shows how materials naturally respond to induced stresses in a “quantized“ manor. The process’s load/deflection’s sawtooth curve with its exponential fit, in the engineering, study shows in detail the bifurcation area between quantum jumps. The effect has been used to make light weight structures and shock mitigating/recoiled reduction systems and earthquakes isolation systems. The model shows why with the, exponential load increase and loss of resistance at phase change, quantum jumps are so very fast.
@ true that! First saw the effect, all knew was the frequency and energy relationship. The first ten years I was trying to make light weight structures with it. The next ten years I found in a engineering handbook on this column thing. It’s the old pushing down on a wooden yard stick buckling effect. The last ten years people have been telling me it looked like this wave function thing in modern physics. Had no idea what either of those things were. Now, Just trolling 🧌 it in front of the mostly elitist professionals so they can tell me it’s a bunch of crap 💩!
now i know that electrons seems to be blue and not smiling. because It could not find an exact place to stay? your smile instead indicates that you are in the right place. Best wishes from Germany.
a fun little classical model i like a lot, because it works pretty much like matter for everything from gravity to optics to whatever, except involved electrodynamics and so on; Is light in a box. lets imagine that we have a semi transparrent baloon filled with light and that only relfects and transmits light, so we essentially have a little greenhouse filled with light and inflated by the radiation pressure inside, lets also imagine for the sake of the agrument that the radiation pressure inside is constant, so it stays inflated. the light inside is going to carry all the momentum of the mass, and it is a mass simply by virtue that its state of motion is alterable, it behaves just like normal matter, e=mc^2 and all, to change its momentum we just scatter some light of it, some of it gets absorbed by the baloon and some scatters off it, and its packet of light waves inside is now changes, in the previous staionary reference frame, where it has over all 0 momentum in the radiation inside, it now has some momentum in a different direction, because of the transmission and relfection properties it also contains more energy but scatting it back and forth smoothly and slowly does not change its stable mass, because if it is carrying too much light with respect to the exterior more will leave than enter and so on. let the surface fluctuate with the light inside and let there be uncertainty in the content of the light inside, thens subject it to some light you scatter off it to measure its position or momentum, find me the uncertainies :P. it is a classical model so what gives?
😢 Trying to understand the Sun which I have learned that my thoughts are faster than the speed of light of a response with witnesses. Would love to be wrong for beliefs of entanglement. No fear of looking at it ,
there is no such thing as superposition. everything can be calculated. i hate it when physicists talk about probabilities. its just a 100 year false mode of thinking. heisenberg was a moron. its not going to collapse. there is this thing called bohmian mechanics
you can view the Dirac Delta Function as the Dirac Delta Distribution: it is an infinitely narrow Gaussian. Fig. 2 en.wikipedia.org/wiki/Dirac_delta_function
Voice-work with please, is too crunchy when you modulate-dont do this, you have lovely natural voice, do not low it or make high and five this crunchy terrible sound like old motor or ciuch , old frog. You need speak loder if you modulate it and do not hold tour voice in deep throut-move it to teeths, take few lessons, it could be a reason. Make nice, smooth voice. Dont repeat the same things but hudge plus for details for begeeners , no one tells it-only you, but you need better form, graphs...
Here is one of the "End Points".. Where is "The Green" in "The SpaceX Launch"...???... "Google A Rainbow Picture"...Take Care...Bye... Look at "The Rainbow" as "A Gentle Pendulum Clock"... Green Arrow In Or Down.. Orange Arrow Up Or Out.... "Yellow Is The Tick".... SpaceX Launch Speeds Up The Pendulum.... "Squeezes The Rainbow Together"... Green Goes "In" The Engine.. You Can Only See Orange "Out".. Take Care...Bye...
No, it’s still fairly fringe theory. Zeno's paradox is not a real thing - it only existed because the ancient Greeks didn’t understand infinite series, I.e how an infinite number of terms could have a finite sum.
Zeno's paradox can be solved using limits in calculus without using discrete spacetime. A lot of physicists think spacetime _might_ be discrete and there are some tentative theories suggesting this like loop quantum gravity, but none of them have evidence for them and are just speculation.
Heat waves. Mass disolving internal magnetic fields as external heat energy outside of entanglement of mass as campfires. Lightning burns through atmospheric gasses disolving into external heat energy outside of entanglement of mass as fire bouncing off atmospheric gasses lighting up the sky instantaneously. Campfires slowly disolving internal magnetic fields disolving into external heat energy as fire. Electricity is rapid heat exchanging through mass faster than earth's quantum internal magnetic fields grounding currents through its nucleus or core where external heat energy is strongest as molten masses held in centrifugal force cycling circulation as mass. Mass occupies space within mass as outward force of pressure known as magnetism weightlessness outside of a farther reaching greater magnetic field. External magnetic fields spinning all external heat energy within its field outside of the nucleus or core is a repulsion external magnetic field. External magnetic fields spinning all external heat energy away from renewable heat energy of mass causing mass to disolving into external heat energy outside of entanglement of mass as outward force of pressure no longer holding internal magnetic fields grounding currents through its nucleus or core. External heat energy outside of entanglement of mass doesn't ground currents it strips away renewable heat energy singularities away from internal magnetic fields. They disolve from within as outward force of pressure of cold repulsion of occupational space. Conservation of heat energy singularities outside of entanglement of mass as fire. Fire is external heat energy outside of entanglement of mass. Accumulation is disolving internal magnetic fields as external heat energy outside of entanglement of mass as fire. Stars decay their atmospheric gasses as fire lighting up the atmosphere like a filament bouncing off atmospheric gasses still present.
9:16 Small correction: probability is the area under the square of Psi, not the square of the area (i.e. square Psi first, then find the area)
Nice video, though!
You’re absolutely right! Sorry about that!
@Lachy314yup!
@@LookingGlassUniverse but how does ψ2 represent probability of finding the e - and how is that different from radial probability distribution function (4πr2ψ2 )
"it's a common misconception in quantum mechanics that everything is discretised"
^ how to become my favourite UA-camr in one easy step
It is called quantized.
Nicely explained. Just one request, not just to you, but to all physicists: Please stop saying that a particle _is_ in multiple places at once and make it clear, that it is about _probabilities_ of being in any of those places. It took me (not a physicist) a while to learn about that distinction, and although that oversimplification might be helpful to spark interest in quantum mechanics, it's still misleading. It's those oversimplifications that often prevent real understanding of a topic.
I think particles are. in superposition, it’s not merely a probability thing
@@LookingGlassUniverse Thank you for your answer! Please don't take my input as an attempt to correct you, that's not my place. I'm a layman and I'm well aware of Dunning-Kruger. But I'm trying to understand.
I watched a bunch of videos (and will continue doing so), and one that helped me a lot understanding the superposition was FloatHeadPhysics' video about Schrödinger's cat. My understanding from that video is, that a superposition is something, that can not really be expressed in terms of classical physics. Saying that a particle "is" in a place (or multiple places at once) is an expression rather associated with classical physics, and thus misleading when talking about quantum physics.
I had other moments where I stumbled about simplifications that prevented me for a long time getting real understanding, so I might be a bit sensitive in these regards. For example I understood the GRT when I understood the idea of bent spacetime, and that gravity is not an actual, but rather a fictional force (similar to the centrifugal force). It was Vsauce's video "Which way is down" that gave me that heureka moment.
So when I'm nitpicking here it's really just about trying to understand it right, and I very much welcome your input on this. Besides: It's really amazing that people like you share their knowledge in the way you do, it's highly appreciated!
Edit: The keypoint in FloatHeadPhysics' video is around 10:38.
@imagiro1
I saw that video too lol
Honestly to say a particle is in all places at once before measurement isn’t necessarily incorrect, and in fact there is no way to prove this as incorrect because upon measurement a definite state is formed, eliminating the prior superposition. I’ve read that a safer interpretation of position and momentum is that particles actually don’t have any position or momentum prior to measurement, and that measurement itself creates position and momentum states. Who knows with this stuff, you know? Gotta love Mahesh though
@@austinlincoln3414 Yeah, I like the idea of a particle not actually having any of these properties until measured, but I didn't dare to say it like this - and I don't know if it is valid to say it like this.
But as I said before, I think it makes a difference for someone who tries to capture the idea of superposition, to think of a particle actually being in multiple places at once or perceiving superposition as a new state, that can't be expressed through classic physics. To me it was quite helpful. However, of course I can be completely wrong - would not be the first time :)
Btw, you might also like the video "Visualization of Quantum Physics" by udiprod. It was the first video that gave me what I needed to actually have ideas about quantum mechanics that turned out to be the basic idea behind quantum field theory, plus some other insights.
Yet again, I'm grateful to everybody who shares such a knowledge, the more the better. And Mithuna does a great job. I hope she will still be involved in this conversation.
@imagiro1
Thanks for the recommendation! I’ll have to check it out. I wish I understood quantum mechanics more or could do the maths, but I haven’t gotten to that point yet. I read some of this book called introduction to quantum mechanics that was pretty good. That was where I heard about particles having no states until measurement. Also is your pfp from 2001? That’s dope
The wave function reminds me of the state of fan blades that spin so quickly that the position of the blade is blurred, where it appears to occupy everywhere on its path simultaneously and therefore simply appears to be in a state of superposition. Not that it is of course but rather it is our inability to keep up with its precise location at any time.
Great analogy
Sadly, simple explanations such as it oscillating between states quickly does not work. This can be shown in Bell's theorem, as it would violate the speed of light limit. The only way to get around particles existing literally in multiple states at once without violating the cosmic speed limit is to just treat the outcome of any physical interaction at all (even between two particles) as both random and relative to a chosen frame of reference, which is just relational quantum mechanics.
I like that
The analogy is good for visualizing the "blurriness" in position, and because you chose motion as the source of blurriness, you'll notice that it's also a good analogy for vilualizing the "blurriness" in momentum (the faster they spin, the harder it is to know how fast the blades spin).
BUT! That is exactly why it's not a good analogy for quantum mechanics! In QM, it should be easier to see how fast the blades spin if they are blurry, and if the blades are perfectly clear, it should be impossible to tell how fast they spin.
@@Heulerado I am still struggling to understand the meaning of these words, like collapsing a wave function, and the concepts. i think the crystal only made it worse. so far, the way i see it is, "i have two measurements i can read, but i can only read one thing at a time. i can tell you the position at some exact moment in time, or i can tell you the rate over a given time span, doing either of which i will, weirdly, call "collapsing the wave function". and these confusing words and poor examples are why people think photons are watching to see if you are looking at them, and changing their behavior if they see you spying on them.
Grateful for this as I'm reviewing for a Quantum Mechanics final! Thank you!
Oh wow! just found this. I'll have to watch them all. Ty for your efforts!
Pure Gold - Camly Explained with a very positive attitude. Thank you!
Good explanation. I particularly like your analogy with the crystal. Excellent.
Very helpful and informative, thank you so much. Its refreshing to have this type of material presented in a clear and common-sense manner. Thanks again.
I think you're aware of this, and you were just trying to avoid the use of complex numbers for the sake of simplicity, but I think it's important to note that at 15:39, the Fourier transform of a delta peak is not a (co)sine, but a complex exponential. Conceptually this is important, because a cosine-shaped wave function suggests that the electron cannot be at certain positions, whereas a complex exponential suggests it can be anywhere.
Also, the statement that 'it's going to be traveling at the speed given by that momentum' is a bit dubious. The phase velocity of a single momentum plane wave is given by c=E/p (follows from Einstein's E=hf, de Broglie's p=h/lambda, and c=lambda*f). It is the group velocity v=dE/dp of a collection of plane waves ('wave packet') that will travel with a velocity v=p/m.
The spike is the Dirac delta function?
I think peetty much yeah. Issue is the uncertainty principle would make it have infinite possible velocities, with nin-decaying probabilities.
Yup!
Maybe the speed of light in free space being finite helps prevent the spike from being the Dirac delta [extended]function.
Maybe the impulse p's velocity used in the Uncertainty Principle is *ACTUALLY* v/√(1 - v^2 / c^2) to avoid the spike becoming a Dirac delta [extended]function by complying with special relativity's universal speed limit of v < or = c.
Maybe the impulse p's velocity used in the Uncertainty Principle is ACTUALLY v/√(1 - v^2 / c^2) to avoid the spike becoming a Dirac delta [extended]function by complying with special relativity's universal speed limit of v < or = c.
You have clarified so much for me. I was struggling for a while with these concepts. Thank you for making this. I'm excited to check out the other videos!
That was a terrific video. It's the first time as a lay person to get that kind of detail about what a wave function is. I do have a 4-year degree in computer science and electrical engineering from a long time ago, but even now I could roughly understand what's going on. It makes me wonder why so much hand waving is being done instead of just giving some details like this. I understand that plenty of people have no college-level math so showing them these things in a mathematical way would be useless. But you also managed to present intuitive description.
So I guess in summary I wonder why a video like this hasn't been done in the past, or somehow I missed it despite watching hundreds of videos related to this.
OMG i sooooo needed this video right now, you came just in time!
When you hear someone saying "here's an 'anology' for a wave function..." and see a piece of salt-like crystal at her hand, your expectations makes a top until....well the anology completes with the Fourier transform :-) Don't get me wrong. I am an electrical engineer all too familiar with F. transform and even took a lecture (40 years ago) in quantum mechanics where the uncertainity principle was derived in an amazing and supring way using the Fourier transform as far as I recall, I still couldn't help smile your mundane(!) anology :)
Great work, Mithna! Keep up the good work!
9:26 it's not the area under the curve squared, it's the area and the square of the curve.
Thanks for producing this series. It's quite helpful.
Wonderful explanation! 1:37 Your analogy with the glass cube leads me to believe that you would enjoy my visualization of the fractional fourier transform (can be found on my channel)
Refreshingly illuminating and wonderful presentation.
Wow, really really good explanation of the topic. Absolutely brilliant. Subscribed.
Thank you for the clearest description of the wavefunction on UA-cam!
Could you please do a video on Jacob Barandes' latest work? 🙏
This was fun! I love that you present some physical reasoning for the Fourier transform to pop up here
9:08 Is it (area under the graph of the function)² or area under the graph of (function²)?
Oops, you’re right! Square the function first then integrate. Sorry about that!
Well, technically it is the area underneath the function defined by the product of the conjugate of the wavefunction and the original wavefunction; and/or the area of the function defined by the square of the magnitude of the wavefunction. The wavefunctions take a position in space and time and returns a complex number, a linear combination of 1 and sqrt(-1)
@@codetoilThe presentation would be spoiled by going into that much detail!
Thank you a billion times! Your videos are so interesting! It would be great if you were able to walk us through the application process where the equations are actually used to create a product.
@ a product that requires the use of a quantum equation in order to be produced maybe a computer chip or a laser etc.
I wonder if it would be fruitful to explore the wavefunction in other bases besides position and momentum. Perhaps a density basis, or energy density basis. Or energy-frequency dual basis.
Can you make a video on making sense of time evolution of a wave function, specially for free particle.
Yup! I’ll do the schrodinger equation soon
Great presentation with easy to follow mathematics aiding clear communication.
This is probably the best explanation of the Wave function I've even seen online.
The course youre offering looks very enticing after watching your teaching skills.
Though the link says the product is not available. Is it still going to be offered?
Thank you so much! The next cohort starts in the first week of January :)
I love that crystal you have there, where can I get one? It's so relevant to crystal growth. 😮 Wonderful class today, Thank You.
It's a calcite crystal! It's hard to get a nice clear one- I got this one on Etsy
Ooh dear, a c r i s t a l ... I can see certain people banging their tables and shouting "I knew it has to do with cristals!!!"🙄...Sorry just couldn't resist. 😉I love this series! Public education at it's best! AND, the "dead" declared social media alive and kicking;-) Wonderful❤
the crystal analogy is better than you let on: we do see projections onto a space axis or a momentum axis, and they are 90 degree rotations of the projection direction. If you don't believe me, see "fractional Fourier transform".
The electron is only in a position to a reference point, right? as our earth and sun are travelling at great speeds, how does this relate to the electrons existence?
Why do crystals and chairs and dogs and trees appear to us as objects that persist through time? I understand that particles locally coalesce when measured but what keeps the particles in the chairs and trees coalesced and consistent through time? Is it a constant interaction with the environment or do they coalesce once and then persist without experiencing a delta in energy that destroys the persistence? Or is there some information encoded in the wavefunction, if it is universal, that causes objects to persist?
If our sun stopped and earth and the universe we are attached to stopped all movement, would anything including matter and electrons exist?
Just because an object makes waves or an interference pattern, doesn't mean that the state of it has a wave-function. A wave-function has nothing to do with a wave at all. The wave-function is not meant to describe the physical trajectory of a particle, but to encode the probabilities of their state. The interference pattern, a measurable phenomenon, strongly suggests a wave-like behaviour, but it has absolutely nothing to do with a wave-function, and it isn't proof that light moves in a wave-like trajectory either. It has behaviour like a wave, but it does not necessarily move exactly like a wave. A wave-function does not describe reality at all. It is a mathematical tool to calculate the probability amplitude of a particle's state, not a physical wave. Quantum mechanics allows us to calculate the likelihood of where a particle might be, but it does not claim to describe the actual path that the particle takes or where it exactly is. You are only calculating where it might be, where it probably is. This is often misinterpreted and overstate to the general public by physicists, particularly when it comes to the double-slit experiment, saying things like "particles decides how to behave" and other such nonsense. No, they don't decide how to behave. They just behave like they do. Figure out how they do it instead of making a particle seem like it is reliant on cognitive abilities to behave. You should be very sceptic when someone invokes quantum physics. It means that they are talking about something that might be true, but there is also a probability that they don't know what they are talking about at all. People do this with AI as well, but that is another subject.
Furthermore, there is no collapse of the wave-function, simply because there is no physical wave-function. When it is said that a particle is in a superposition, it's only so because the probability of that particle can be in either of two states, and the probability says that it can be in both states at once too, but it never is in two states at the same time in reality. Only information can be in two states at the same time. Superposition is a a combination of outcomes, that is all. When measured, the particle is in one state, and one state only. By that I mean states that are not opposed to one another, and contradictory, so that it becomes a paradox. If you use quantum physics to calculate something that is exactly the same result as in nature, then that means you can also find the result of that very same thing with classical physics. Just because you use quantum mechanics to describe an object, does not make quantum physics real. Just because it has unparalleled accuracy doesn't mean the calculation you made is what reality is like. It just makes the probability of what you found to be true, and as such, you can find that to be true with classical physics as well. What you did was to use probability as a tool to find the actual real object. Good. Now that you found it, you can find it again by observation and measurements, unless it's a photon, because nothing else in the universe moves as fast as it, so only another photon can track or observe another photon.
When you say that the infinite spike is not a real function do you mean to say that the function must be continuous? If so, is that function necessarily nonzero everywhere? does that mean that there is a small but nonzero chance of finding an electron miles away from the testing station? Is this what quantum tunneling is?
No, it's not about being continuous. It's just that a function f(x) here must give a number for every x, while infinity is *not* a number.
The wave function does not have to be non-zero everywhere. In the particle in a box scenario the wave function is zero outside and on the sides of the box.
I like your channel a lot but I believe you have a subtle misconception about what the wave function is. The wave function isn't a/the state in a Hilbert Space it is a representation of the state in a chosen coordinate space (may be position or momentum or something else). It is inherently tied to a coordinate system. Yes it contains all the information of the state but it isn't the state. The same way the coordinates of a vector represent an abstract vector. This also allows the wave function to be a Dirac Delta since it is only defined in the spectral decomposition of the state and is always integrated over. en.m.wikipedia.org/wiki/Quantum_state
I agree that physicists use the word "state" rather than "wavefunction" for the obvious reason that it's, uh, not a function. When we do use "wavefunction", we usually give it its full name in whatever basis it may be ("the wavefunction in the momentum basis", or "the position wavefunction"), unless it's obvious from context, etc. But I think it's valid (or at least instructive, if taught like she has in this video) to generalize it and talk about searching "THE wavefunction", which we can project into the various bases. That would be describing the state vector |ψ>, like the video says.
The big problem is that it's, uh, not a function. Physicists are not new at giving things the exact wrong names, but I think we shouldn't be encouraging it. So a big disclaimer would've been nice.
@@Heulerado Thx for the reply yes some physicists mix the terminology. I believe this has the historical reason that and most of the original equation for QM also work for the representation of the state. Hence the distinction is not necessary in this context. But i have a particular issue with the following scene at 12:31. Where she clearly states the opposite of what is correct. Which will confuse people if they try to really understand it.
As I understand it, MWI proponents like Looking Glass Universe (which I do not support MWI personally) argue that the wave function is only relative to a chosen coordinate space because ψ is just a limited perspective of a much bigger Ψ referred to as the universal wave function. If you change your coordinate system you change the perspective that you are looking at Ψ from so that changes ψ. Hence, for them, it is still describing the ontological state of the system, but just from a particular limited perspective, kind of like describing one side of a box at a time due to your perspective because you cannot see the full box at once.
@@amihartz Yes, measurement basis effects the output probability distributions and measuring in the sub-hilbertspace as you are suggesting is described by the Generalized Measurement via partial traces (en.wikipedia.org/wiki/POVM) in open quantum systems. As far as i know one needs an open quantum system for the MWI interpretation to work and so far I don''t see how these interpretations are at odds. AFAIK one can write down a version of MWI using still the abstract vector notation, no problem. Can you point in the direction where this is a problem?
did you take any chemistry classes
Where physicists frequently drop the ball in their explanations of how a particle can be in a superposition of quantum states is in failing to make clear that the multiple "places" they're referring to are not physical locations in 3D space. These are coordinate positions in the abstract, multi-dimensional realm known as Configuration Space, where the quantum wave-function is defined. Moreover, the "weights" associated with these coordinate positions are not merely negative, they are complex-valued probability amplitudes, a completely non-physical aspect of the quantum wave-function. Where such a superpostion becomes physically observable is at the point of measurement, where it is probabilistically projected into a particular location in 3D space in accordance with Born's rule (which calculates the conjugate square of the superposition's probability amplitudes to produce a real-valued probability of observing the particle in that location).
In short, subatomic particles do not occupy multiple locations in 3D space; they manifest as a superpostion of locations in the abstract, multi-dimensional realm of Configuration Space.
As a person who had to do the full Fourier transform with pen and paper, I can say it was NOT easy 😅
Great explanation! 😎👍
When you say the electron could be anywhere within the box, and not discretized to multiples of a tiny Planck length, you are suggesting we've measured that and know it is true by measurements. But we can't measure a position that accurately. Isn't it true that in fact space MIGHT be discretized to the Planck length?
None of this is relevant.
12:32 The left hand side is NOT the wave function. It’s the state vector (or the ket vector). It’s not even a function!
The function \psi(x) on the right hand side is the position basis wave function. It’s called position basis wave function because the right hand side is the base decomposition of the state vector by the position basis.
I've never understood why people often call the state vector the "wave function" when it's not literally a function. Although, maybe it is because my learning is through quantum computing and not through traditional physics education. I always associated the state vector with |ψ⟩ and the wave function with ψ(x)=⟨x|ψ⟩.
Whether you call it a vector or a function is irrelevant. It's an element of a Hilbert space and the only meaning of that is really just unitarity, which physically is completely trivial.
@@schmetterling4477 State vectors and wave functions are different. State vector is written |\psi> and position basis wave function is written . They are completely different mathematical objects.
@@vonneumann6161 Are we discussing the difference between a physical vector and its coordinate representation now? That is completely irrelevant in this case since there are no physical vectors in quantum mechanics. All of this plays out in a completely abstract vector space. Unlike physical vectors none of it exists in reality.
That was very clear, I like primers like this to get a sense of the "knowledge landscape". That made quite a few things click.
The wave function \psi is complex valued. Probabilities are defined by the square |\psi|^2. Obviously by passing from \psi to |\psi|^2 we have wiped out a lot of information (namely the phase). If this were the whole story, QM would work with ordinary real valued probability densities instead. "Superpositions" of probability densities can also easily be defined: every convex combination of probability densities is another probability density. You need to explain why this is not enough for QM, i.e. what the significance of the phase is. In particular in view of the fact that \psi and a*\psi
describe the same state , for every scalar a with |a|=1.
There are no probabilities in quantum mechanics. That's just a misnomer. What we are estimating are event frequencies. For those we need the Born rule because they depend on the spectral absorption function of the measurement system. The |psi|^2 case is just a lazy way to write the Born rule for a unity measurement projection operator. There is no actual measurement system that implements such an operator. It's a convenient fantasy for theorists, no more, no less.
If we design a certain quantum cellular automaton and then install a quantum particle in it and a device that measures it, how complex does this device have to be for the wave function of this particle to collapse (how minimal can an observer be whose components are also quantum)?
No one really knows where exactly this threshold between classical and quantum is. And why. And what exactly is the collapse and how it happens. See "measurement problem".
@@thedeemon I know, but can we learn some thing new from some toy model that even not trying to emitate nature but have quantum behaviour?
@@YarUnderoaker oh yes, such a device would help us a lot in understanding these questions.
@@thedeemon The transition from quantum to classical is given by repeated irreversible energy transfers. We have known this very well since the late 1920s. That you do not know this is merely a function of your intellectual laziness. All of this can be found quite easily in the physics library.
@@schmetterling4477 Oh c'mon, multiple books (even quite recent) have been written on this subject of measurement problem still not being solved. Update your physics library.
I keep hearing the term "collapse"... what does that mean?
The wave no longer describes the state of the particle, because a measurement's specific value is known. General to specific.
When something collapses, like a building, it goes from having large extended structure to a pile of rubble much smaller than its original size. When a wave function "collapses" it's reduced to one of the basis states. So if just measured where the particle is, the answer you'll get is "at x=1" and the state it's now in is the wave function for a particle at x=1. This will in general be a much simpler state than before you measured it. So it's collapsed in the sense that it's squished down to a spike at just one place on the x axis.
The wave function has been measured/quantized/communicated to have practically settled into a quantum state called an eigenstate (one of the countably many dictated by the measuring apparatus and the object being measured) which can be made by a subsequent measurement to produce a specific rational number called its eigenvalue.
Every eigenvalue isn't a point but an interval because the number 1 = 1/3 × 3 = 0.33... × 3 = 0.99... is between 0.9 and 1.1 = 2 - 0.9, between 0.99 and 1.01 = 2 - 0.99 and so on. There is *ALWAYS* a gap between the two bounds. All other rational numbers can obtained by scaling the number one with a nonzero rational number.
As one produces the eigenvalues from eigenstate and the eigenvalues always have a fuzziness due to the limited resolution of the measuring apparatus, there is an inherent uncertainty in the eigenstate.
A wave function which has "collapsed" means that it is practically in an eigenstate with a fuzziness not measurable by the apparatus but can be consequential via positive feedbacks in subsequent measurements (i.e. the Butterfly Effect in Chaos Theory).
A classical measurement had occurred meaning there is now a new distribution of where the particle could be based on each infinitesimal of the basis or whatever and how each infinitesimal would react classically. At least that’s what Landau told me in his cool special books
I've always been curious how an (e.g.) electrons wavefunction relates (or doesn't) to it's associated electromagnetic waves. Ditto for a photon. Are these entirely unrelated? Which feels unlikely, but I've never seen it addressed.
Amazing question. The EM field is pretty intimately related to “photons”- vibrations of the field are what leads to light. This topic is worthy of a whole course in itself!
Very well explained. Thanks much.
I had no idea you were still making videos. The last one youtube showed me was from 9 months ago!
First class explanation. I get it!
If all schools had the same level of educational representation as this, the world would be filled with much smarter people.
This was outstanding!
you want PhD level maths and physics to be taught in schools? There's a reason for the bell curve. What a ridiculous comment
@@salerio61 My point wasn't the level of education, but the level og educational representation. Two very different things.
No one expects this level of academic expertise to be taught in elementary, middle, or high schools. But when a presenter has strong academic qualities and the ability to visualize in a way that gives students a better chance of learning a subject, regardless of level, a good foundation is laid for a broad understanding of the subject among students.
Hmm... good pedagogy, keep at it!
Thank you.
This is brilliant thank you
So position and momentum are inextricably linked at a fundamental level?
Why would they be?
@@schmetterling4477 Because of the uncertainty principle: en.wikipedia.org/wiki/Uncertainty_principle
@@davidzuccaro1906 The uncertainty principle has nothing to do with quantum mechanics to begin with. It's just a general property of Fourier transformation-like operators and applies just as well to water waves. In quantum mechanics these quantities aren't describing the position and momentum of one system to begin with. They are describing distributions of positions and momenta of a quantum mechanical ensemble... which is an abstract that nature doesn't know anything about. For an actual individual measurement you can get any combination of position and momentum that you want.
Can you do a piece on the nature of WF collapse and what constitutes an observation (like why doesn’t it collapse sometimes and not others - for example recording the result on a computer but deleting the information before checking it? Turning on and off photon detectors so they are keeping a record/not keeping a record? Removing them? Adding detectorsbut leaving them unplugged etc Does this maintain interference?). This is not really clear in all the QP videos I’ve seen. I feel this is a very misunderstood aspect of quantum mechanics.
It would be interesting since she believes in the Many-Worlds (multiverse interpretation) and I have never been able to visualize what it even _means_ to make an observation in terms of Many-Worlds as it is not clear to me where the observables even are in such an interpretation. In relational quantum mechanics, it is very simple, an observation is just an interaction between any two physical systems _from the perspective of one of those systems,_ i.e. the realization of the properties of the system does not occur for physical systems not part of the interaction, meaning that what properties of a system have been realized or not depends upon reference frame.
Wave function collapse is a meme on the internet. It's not a thing in physics. You won't even find that term used in well written textbooks like Sakurai. It has no discernible scientific function. I haven't even seen a precise definition of what it is supposed to mean.
@@schmetterling4477to me it means how observations (single outcome) don’t tally with what QM seems to say - which is the universe is a wave function yet we don’t see superpositions.
@@jamesmather7896 You won't find a single quantum mechanics textbook that says that the universe is a wave function. Wheeler liked to talk about "the wave function of the universe", but that is a horrible idea that leads to nowhere. The universe does not have a wave function. A wave function is a mathematical abstract that allows us to estimate experimental event frequencies by APPROXIMATING an actual (serially performed) quantum experiment with an imaginary (non-existent!) ensemble of identical copies of the same experiment. Many theorists don't seem to know this (or they just don't care about this level of detail) and then they are talking nonsense like Wheeler did. Laymen do, unfortunately, have a tendency to amplify the nonsense in their minds while completely suppressing the actual science facts. :-)
@ then how do you explain single outcomes in observations instead of us observing the superposition of the wave function? There is no doubt that there is some addendum to Quantum Physics as that is what we observe - call it what you will: decoherence / WF collapse / measurement / interaction etc.
Is there an aspect of the wavefunction for energy?
Yep! The basis for that is expressed in terms of states with definite energy. If you want to know more, look up the quantum Hamiltonian.
@ Thank You!
i am getting to understand psi , thank you.
Great explanation, but I have to nitpick about the math. A few times you say a function that’s zero everywhere but infinite at one point “isn't even a real function.” No, it’s a perfectly fine function mathematically! Maybe what you mean is that it’s not continuous, or its range isn’t finite?
No, that's not a function but a distribution. The mathematicians had this figured out in the 1920s, which is why von Neumann could write his book about QM in 1932 without needing any new math.
It was very nicely described and I get it but what IS the wave function though? Is it just a mathematical model that works or is it representing something physical in reality?
She believes in a class of interpretations called MWI which argues that the wave function is not only a literal physical entity but there is only actually one of them in existence, and in fact it is the _only_ thing in existence, i.e. the entire universe is just one giant infinite-dimensional "universal wave function" and the little ones we perceive are just due to us only perceiving a part of it at a time from a particular perspective.
No one knows. It's something different physicists take different positions on. Of the various interpretations of quantum mechanics, some say it's real and others say it's not; but even the ones that say it's real are a bit vague on where/how the wve function is, since the function evolved in a many-dimensional configuration space, not 3d physical space.
Given that the wave function can interfere with itself (as in the double slit) it seems awfully likely that it is physically real to me. But we are a long ways off from understanding how that can be or how it works.
@@erinm9445 I see. It's interesting though because you could train some neural network to make correct physics predictions but would you call that science? Probably not right because it would not tell you anything about the physical world.
@@neobaud513 Why did you ignore my comment?
@@amihartz I read it but it didn't really answer my question
When you say that, actually the electron is spread out, isn't that an interpretation of the math, a hypothesis, and not necessarily what is actually the case? Don't we not really know what is actually the case, we only know what the outcomes are that we measure? And we know they're well predicted by the wave function equation, so we know that whatever is the actual case, it must fit with that wave equation. Isn't it even possible that until measured, the electron doesn't exist in physical space at all?
यह अपारदर्शी और पारदर्शी ऊर्जा है जो केवल ईस आवृत्ति में होती है जैसे अवकाश या वायु या प्रकाश लेकिन दर्शीय ऊर्जा जैसे तरल और पदार्थ में क्या होता है? अवस्था के अनुसार आवृति मे बदलाव आता है और इसकी डिजाइन और पैटर्न भी बदल जाते हैं। 0=(-0/-0)
So the position vector is a vector in a basically uncountably infinate dimensional space!?!
The physics gods had me in mind when they made this place. Realtivity and its differential geometry and quantum with its mind bending abstraction that somehow logical if you think about it hard enough. both my favorite things.
Basically, yes, but the dimensionality is really irrelevant. All important properties are following from the unitarity condition which specifies that we aren't losing any members of the quantum mechanical ensemble during its evolution.
doesn't every finite system have discrete eigenvectors and eigenvalues?
No, it can still have countably infinite eigenvalues/vectors. Eg, the harmonics that form the energy eigenstates for the infinite square well.
@@LookingGlassUniverse Aren't these discrete?
@@Kraflyn I think you're right. Discrete values can be countably infinite.
Is the wave function physical?
There are different interpretations of quantum mechanics, and they give different answers to this question.
No. It's a mathematical abstract. It is ontologically equivalent to a probability distribution. There are lots and lots of such abstracts in the mathematical description of nature. Real numbers, for instance, are also just abstracts. Nature doesn't implement them.
But why male models? --Zoolander
Psi is a graphical depiction of alternate worlds branching off. The ancient Greeks were ahead of their time.
10:23 if you csn say chi, then you can say psi as opposed to chai. 😂 do re mi
brilliant video thank you
Please, Mithuna, take a look at the work of Prof C.S. UNNIKRISHNAN, in arXiv, Reconstructing QM without foundational problems!!!!!
Yes, that is complete nonsense.
@schmetterling4477 why?
@@2nd_foundationStandard QM doesn't have foundational problems. It's one of the most trivial theories. :-)
@@schmetterling4477Thank you very much for your answer, but I don't see what are your arguments. Yes, you should read the article in detail, the important finding is on page 8. The incompleteness of QM, it is known as a reference Penrose is explaining this in many UA-cam videos; the scandal of QM was written by Nico van Kampen in 2008 etc. prof UNNIKRISHNAN described the foundational problems of QM in section 1, also related to experiments, in section 3 he is showing how this action wave formulation resolves the foundational problems of QM. In Prof Unnikkrishnan own words "The only way that the action principle can work is if the action is manifest in dynamics as a periodic entity in both quadratures, capable of interference. THEREFORE Hamilton mechanics (equation) that does not use this periodicity is incomplete. The equation in the article is the correct one. Then there is a single action mechanics- there is no division into classical and quantum mechanics ( since the QM measurement problem was also resolved in this way, there is no need for classical apparatus either, unlike in the current theory). QM is not wrong but it is right about ONLY statistical results. And its foundational premise is flawed." In any case if you don't have solid arguments based on experiments, please read in detail the article and references, don't let you trap, if not good luck with your views of reality.
@@schmetterling4477Thank you very much for your answer. The important finding is on page 8. The incompleteness of QM, it is known as a reference what Penrose is explaining in other youtube videos; another one: the scandal of QM was written by Nico van Kampen in 2008 etc. prof UNNIKRISHNAN described the foundational problems of QM in section 1, also related to experiments, in section 3 he is showing how this action wave formulation resolves the foundational problems of QM. In Prof Unnikkrishnan own words "The only way that the action principle can work is if the action is manifest in dynamics as a periodic entity in both quadratures, capable of interference. THEREFORE Hamilton mechanics (equation) that does not use this periodicity is incomplete. The equation in the articleis the correct one. Then there is a single action mechanics- there is no division into classical and quantum mechanics ( since the QM measurement problem was also resolved in this way, there is no need for classical apparatus either, unlike in the current theory). QM is not wrong but it is right about ONLY statistical results. And its foundational premise is flawed." . so take care and good luck !!!!
Wavefunction is not just probability. Physical phenomena require units.
The units are the bases.
@@mooseyard Wavefunction is probability per unit of volume.
Funny to think about it it...pretty deep stuff to think about it...
Thanks Madam.
To be honest she has such beautiful face, voice and especially eyes that it is difficult to concentrate on the information provided. Still she explains so clearly that even I understood everything. Another problem is to remember))
And you fell for a pretty face... again. ;-)
@ no, she does explain clearly
@@ArnMH81 Dude, she starts you off with an animation of the double slit. The double slit has absolutely NOTHING to do with quantum mechanics. It's a simple classical wave experiment from 1801. You can do it at home with water waves in your bathtub. If you are willing to think for more than a minute about it, then you will also notice that Planck's constant does not show up in the double slit at all. How in the world is an experiment that doesn't contain Planck's constant OR at the very least two-quantum correlations like entangled quanta supposed to be about quantum mechanics? ;-)
@ she has another video when she performs double slit at home. I actually did it myself after her video. Anyway I guess the point is not to make sure that double slit experiment is not some lie. It has been done millions of times. But in terms of accessibility of the information provided she is very good. Ok I don’t deny, her voice and looks influence my opinion. I am bias if you need truth )). Still I think she explains quite simple. For example the math for wave functions ant operators- some of them I didn’t want to go deep into because I had that feeling of insurmountability. After her explanation I realized that I was able to get it. And I studied further.
@@ArnMH81 The double slit is a demonstration of classical wave behavior. As such it is mildly useful. Beyond that it teaches nothing. In the context of QM it teaches us nothing and it is not being used in the professional teaching of QM for a number of reasons. For one thing it is not even unitary. Why ask a beginner who is just about to learn about wave functions to toss them into the bin right away because the sub-trivial example doesn't work with them? Very, very poor teaching, indeed.
This is great, yeah if it were truly discreet we'd probably be done w physics and know it all by now!
Goedel wants you to hold his beer, right now. ;-)
@schmetterling4477 no I mean a discreet quantum universe can't really exist. The inverse extrapolation of Gödel is that his theorem is a complete description of everything, a TOE
@@MichaelPaulWorkman Quantum mechanics is much easier than discrete systems. You are simply barking up the wrong forest of misconceptions here.
1:28 somethig called the wave function - to my ignorant eyes it looks like a pitchfork TBH
It's misleading to say the wavefunction is the "state" of a system. It's actually a representation of our knowledge of what would be observed if the system were measured. (See the Born rule.)
Our knowledge of the system state is usually limited & incomplete, so the predictions of measurement results are usually probabilistic, with the probabilities summing to 1.
"Superposition" is a fancy way of saying there are at least two possible results if the system were measured. In other words, each of the probabilities is less than 1.
States in QM are abstract vectors in Hilbert spaces, and projecting a state vector onto the position basis gives the wave function. The wave function contains all the information of the state in the form of complex coefficients, so it's perfectly fine to call the wave function the state.
>simplelife1021 : The claim that the known information is the _complete_ information is an unproved metaphysical claim promoted by Bohr, which was disbelieved by Einstein, Schrodinger, Bohm, Bell et al.
Also, a projection can lose information. Consider a photon that passed through a horizontally-oriented polarizing filter. Projecting its state to a vertically-oriented basis wavefunction (in order to predict the probabilities of the possible results of vertically-oriented measurements) suggests the photon is in a superposition of up & down polarizations, which loses the info about its actual horizontal polarization prior to the measurement.
@simplelife1021 : The claim that the known information is _complete_ is a metaphysical claim promoted by Bohr, disbelieved by Einstein, Schrodinger, Bohm, Bell et al.
A projection can lose information. Consider a photon that passed through a horizontally-oriented filter. Projecting to a non-horizontal basis loses the info about its horizontal polarization.
we called psi, "The chickenfoot operator" back in university
So.everyone.is at two places at once or more
The wave function have nothing to say about the elementary particles contained information and behavior.
If something is hidden behind complex abstract definitions or math formulas, is wrong.
The truth is always easily understandable but not always easy to live.
Why are you telling us that you weren't paying attention in school? ;-)
For it is written
Vs
Life we're living
Alphabetic effects above neck => It's this way it's that way
Mathematics is rooted in much the same way
Doing my best to stimulate content
Keep up the good work
You can actually do a Fourier transformation mechanically:
ua-cam.com/video/6dW6VYXp9HM/v-deo.html
what is the electron doing?
well not their homework. lol
any video on Willow?
glass
thank you 🙏. really clear. and really interesting. however - not very intuitive 😢😢😢😢. to begin with - how is this ever useful? please 😢
It's a "resource." It's *USELESS* until it's connected via "technology" to applications.
Most 20th century technology is based on quantum mechanics. Semiconductors / chips / transistors, atomic energy, radio communication…
This video shows a natural quantized effect using a version Euler’s Contain Column Theory?
This video of an engineering test models the quantum physics problem called “Particle in a Box”. The wave function is used to predict the location of quantum particles, atomic structure and molecules.
ua-cam.com/video/wrBsqiE0vG4/v-deo.htmlsi=J5jCMxq3pRe_P-gG
Very stable trusses and truss core panels have been fabricated using the process shown in the video.
The video shows how materials naturally respond to induced stresses in a “quantized“ manor.
The process’s load/deflection’s sawtooth curve with its exponential fit, in the engineering, study shows in detail the bifurcation area between quantum jumps.
The effect has been used to make light weight structures and shock mitigating/recoiled reduction systems and earthquakes isolation systems.
The model shows why with the, exponential load increase and loss of resistance at phase change, quantum jumps are so very fast.
Too many big words .
@ true that! First saw the effect, all knew was the frequency and energy relationship. The first ten years I was trying to make light weight structures with it.
The next ten years I found in a engineering handbook on this column thing. It’s the old pushing down on a wooden yard stick buckling effect.
The last ten years people have been telling me it looked like this wave function thing in modern physics.
Had no idea what either of those things were.
Now, Just trolling 🧌 it in front of the mostly elitist professionals so they can tell
me it’s a bunch of crap 💩!
now i know that electrons seems to be blue and not smiling. because It could not find an exact place to stay? your smile instead indicates that you are in the right place. Best wishes from Germany.
a fun little classical model i like a lot, because it works pretty much like matter for everything from gravity to optics to whatever, except involved electrodynamics and so on; Is light in a box. lets imagine that we have a semi transparrent baloon filled with light and that only relfects and transmits light, so we essentially have a little greenhouse filled with light and inflated by the radiation pressure inside, lets also imagine for the sake of the agrument that the radiation pressure inside is constant, so it stays inflated. the light inside is going to carry all the momentum of the mass, and it is a mass simply by virtue that its state of motion is alterable, it behaves just like normal matter, e=mc^2 and all, to change its momentum we just scatter some light of it, some of it gets absorbed by the baloon and some scatters off it, and its packet of light waves inside is now changes, in the previous staionary reference frame, where it has over all 0 momentum in the radiation inside, it now has some momentum in a different direction, because of the transmission and relfection properties it also contains more energy but scatting it back and forth smoothly and slowly does not change its stable mass, because if it is carrying too much light with respect to the exterior more will leave than enter and so on. let the surface fluctuate with the light inside and let there be uncertainty in the content of the light inside, thens subject it to some light you scatter off it to measure its position or momentum, find me the uncertainies :P. it is a classical model so what gives?
😢 Trying to understand the Sun which I have learned that my thoughts are faster than the speed of light of a response with witnesses. Would love to be wrong for beliefs of entanglement. No fear of looking at it ,
there is no such thing as superposition. everything can be calculated. i hate it when physicists talk about probabilities. its just a 100 year false mode of thinking. heisenberg was a moron. its not going to collapse. there is this thing called bohmian mechanics
Awh, you are so cute when you think of yourself as a genius. ;-)
maybe Feynman's approach from his lectures would fit better, so that peeps understand why square
you can view the Dirac Delta Function as the Dirac Delta Distribution: it is an infinitely narrow Gaussian. Fig. 2 en.wikipedia.org/wiki/Dirac_delta_function
Think too much
Voice-work with please, is too crunchy when you modulate-dont do this, you have lovely natural voice, do not low it or make high and five this crunchy terrible sound like old motor or ciuch , old frog. You need speak loder if you modulate it and do not hold tour voice in deep throut-move it to teeths, take few lessons, it could be a reason. Make nice, smooth voice. Dont repeat the same things but hudge plus for details for begeeners , no one tells it-only you, but you need better form, graphs...
Here is one of the "End Points"..
Where is "The Green" in "The SpaceX Launch"...???...
"Google A Rainbow Picture"...Take Care...Bye...
Look at "The Rainbow" as "A Gentle Pendulum Clock"...
Green Arrow In Or Down..
Orange Arrow Up Or Out....
"Yellow Is The Tick"....
SpaceX Launch Speeds Up The Pendulum....
"Squeezes The Rainbow Together"...
Green Goes "In" The Engine..
You Can Only See Orange "Out"..
Take Care...Bye...
Pretty sure you're wrong and it's not a misconception. Spacetime is discrete, else you run into Zeno's paradox of motion.
No, it’s still fairly fringe theory. Zeno's paradox is not a real thing - it only existed because the ancient Greeks didn’t understand infinite series, I.e how an infinite number of terms could have a finite sum.
That isn't an actual Paradox
@@mooseyard okay, well I've seen it argued that if spacetime weren't quantized that the universe would be full of light. What about that?
Zeno's paradox can be solved using limits in calculus without using discrete spacetime. A lot of physicists think spacetime _might_ be discrete and there are some tentative theories suggesting this like loop quantum gravity, but none of them have evidence for them and are just speculation.
Heat waves. Mass disolving internal magnetic fields as external heat energy outside of entanglement of mass as campfires. Lightning burns through atmospheric gasses disolving into external heat energy outside of entanglement of mass as fire bouncing off atmospheric gasses lighting up the sky instantaneously. Campfires slowly disolving internal magnetic fields disolving into external heat energy as fire. Electricity is rapid heat exchanging through mass faster than earth's quantum internal magnetic fields grounding currents through its nucleus or core where external heat energy is strongest as molten masses held in centrifugal force cycling circulation as mass. Mass occupies space within mass as outward force of pressure known as magnetism weightlessness outside of a farther reaching greater magnetic field. External magnetic fields spinning all external heat energy within its field outside of the nucleus or core is a repulsion external magnetic field. External magnetic fields spinning all external heat energy away from renewable heat energy of mass causing mass to disolving into external heat energy outside of entanglement of mass as outward force of pressure no longer holding internal magnetic fields grounding currents through its nucleus or core. External heat energy outside of entanglement of mass doesn't ground currents it strips away renewable heat energy singularities away from internal magnetic fields. They disolve from within as outward force of pressure of cold repulsion of occupational space. Conservation of heat energy singularities outside of entanglement of mass as fire. Fire is external heat energy outside of entanglement of mass. Accumulation is disolving internal magnetic fields as external heat energy outside of entanglement of mass as fire. Stars decay their atmospheric gasses as fire lighting up the atmosphere like a filament bouncing off atmospheric gasses still present.