Isn’t all equations are just play of concepts symbolized by numbers or letters and they get to divide, multiply, add or subtract?!? And End up being on both sides of equation = !!
I have a physics PhD, and while of course nothing here was new to me per se, I really appreciate the way you tie the story together. The undergraduate physics curriculum is more focused on establishing the model and using it to make predictions that illustrate its key features (like the double slit results), but the "origin story" doesn't often get explained. Thanks for that!
Except that this is not the correct origin story. Schroedinger simply guessed an equation. It turns out that it was the wrong equation and he never understood where it came from. The more rational approach to quantum mechanics can be found in Heisenberg's matrix mechanics papers and then, as a culmination, in 1932 in von Neumann's book.
This video is absolutely top drawer. The animations, the lesson, the historical pages, the actual visualization of how the imaginary unit shapes the wave function. Just brilliant. I have to get the book.
When I learned (decades ago) that multiplying by i causes a rotation in the complex plane, I wondered why no one ever seems to talk about that in quantum mechanics. Thanks you so much for the beautiful animated images and the detailed explanation. Wonderfully done, and it explains so much I never learned in my advanced physics courses in college.
As soon as we mention the word "unitarity" in the first QM 101 class we are talking about nothing else than rotation. That's basically what it means, most physics professors just don't care to explain it and most students don't notice on their own.
Seriously. I was so lost in my undergrad classical & quantum mechanics classes (but managed to eke out good grades because everyone else was lost also) but these kinds of videos make it so simple to understand.
The sophomore / junior physics text by David Griffith ‘intro to quantum mechanics’ has a very good discussion on how the solution to Schrödinger’s equation and even computes the solution with principal quantum number n=1, to arrive at the spherical s orbital which is the ground state for hydrogen. Watching this video took me back 20 years to undergrad years. My quantum mechanics professor, Dr Kwon Lau passed away a few years ago, but his lessons are still alive in my mind.
The unfortunate thing about that text though is it paints a very different picture of quantum mechanics than what is truly happening. It’s very difficult to teach this properly, of course, but to me it doesn’t do enough to distinguish the coordinate dependence of the wave function as a result of the configuration space from the real spatial behavior of the particle. Much of the intuition built up ends up being incompatible with a system of multiple particles. In the text there is a sense that the expectation value of a quantity is in some sense a description of the behavior of the particle with physical content, but in truth it only the average of measurement outcomes.
Something worthy to note is that the Born rule was adopted solely because it worked. Originally, Born thought the probability to be |Ψ|, but Ehrenfest’s theorem didn’t corroborate this. A next reasonable step for amelioration came with simply squaring it, as Born did, and it just so happened to work. How amazing is that?
Yes and no I would say for the success of the Born rule. For "single-particle" experiments, it does not apply. Some works concretely discuss this particular point.
I love how clearly and simply you explain everything this was so much better than any of my Physics professors when I was in college. I’ve been watching your videos since high school and they’ve always been so fascinating and illuminating. Thanks for all you do!
This is quickly becoming one of my all-time favorite physics channels. Excellent balance between depth and clarity, really nice visuals, and a very pleasant voice and speaking style.
The video is a gemstone. I remember university at that time physicists were struggling with the Schrodinger some of them knew about hydrogen solution but almost nothing to derivation on this topic. Great work. Aside the true paradox is when most people refuse to accept imaginary numbers then conversely to me is very much more obvious to use them.
Very good explanation and visualization, but one thing I'd like to add. It's not matter that is a wave. It is the interaction with matter that is described by a wave.
@@ivocanevo You have a physics PhD who has measured trillions of quanta. Not once was there a problem. A quantum is simply a small amount of energy that was originally in a quantum system and that at the end of the measurement is in the detector system. There are no problems with that, neither in theory nor in the lab. It takes a theorist who has never been in the lab to imagine that this is somehow problematic. ;-) If you want to see examples of microscopic "explanations" of how measurements work in detail, read Mott 1929 or von Neumann 1932 (chapter 5, I believe). All of this was understood in the mainstream literature very early on. ;-)
I'm just a lowly engineer, but it seems to me that a lot of the discomfort around imaginary numbers arises due to their name. There really isn't anything imaginary about them; they are a very natural extension to numbers, helping us represent the ways that nature breaks out of single dimensions (physical or otherwise). In this case, that shift into the complex plane leads us into this hidden wavelike aspect of material interaction. There's no need to feel concerned... it's a good thing!
I was first introduced to the Schrodinger eqn in a physical chemistry class, which did NOT cover the derivation, so it is really cool to see the correspondence with the classical wave equation. It almost seems obvious now
What a great video. It has pulled together all my reading about this subject into a beautiful summary. I cannot recommend this enough. I will watch it many more times over, for its clear and beautiful presentation. Keep up the excellent work
I really hope expanding to international shipping works out well! Looking forward to one day hold your book in my hands. Love your style, keep up the great work.
In electrical engeneering, we use the same tricks, replacing trigonometric functions that represents alternate voltages and currents, with imaginary exponemtials. Its just a way of avoiding trigonometric algebra. The i on the fornulas means that some quantities are 90 degrees "displaced" in time (related to a implicit frequency oscilarion assumed). Its just "syntax sugar" as computer guys says. The conpiled code (reality, experiments) is the same (real values oscilating in time) that needs 2 real numbers to describe (amplitude and phase), that can be represented using a trigonometric A.cos(wt+phi) or using imaginary exponentials A.exp(i.phi) in a certain ff (implicity assumed). People use to misunderstain, for exemple, imaginary electric POWER. Its just energy oscilating in time, written in a different way
It is not sugar, it is "meat". In EE you have a phasor rotation. In QM you have spin planes. All the uses of "i" in elementary physics are real geometry. People do not get taught this because of bad traditions. In QM it is far more insightful to trace the appearance of Schrödinger's "i" from the proper Dirac fermion (mathematically represented by a spinor field). In the proper Dirac theory there are no ℂ numbers needed, just real spinor fields valued in the real spacetime algebra (Dirac "matrices" are categeory-theoretically mapped to the frame basis vectors in the spacetime algebra for the fermions co-moving frame, so they are real vectors, no longer needing to be treated as uninterpreted ℂ matrices). The unit imaginaries are the bivectors or the spacetime pseudoscalar (elements of real geometry describing rotations an boosts). This reveals the Schrödinger i comes from 1st dropping to non-relativistic approximation then turning off the magnetic field in the Pauli equation. So it is still a spinor one is describing (and instruction to rotate the frame fields for observables). Not a raw uninterpreted ℂ number. When you know this, you realize Hilbert space representations are unnecessary. See also Jacob Barandes' work. (Hilbert space is unphysical.) In the proper spacetime algebra with spinors we still get interference from superposition, due to the bivector algebra (really, the full even subalgebra).
There is a huge difference, and that being the _i_ in quantum mechanics cannot be removed; it is fundamental to the framework, where it is not in EE uses. The fact you do not know this exposes you massive lack of understanding; it is not mere 'syntax;' imaginary numbers literally cannot be removed from QM unlike the EE example you gave
@@pyropulseIXXI I'm not sure how you concluded that..? It sounded like Claudio Costa was only talking about his personal knowledge of how i is used in EE, and not overapplying it to QM.
@@pyropulseIXXI I If you say so... But, please, tell me what is the fundamental meaning of i in QM that cannot be replaced by other math notations like matrices, bra-kets, or trigonometric functions? What is i, if not just math notation, syntax sugar (or meat) used to simplify explanations, calculations, and reduce formula notation ? You think its a phisical entity?
One of the best scientific videos I've ever seen. Year ago I studied physics to the point of solving the electron in a box but I never understood where the equation came from.
I love these videos for the physics (that I barely get) and the history - it's all a human endeavour. The example presented @12:28 through 16:40 is especially informative - and I was even ahead it for some parts (minor miracle).
The i in the Schroedinger equation is mapping the Jordan algebra of hermitian operators (observables, such as the Hamiltonian H) onto the Lie algebra of infinitesimal symmetry generators (such as the generator of the dynamics iH). In a certain sense, the i is traditionally written on the "wrong" side of the equation from this pov. It should be dψ/dt=-(i/ℏ )Hψ, but of course it's the same.
Very many thanks! The visualizations are both clear, and visually pleasing. I'm certainly going to get the book! If only you'd been around when I was doing undergrad physics, some ... decades ago! Please keep on making video explanations.
Very good presentation of title " Imaginary number are real " the vivid demonstration with numerical simulation make an impression that you are good communicator hence the book. Your this video help me to my recent work on "event quanta " . Thank you Good luck I got some questions arises, I hope you will cover in the edition or next video.
Really loved this video! The walkthrough of Schrodinger's argument was new to me... Usually I see something about the space and time translation operators
Thank you so much for the amazing presentation of the subject. The phase of the wavefunction is as important as its modulus. For instance, the electric polarization of ferroelectric crystals is directly related to the "Berry phase" of the Bloch wavefunctions and has nothing to do with their square modulus.
Imaginary numbers AREN'T real (you cant measure it in an instrument or put it in a plate), but it has undeniable influence on reality that reality alone cant explain. Like a hidden mathematical hand outside reality or an incomprehensible divine being manipulating behind the scenes.
I don't think you need to worry about the detail or pace. We can always rewind the video until the it sinks in. Sometimes 4 or 5 rewinds or more are needed.
It is sort of 'coincidence' that multiplication of complex numbers is isomorphic to SO(2). So it should not be a surprise that complex numbers emerge in problems where rotation is involved. We could write *i* as 2x2 real matrix, or solve a system of equations for coordinates, and results will be the same. It is just about simplicity of representation of the same object.
Ah, the human species really likes to cultivate mystery. Some mathematical formulations have been called 'complex numbers', with its 'imaginary' part. "Imaginary" means "That does not exist" when speaking of Reality!! They could just as well have called this part 'auxiliary' but it is less spectacular and inspires less respect and admiration. Very good explanation by the way. The best I have ever seen.
Another way to name imaginary numbers are lateral numbers. I like the latter because it removes the "fakiness" of thinking about something "imaginary" and instead accept that the number-line can be extended laterally
The whole issue of complex or even hypercomplex numbers in nature feels resolved after having studied Clifford algebras, Lie algebras, and representation theory. It's some pretty heavy math but once understood boils down to a simple idea. The "ordinary" real numbers are just the simplest kind of number, those which have faithful representation in 1 real dimension. The complex numbers require at least 2 real dimensions for a faithful representation, but they are still numbers in this broader, more general sense. In the greater scheme, even the complex numbers are relatively simple, since they still commute. I tend now to think of a "number" as any member of a valid algebra, and I think that nature sort of works like that too.
In this case the complex numbers don't come from a Lie algebra. They come from unitarity. Why? Because the Lie algebras underlying quantum mechanics are exactly the same that can be found in classical physics, where nothing ever requires a complex number. Here the complex numbers are NOT just a convenience tool. They are what distinguishes the quantum mechanical solution of Kolmogorov's axioms from the probability theory solution, which is VERY different. The most interesting thing that happens in probability theory are the central limit theorems and that's about it. Quantum mechanics, OTOH, creates an entire universe of matter and radiation... all because of that little "i". Take it away and things are about as interesting as watching paint dry.
@schmetterling4477 I don't disagree with anything you said. I merely pointed out that there is nothing deeply weird about complex numbers, they are just numbers. You don't actually need to use i or complex numbers per se, you can express all the algebraic logic using for example 2x2 real matrices in place of all the complex numbers. Because its the algebra that matters and not the symbols themselves. *Anything* which squares to minus 1 can replace the imaginary unit in Euler's formula and recover all the same behaviour. Complex numbers are just elements of the Cl(1,1) algebra and they also meet the requirements of a Lie alegra, but as for the *reason* they appear in QM, well that's a separate question. The Dirac matrices for example come from requiring 4 special symbols which obey certain algebraic relations which cannot be met by using 1x1 real matrices (real numbers). Typically we use a 4x4 complex representation for these, but the point I'm stressing is that these are just representations. The algebraic properties of these things are what really matters to nature, that is the squares and commutation relations. Edit: correction, complex numbers are not in Cl(1,1), I forgot about the anticommutation rule, but hopefully the underlying argument is still clear.
@@disgruntledwookie369 No, there is indeed nothing weird here. It's all about rotations and complex numbers are just a convenience tool for the two dimensional case. What is weird is that we still don't teach where all of this comes from in undergrad QM courses. We are breeding generation after generation of physicists who are puzzled by a rather trivial formalism that, at the end of the day, is simply a partition of unity that uses multi-dimensional versions of Pythagoras.
In this episode the math went over my head. I could juuust about keep up with your episode on Euler (GENEROUSLY helped by the graphics!), but his one was tough to chew.... But I had fun anyway, and while I couldn't follow the equations, that the phase is represented by i clicked profoundly.
I know this wasn't the focus of your video, but your visualizations of the propagation of the wavefunction through the slits reveal many features I hadn't considered before despite seeing the double slit experiment explained many, many times. The reflection of the wavefunction is almost completely neglected in any treatment, but a huge amount of the probability is contained in that component. Yet, even with the single slit, the reflected wave is not a smooth distribution, there are zeroes in the probability at certain angles. This animation also makes the action of the slit itself very clear. I never thought about it before but the slit acts as "particle in a box" filter for the wavefunction as it propagates, decomposing the Gaussian wavepacket into the discrete modes along the axis parallel to the slits, and this is seemingly what causes the interference in the reflected wave. It just goes to show that even the single slit is weird in quantum mechanics. This is despite the fact that your example uses a slit size that is large compared to the deBroglie wavelength, making the interference in the transmitted wave negligible.
The reflected wave is the various trajectories of the particle missing the slits and bouncing backwards. In real life, one would only have a detector on the opposite side of the barrier, with nothing to catch the reflected particles. It can be assumed that the reflected wave exists but in the context of an actual experiment you wouldn’t bother trying to find out where the duds ended up. But as you mentioned, if one did wish to set up a detector behind their electron gun, they would be able to see the inference on that side as well.
@@howiedewitt6223 Well said, if you were to setup a detector on the reflected side, you would find even the reflected trajectories for the single slit are highly non-classical!
It’s also worth noting that researchers have looked deeply into whether the imaginary numbers that arise in QM are essential or if the same predictions could be made without them… Still an active area of exploration to this day, but they seem essential!
@@worththekeeping Any mathematician will tell you that that's just a trivial isomorphism. Everything that can be done with complex numbers can also be done without them.
@@schmetterling4477 ya but mathematicians think everything is trivial once they’ve proven it😉 My understanding is that it is nontrivial to reformulate QM using only real numbers and that efforts to do so usually run into problems with consistency
i Captures all important unknowns acting as a curl. And makes a compromise causality of these: chance. In Schrödinger's model of matter waves, causality, constant energy and separation of space and time we apparently need this compromise of i. If matter is to be considered a wave then there appears to be more to the puzzle than our physical universe. And it acts as a curl.
Hey! Stellar video as usual? Just curious what is the target audience for the book? Like what type of background does a person need to get most out of it? Thanks!
Isn't it ('i') a 90 degree rotation, pi/2 rads, and then hbar doing a second normalisation of that 'pi' rads thing ;-) I expect we'll be going round in circles on this. Looking forward to hearing what's actually said !
Thanks for this excellent explanation and clear illustrations! One gripe: please don’t put arrows on the negative direction of the Cartesian/complex plane.
It does, does it not? But it isn't. The phase is extremely important and it causes endless convergence problems when we have to integrate these functions in more complicated field theory problems.
Is there any way the Born rule can be derived more logically? The idea that the wavefunction has all that predictive power and yet suddenly "collapses" and turns into random probabilities at the very end has never sat right with me. Where does that "randomness" come from? It's all so elegant up to that point, and then Born comes along and shoves us back into a real-number classical-physics picture kind of unceremoniously I've always wondered if there is a deeper reason why particles are only detected at one point rather than across the whole wave, and there are a lot of simplifications to the picture that most physics models doesn't seem to want to touch. For example, doesn't the detection surface also have billions of its own bound electrons, each with their own wavefunction? Wouldn't they interfere with the approaching electron as it gets close, and wouldn't that influence the detected scattering pattern? My thought is maybe the "noise" generated from everything already there in the scene ultimately destructively interfere with most of the approaching wave, "ruling out" most of it until there's only a single (apparent) point particle at the end. Maybe the Born Rule is just a quick-and-easy shorthand for accounting for the interference from that random noise, like maybe the probability distribution of the electron's possible "true" wavefunctions given an arbitrary noisy environment shakes out to be the square of its "pure" wavefunction without any interference. I've always wanted to follow this hunch, but it's very computationally expensive to model those effects with any reasonable degree of accuracy, and I'm not even a physicist myself so I wouldn't even know where to start. But that's always been an elegant possible alternative to Copenhagen, Bohmian, or Many-Worlds to me -- a "no-collapse" model that doesn't have to give birth to a multiverse because, accounting for everything possible, all alternate possibilities cancel out to zero, just in a way so complicated and exact that it's much easier just to work with probabilities.
The Born rule is an abstract description of the absorption spectrum of the detection system. It is easier to "grok" if you consider a spectroscopy experiment. Imagine you have an atom in a large, perfectly reflecting cavity. You make a small hole into it and couple that to an optical fiber that leads to an interferometric spectrometer. That spectrometer has absorption lines that depend on the distance between the mirrors. The spectrometer can only absorb photons that have the resonance frequency of the spectrometer. So now the atom emits a photon of a certain energy. That energy bounces around the reflecting cavity and couples to the spectrometer. It either gets absorbed in the spectrometer or it doesn't. If it gets absorbed, then this energy is no longer in the atom/cavity system. If it doesn't get absorbed, then the atom gets re-excited with it and effectively stays in the original state forever. The projection operator in the Born rule projects onto states that belong to exactly the absorption energies of the spectrometer. The final system can only take on those particular states because otherwise it simply can't shed its energy. It is unfortunate that we don't teach physical pictures like this when we teach the Born rule. It seems like a completely abstract and arbitrary thing, but in reality it is the mathematical expression of the fact that the absorption spectrum of the measurement system matters a great deal for the final state of the coupled system. You can generalize this to the usual location/momentum/spin measurements once you realize that "a quantum" is a combination of energy, momentum, angular momentum and charges. Historically one can find these energy/momenta as indices into Heisenberg's matrices in his matrix mechanics papers. In that representation it's easier to see that what we are doing is to consider "transition probabilities" from the quantum system into the measurement's systems absorbing energy/momenta channels. In the Schroedinger equation representation the locations/momenta become test functions for linear operators and that obfuscates the physical meaning of the math quite a bit. I believe von Neumann explains this (in a somewhat hard to understand way) in his 1932 book on quantum mechanics that marks the beginning of the modern "shut up and calculate" era. It's all perfectly legit... you are just not being told what the physics behind the symbols really is in a typical undergrad course on QM. The "randomness", by the way, is not randomness. Random numbers do not conserve energy, momentum, angular momentum and charges, i.e. quantum mechanics can not be random for trivial reasons. What happens, instead, is that the measured result is entangled with some "far away" system (the source of the energy) that in general resides in some spacelike separated region of spacetime. Nature has therefor made it impossible for us to know the state of that system, hence the local measurement looks random. It isn't, but the lack of "god mode" in a relativistic universe makes an introspection into the total state of the universe impossible.
Great explanation, excellent graphics. Even when sending electrons though the double-slit one at a time, the interference pattern will build up - unless the electron is observed to pass through a particular slit, after which it behaves like a classical bullet. As Feynman said "No one understands quantum mechanics"! I'm an architect, but really appreciate the walk through of the equations.
Feynman was joking. Quantum mechanics is probably the most trivial theory in all of physics. We just don't teach it well and that perpetuates all of these myths. It's like the question "Why did the architect put that beam there? It's such a great artistic choice!". In reality he didn't. The structural engineer did to prevent the building from collapsing. :-)
@@schmetterling4477there is nothing trivial about the ontological questions raised by quantum mechanics, and Feynman discussed these topics at great length in person and in his writing. He was not joking and frankly your comments make you sound like someone who is aware of many concepts but has not been able to connect it with practical understanding of the physical world. If you believe you have a trivial explanation of notions like entanglement I invite you to share it here. Please go ahead and derive it from first principles.
@@howiedewitt6223 Quantum mechanics is simply the system theory of systems that go "click-click-click" like a Geiger-Mueller counter. One can derive it in like a dozen pages of slightly above high school level math from Kolmogorov's axioms and if you can overcome your intellectual laziness then you can do a literature search for papers in which it has been done. I just gave you the search terms. These papers are boring like watching paint dry. You are basically solving a partition of unity problem with Pythagoras (sin^2 + cos^2=1) instead of with the usual p+(1-p)=1 formula of ordinary probability theory. Did Feynman know this? I don't know and I am not sure he did based on what he wrote about QM, but all of this was already known to people like von Neumann in 1932... which is a year before Kolmogorov even published his axioms. Why is quantum mechanics the way it is? Because of special relativity. That's undergrad intuition. All you have to do is to stare at a spacetime diagram for a few minutes and then it should become clear why relativistic systems can not be deterministic in the classical 19th century sense.
Yes, that is what we teach in high school. Not sure why anybody needs a refresher on the internet. If you went to a good public school then you have seen it. It seems to be taught in Algebra II in 11th grade. That's roughly where it belongs.
I am currently a student in physics at Leiden University and have learned the derivation from ground up using the ansatz for said plain wave. It is incredible just how profound the equation is and the solutions. We have not yet learned to solve for the hydrogen atom but that will come in due time in next semester as it requires much more tools. Was wondering when you would bring those things up in the video and you did eventually do it. Great job. Sharing it with non-physics friends now so they get an idea of how to do it. Not everyone needs just 6 lines to derive it ;)
The equation isn't profound. It's not even physical. It does not conserve energy and momentum, which you will notice as soon as you learn to solve the particle in a box problem. It's a toy quantization procedure that is useful for many applications in which its fundamental flaws are of no importance and for which physically correct equations are waaaay overkill.
@@schmetterling4477what exactly do you think unitary evolution represents 😂😂😂 what could it possibly mean for the Hamiltonian to be time independent as it is in the box problem? Might it have something to do with energy conservation? 😂😂😂😂😂
@@howiedewitt6223 The problem is not the time independence. The problem is that the potential is fixed. There is no back reaction that could preserve momentum. We are simply not modelling the momentum transfer on the box. That can only happen the same way as in classical mechanics by having V(r1-r2) and at least two particles in the system. But now you are getting in trouble with the Born rule, though, because what are you measuring? The particle in the box or the box or both? The formalism is simply not self-consistent. You can squint and ignore and then the world of the Schroedinger equation is wrong but useful. If you decide not to squint then you will slide down a slippery slope that won't end before you hit quantum field theory, which, as I said, is waaaaay overkill for most applications. So, yeah, go and learn how to do it the wrong way because you will need it for all of your applied physics classes like atomic physics, nuclear physics, solid state etc.. Then take a quantum field theory class or two or eight and learn how nature actually does this.
Excellent video! From a quantum mechanical physicist. I always feel bad about the name given to SQRT(-1) =i "IMAGINARY NUMBER", it confuses its significance often.
Imaginary numbers are no more real than "real" ones. In particular, any equation involving imaginary numbers can be written down as two equations with only real numbers. In the case of Schrodinger equation, you can have two equations mixing amplitude and phase, both linking to other known equations. As they are equivalent, there is no telling which one is more real than the other, whether it maths the maths simpler or not.
0:50 It's actually -i h, not i; not that it matters qualitatively, but as the equation stands, it is a mashup of both the dimensionless and non-dimensionless versions.
I love this explanation, I'd love if you could do an explanation on Barande's stochastic processes for quantum mechanics. He sounds really convincing but I have no frame of reference to understand his math
Quantum mechanics is not stochastic. Nothing in physics can be stochastic. Stochastic processes do not conserve energy and momentum. Some theorists who are eternally stuck at the level of the free non-relativistic particle and some semi-classical potentials love these approximations because they can be calculate with relative ease, but I don't see any other reason for even considering them. They do not represent actual physics. Now, it is important to understand the difference between reversible and irreversible processes, but that is almost entirely orthogonal to stochastic model.
There are a ton of videos discussing the surprise that complex numbers seem to be a fundamental part of reality. What is missing, and what I really would like is to see is a video explaining that this does not mean that complex numbers are "more physically real" than real numbers, but just that they are a convenient tool for describing operations on the circle, and the circle seems to be more fundamentally real than intervals. It is like saying "quaternions seem to be fundamental to reality" because they are very convenient to describe 3D rotations. Accepting that C in physics is more about the geometry and topology of the circle than the algebra of a field also helps ease the way to considering other fundamental topologies, such as SU(2) and SU(3), or even Calabi-Yau manifolds etc. Tl;dr: C is not special. The circle is special, and C gives a coordinate system on the circle.
Yeah, but you still didn't tell us where "the circle" came from. You are halfway there, but you still lack the full understanding. Hint: What is unitarity. THAT is where the unitary groups come from. And why do we have unitarity in quantum mechanics? Your turn. ;-)
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The way you combined the wave phase perspective to the interference was very well done. Great video ❤
Isn’t all equations are just play of concepts symbolized by numbers or letters and they get to divide, multiply, add or subtract?!? And End up being on both sides of equation = !!
Nice animations 😉
The man himself. Manim is amazing and I'm glad to see it being adopted by other math and science creators!
@3blue1brown Nice animations
You and other science guys are really crazy about knowledge of science and other fields.❤
lol thanks 😁
@@dr.shrimppuertorico2749 Grant: you bet
The animation and explanation of interference being caused by the phase of the wave function was so well done!
I especially like the interference in the reflected wave off the single slit.
I have a physics PhD, and while of course nothing here was new to me per se, I really appreciate the way you tie the story together. The undergraduate physics curriculum is more focused on establishing the model and using it to make predictions that illustrate its key features (like the double slit results), but the "origin story" doesn't often get explained. Thanks for that!
Except that this is not the correct origin story. Schroedinger simply guessed an equation. It turns out that it was the wrong equation and he never understood where it came from. The more rational approach to quantum mechanics can be found in Heisenberg's matrix mechanics papers and then, as a culmination, in 1932 in von Neumann's book.
@@schmetterling4477 "It turns out that it was the wrong equation and he never understood where it came from."
What?
@@kellymoses8566 If you read Schroedinger then it is highly doubtful that he understood that quantum mechanics is a relativistic ensemble theory.
loser nerd
This video is absolutely top drawer. The animations, the lesson, the historical pages, the actual visualization of how the imaginary unit shapes the wave function.
Just brilliant. I have to get the book.
When I learned (decades ago) that multiplying by i causes a rotation in the complex plane, I wondered why no one ever seems to talk about that in quantum mechanics. Thanks you so much for the beautiful animated images and the detailed explanation. Wonderfully done, and it explains so much I never learned in my advanced physics courses in college.
As soon as we mention the word "unitarity" in the first QM 101 class we are talking about nothing else than rotation. That's basically what it means, most physics professors just don't care to explain it and most students don't notice on their own.
I wish I had this visualization so many years ago when I was in physics...
I really think some 9 year old ipad kid is gonna discover these videos one day and proceed to turn into the next John von Neumann.
Seriously. I was so lost in my undergrad classical & quantum mechanics classes (but managed to eke out good grades because everyone else was lost also) but these kinds of videos make it so simple to understand.
EXACTLY
The sophomore / junior physics text by David Griffith ‘intro to quantum mechanics’ has a very good discussion on how the solution to Schrödinger’s equation and even computes the solution with principal quantum number n=1, to arrive at the spherical s orbital which is the ground state for hydrogen. Watching this video took me back 20 years to undergrad years. My quantum mechanics professor, Dr Kwon Lau passed away a few years ago, but his lessons are still alive in my mind.
Still my favorite homework problem of all time, found in this textbook: model the Earth-Sun system as a hydrogenic atom.
The unfortunate thing about that text though is it paints a very different picture of quantum mechanics than what is truly happening. It’s very difficult to teach this properly, of course, but to me it doesn’t do enough to distinguish the coordinate dependence of the wave function as a result of the configuration space from the real spatial behavior of the particle. Much of the intuition built up ends up being incompatible with a system of multiple particles. In the text there is a sense that the expectation value of a quantity is in some sense a description of the behavior of the particle with physical content, but in truth it only the average of measurement outcomes.
Something worthy to note is that the Born rule was adopted solely because it worked. Originally, Born thought the probability to be |Ψ|, but Ehrenfest’s theorem didn’t corroborate this. A next reasonable step for amelioration came with simply squaring it, as Born did, and it just so happened to work. How amazing is that?
Yes and no I would say for the success of the Born rule. For "single-particle" experiments, it does not apply. Some works concretely discuss this particular point.
I love how clearly and simply you explain everything this was so much better than any of my Physics professors when I was in college. I’ve been watching your videos since high school and they’ve always been so fascinating and illuminating. Thanks for all you do!
This is quickly becoming one of my all-time favorite physics channels. Excellent balance between depth and clarity, really nice visuals, and a very pleasant voice and speaking style.
The video is a gemstone. I remember university at that time physicists were struggling with the Schrodinger some of them knew about hydrogen solution but almost nothing to derivation on this topic. Great work. Aside the true paradox is when most people refuse to accept imaginary numbers then conversely to me is very much more obvious to use them.
Very good explanation and visualization, but one thing I'd like to add. It's not matter that is a wave. It is the interaction with matter that is described by a wave.
Good derivation walkthrough
You really should adress the measurement problem and the question of non-locality in the EPR paper. That would be a delight to see.
There is no such thing as a measurement problem and there is no non-locality in nature, either. There are only people who don't understand physics.
@@schmetterling4477 do we have an Everettian here?
@@ivocanevo You have a physics PhD who has measured trillions of quanta. Not once was there a problem. A quantum is simply a small amount of energy that was originally in a quantum system and that at the end of the measurement is in the detector system. There are no problems with that, neither in theory nor in the lab. It takes a theorist who has never been in the lab to imagine that this is somehow problematic. ;-)
If you want to see examples of microscopic "explanations" of how measurements work in detail, read Mott 1929 or von Neumann 1932 (chapter 5, I believe). All of this was understood in the mainstream literature very early on. ;-)
I'm just a lowly engineer, but it seems to me that a lot of the discomfort around imaginary numbers arises due to their name. There really isn't anything imaginary about them; they are a very natural extension to numbers, helping us represent the ways that nature breaks out of single dimensions (physical or otherwise). In this case, that shift into the complex plane leads us into this hidden wavelike aspect of material interaction. There's no need to feel concerned... it's a good thing!
_Unfortunately, the name stuck_
Gauss hated the word "imaginary", and suggested they should be called lateral numbers.
negative numbers are also imaginary
Real Numbers are not Real. Nature uses Complex Numbers.
@@OmDeLaTara Not according to my bank. ;))
What the heck, your production quality - video and book are top notch! Who the heck are you? what's your background?
Math prof iirc
I was first introduced to the Schrodinger eqn in a physical chemistry class, which did NOT cover the derivation, so it is really cool to see the correspondence with the classical wave equation. It almost seems obvious now
What a great video. It has pulled together all my reading about this subject into a beautiful summary. I cannot recommend this enough. I will watch it many more times over, for its clear and beautiful presentation. Keep up the excellent work
Incredible animations. Thanks a lot!! Brilliant explanation
I really appreciate the build up of logical steps.
Helps to truly grasp the underlying logic.
I really hope expanding to international shipping works out well! Looking forward to one day hold your book in my hands. Love your style, keep up the great work.
In electrical engeneering, we use the same tricks, replacing trigonometric functions that represents alternate voltages and currents, with imaginary exponemtials. Its just a way of avoiding trigonometric algebra. The i on the fornulas means that some quantities are 90 degrees "displaced" in time (related to a implicit frequency oscilarion assumed). Its just "syntax sugar" as computer guys says. The conpiled code (reality, experiments) is the same (real values oscilating in time) that needs 2 real numbers to describe (amplitude and phase), that can be represented using a trigonometric A.cos(wt+phi) or using imaginary exponentials A.exp(i.phi) in a certain ff (implicity assumed). People use to misunderstain, for exemple, imaginary electric POWER. Its just energy oscilating in time, written in a different way
It is not sugar, it is "meat". In EE you have a phasor rotation. In QM you have spin planes. All the uses of "i" in elementary physics are real geometry. People do not get taught this because of bad traditions. In QM it is far more insightful to trace the appearance of Schrödinger's "i" from the proper Dirac fermion (mathematically represented by a spinor field). In the proper Dirac theory there are no ℂ numbers needed, just real spinor fields valued in the real spacetime algebra (Dirac "matrices" are categeory-theoretically mapped to the frame basis vectors in the spacetime algebra for the fermions co-moving frame, so they are real vectors, no longer needing to be treated as uninterpreted ℂ matrices).
The unit imaginaries are the bivectors or the spacetime pseudoscalar (elements of real geometry describing rotations an boosts). This reveals the Schrödinger i comes from 1st dropping to non-relativistic approximation then turning off the magnetic field in the Pauli equation. So it is still a spinor one is describing (and instruction to rotate the frame fields for observables). Not a raw uninterpreted ℂ number.
When you know this, you realize Hilbert space representations are unnecessary. See also Jacob Barandes' work. (Hilbert space is unphysical.) In the proper spacetime algebra with spinors we still get interference from superposition, due to the bivector algebra (really, the full even subalgebra).
There is a huge difference, and that being the _i_ in quantum mechanics cannot be removed; it is fundamental to the framework, where it is not in EE uses. The fact you do not know this exposes you massive lack of understanding; it is not mere 'syntax;' imaginary numbers literally cannot be removed from QM unlike the EE example you gave
I immediately thought of wave polarization when I saw the spatial curvature of the Schroedinger equation.
@@pyropulseIXXI I'm not sure how you concluded that..? It sounded like Claudio Costa was only talking about his personal knowledge of how i is used in EE, and not overapplying it to QM.
@@pyropulseIXXI I If you say so... But, please, tell me what is the fundamental meaning of i in QM that cannot be replaced by other math notations like matrices, bra-kets, or trigonometric functions? What is i, if not just math notation, syntax sugar (or meat) used to simplify explanations, calculations, and reduce formula notation ? You think its a phisical entity?
If you are wondering what is too fast, it's the flashed footnotes that are impossible to pause on!!!!
Sounds like a you problem.
skill issue
Why is De Broglie an "obscure" Frenchman ?
If you're on the desktop version of UA-cam, you can press the period and comma buttons to move forward or backward one frame.
You could set the settng of the video display speed from 1 (normal) to 0.25 (lowest).
One of the best scientific videos I've ever seen. Year ago I studied physics to the point of solving the electron in a box but I never understood where the equation came from.
i cant expres how much i love this chanel it teaches me so much
Can confirm. ^.^
I completed my first course on Quantum Physics just a few weeks ago, wish I had this video before starting the course, very well explained!
I love these videos for the physics (that I barely get) and the history - it's all a human endeavour.
The example presented @12:28 through 16:40 is especially informative - and I was even ahead it for some parts (minor miracle).
The i in the Schroedinger equation is mapping the Jordan algebra of hermitian operators (observables, such as the Hamiltonian H) onto the Lie algebra of infinitesimal symmetry generators (such as the generator of the dynamics iH). In a certain sense, the i is traditionally written on the "wrong" side of the equation from this pov. It should be dψ/dt=-(i/ℏ )Hψ, but of course it's the same.
Very many thanks! The visualizations are both clear, and visually pleasing. I'm certainly going to get the book! If only you'd been around when I was doing undergrad physics, some ... decades ago! Please keep on making video explanations.
Thank you so much for this! This is by far the best video I have watched on the Schrodinger equation
Absolutely splendid!
Very good presentation of title
" Imaginary number are real " the vivid demonstration with numerical simulation make an impression that you are good communicator hence the book.
Your this video help me to my recent work on "event quanta " . Thank you
Good luck
I got some questions arises, I hope you will cover in the edition or next video.
Really loved this video! The walkthrough of Schrodinger's argument was new to me... Usually I see something about the space and time translation operators
Thank you so much for the amazing presentation of the subject. The phase of the wavefunction is as important as its modulus. For instance, the electric polarization of ferroelectric crystals is directly related to the "Berry phase" of the Bloch wavefunctions and has nothing to do with their square modulus.
bro your videos are awesome, just discovered your channel, binge-watch all your content now! Please continue your excellent work
Honorary 14th part of Imaginary Numbers Are Real
Fifteenth because of the Euler’s formula video which arguably should also be an honorary part.
Imaginary numbers AREN'T real (you cant measure it in an instrument or put it in a plate), but it has undeniable influence on reality that reality alone cant explain.
Like a hidden mathematical hand outside reality or an incomprehensible divine being manipulating behind the scenes.
Ever since I saw you lift that curve into the third dimension, I have told every math nerd I know to check out your channel. Just a magical moment!
I don't think you need to worry about the detail or pace. We can always rewind the video until the it sinks in. Sometimes 4 or 5 rewinds or more are needed.
Your videos are always a delight!. So glad you are back on UA-cam.
Very well done. This is the best description of these equations and phenomena I have seen.
My greatest discovery of a UA-cam channel ever. Kudos
So great, can`t wait to recive your book here in Vienna next year.. Thanks for your excellent work.
It is sort of 'coincidence' that multiplication of complex numbers is isomorphic to SO(2). So it should not be a surprise that complex numbers emerge in problems where rotation is involved. We could write *i* as 2x2 real matrix, or solve a system of equations for coordinates, and results will be the same. It is just about simplicity of representation of the same object.
Beautifull. You have gone well beyond just earning a new subscriber. Watching this was a pleasure.
My god, this finally nails quantum mechanics "weirdness" to me 😂 thank you for the fantastic video!
love your work man, one day I will buy your book for sure!!!!
Small mistake in the algebra at 11:30. Forgot to square the h_bar! Excellent video btw.
Ah, the human species really likes to cultivate mystery. Some mathematical formulations have been called 'complex numbers', with its 'imaginary' part. "Imaginary" means "That does not exist" when speaking of Reality!!
They could just as well have called this part 'auxiliary' but it is less spectacular and inspires less respect and admiration.
Very good explanation by the way. The best I have ever seen.
This channel is truly one of the best in youtube
There so many books you read for the video😮i appreciate that bro,thank you so much
WelchLabs gets you from understanding an equation into feeling and seeing the equation...
Another way to name imaginary numbers are lateral numbers. I like the latter because it removes the "fakiness" of thinking about something "imaginary" and instead accept that the number-line can be extended laterally
Your videos are always such a pleasure to watch :), perfect balance in the level of detail and very aesthetically pleasing, well done!
Incredible work, this was super fun to watch
The whole issue of complex or even hypercomplex numbers in nature feels resolved after having studied Clifford algebras, Lie algebras, and representation theory. It's some pretty heavy math but once understood boils down to a simple idea. The "ordinary" real numbers are just the simplest kind of number, those which have faithful representation in 1 real dimension. The complex numbers require at least 2 real dimensions for a faithful representation, but they are still numbers in this broader, more general sense. In the greater scheme, even the complex numbers are relatively simple, since they still commute. I tend now to think of a "number" as any member of a valid algebra, and I think that nature sort of works like that too.
In this case the complex numbers don't come from a Lie algebra. They come from unitarity. Why? Because the Lie algebras underlying quantum mechanics are exactly the same that can be found in classical physics, where nothing ever requires a complex number. Here the complex numbers are NOT just a convenience tool. They are what distinguishes the quantum mechanical solution of Kolmogorov's axioms from the probability theory solution, which is VERY different. The most interesting thing that happens in probability theory are the central limit theorems and that's about it. Quantum mechanics, OTOH, creates an entire universe of matter and radiation... all because of that little "i". Take it away and things are about as interesting as watching paint dry.
@schmetterling4477 I don't disagree with anything you said. I merely pointed out that there is nothing deeply weird about complex numbers, they are just numbers. You don't actually need to use i or complex numbers per se, you can express all the algebraic logic using for example 2x2 real matrices in place of all the complex numbers. Because its the algebra that matters and not the symbols themselves. *Anything* which squares to minus 1 can replace the imaginary unit in Euler's formula and recover all the same behaviour. Complex numbers are just elements of the Cl(1,1) algebra and they also meet the requirements of a Lie alegra, but as for the *reason* they appear in QM, well that's a separate question. The Dirac matrices for example come from requiring 4 special symbols which obey certain algebraic relations which cannot be met by using 1x1 real matrices (real numbers). Typically we use a 4x4 complex representation for these, but the point I'm stressing is that these are just representations. The algebraic properties of these things are what really matters to nature, that is the squares and commutation relations.
Edit: correction, complex numbers are not in Cl(1,1), I forgot about the anticommutation rule, but hopefully the underlying argument is still clear.
@@disgruntledwookie369 No, there is indeed nothing weird here. It's all about rotations and complex numbers are just a convenience tool for the two dimensional case. What is weird is that we still don't teach where all of this comes from in undergrad QM courses. We are breeding generation after generation of physicists who are puzzled by a rather trivial formalism that, at the end of the day, is simply a partition of unity that uses multi-dimensional versions of Pythagoras.
In this episode the math went over my head. I could juuust about keep up with your episode on Euler (GENEROUSLY helped by the graphics!), but his one was tough to chew.... But I had fun anyway, and while I couldn't follow the equations, that the phase is represented by i clicked profoundly.
Thanks! I watched your imaginary number series a few years ago!
Outstanding video! We need to see a double slit experiment which detects those refelections.
I really want your book.
Any one that objected complex numbers at a time is just clueless about the significance of them, its like rejecting the usage of a rotational matrix.
I know this wasn't the focus of your video, but your visualizations of the propagation of the wavefunction through the slits reveal many features I hadn't considered before despite seeing the double slit experiment explained many, many times. The reflection of the wavefunction is almost completely neglected in any treatment, but a huge amount of the probability is contained in that component. Yet, even with the single slit, the reflected wave is not a smooth distribution, there are zeroes in the probability at certain angles.
This animation also makes the action of the slit itself very clear. I never thought about it before but the slit acts as "particle in a box" filter for the wavefunction as it propagates, decomposing the Gaussian wavepacket into the discrete modes along the axis parallel to the slits, and this is seemingly what causes the interference in the reflected wave. It just goes to show that even the single slit is weird in quantum mechanics. This is despite the fact that your example uses a slit size that is large compared to the deBroglie wavelength, making the interference in the transmitted wave negligible.
The reflected wave is the various trajectories of the particle missing the slits and bouncing backwards. In real life, one would only have a detector on the opposite side of the barrier, with nothing to catch the reflected particles. It can be assumed that the reflected wave exists but in the context of an actual experiment you wouldn’t bother trying to find out where the duds ended up. But as you mentioned, if one did wish to set up a detector behind their electron gun, they would be able to see the inference on that side as well.
@@howiedewitt6223 Well said, if you were to setup a detector on the reflected side, you would find even the reflected trajectories for the single slit are highly non-classical!
Wow great points! I did the numerical simulation with QMSolve - they have other interesting simulations as well!
I wish you were teaching these years ago. This is brilliant
Fantastic video! I’m jealous of your studio/workspace as well! So perfect
It’s also worth noting that researchers have looked deeply into whether the imaginary numbers that arise in QM are essential or if the same predictions could be made without them… Still an active area of exploration to this day, but they seem essential!
@@worththekeeping Any mathematician will tell you that that's just a trivial isomorphism. Everything that can be done with complex numbers can also be done without them.
@@schmetterling4477 ya but mathematicians think everything is trivial once they’ve proven it😉 My understanding is that it is nontrivial to reformulate QM using only real numbers and that efforts to do so usually run into problems with consistency
@@worththekeeping What would be non-trivial about converting a complex number a+ib into a vector (a, b)?????
So cool, thank you Stephen! Excellent video again!
i Captures all important unknowns acting as a curl. And makes a compromise causality of these: chance.
In Schrödinger's model of matter waves, causality, constant energy and separation of space and time we apparently need this compromise of i.
If matter is to be considered a wave then there appears to be more to the puzzle than our physical universe. And it acts as a curl.
small error at 11:30 on second line from bottom on the right, h bar squared instead of h bar. very good video thanks a lot!
This helped me understand so much. Thank you for making this video.
Really nice video! I had to buy the book on principal just to show support!
This whole video is superb. Will order the book soon.
19:21 made everything make so much more sense.
nice video! that guitar is tuned down a whole step (or at least the A string) because that was a G :)
Very beautifully and clearly done. Thank you.
Quite possibly one of the best videos I've ever seen on this topic
I'm 15 and barely understand this stuff but from what I can understand this equation is so cool🔥🔥
Hey! Stellar video as usual? Just curious what is the target audience for the book? Like what type of background does a person need to get most out of it? Thanks!
I wish I actually understood what was going on instead of just being in awe of it 😅
Isn't it ('i') a 90 degree rotation, pi/2 rads, and then hbar doing a second normalisation of that 'pi' rads thing ;-) I expect we'll be going round in circles on this.
Looking forward to hearing what's actually said !
Literally beautiful, Thank you !!!
Thanks for this excellent explanation and clear illustrations! One gripe: please don’t put arrows on the negative direction of the Cartesian/complex plane.
Schrodinger's "wave" equation looks more like a diffusion equation.
It does, does it not? But it isn't. The phase is extremely important and it causes endless convergence problems when we have to integrate these functions in more complicated field theory problems.
Is there any way the Born rule can be derived more logically? The idea that the wavefunction has all that predictive power and yet suddenly "collapses" and turns into random probabilities at the very end has never sat right with me. Where does that "randomness" come from? It's all so elegant up to that point, and then Born comes along and shoves us back into a real-number classical-physics picture kind of unceremoniously
I've always wondered if there is a deeper reason why particles are only detected at one point rather than across the whole wave, and there are a lot of simplifications to the picture that most physics models doesn't seem to want to touch. For example, doesn't the detection surface also have billions of its own bound electrons, each with their own wavefunction? Wouldn't they interfere with the approaching electron as it gets close, and wouldn't that influence the detected scattering pattern?
My thought is maybe the "noise" generated from everything already there in the scene ultimately destructively interfere with most of the approaching wave, "ruling out" most of it until there's only a single (apparent) point particle at the end. Maybe the Born Rule is just a quick-and-easy shorthand for accounting for the interference from that random noise, like maybe the probability distribution of the electron's possible "true" wavefunctions given an arbitrary noisy environment shakes out to be the square of its "pure" wavefunction without any interference.
I've always wanted to follow this hunch, but it's very computationally expensive to model those effects with any reasonable degree of accuracy, and I'm not even a physicist myself so I wouldn't even know where to start. But that's always been an elegant possible alternative to Copenhagen, Bohmian, or Many-Worlds to me -- a "no-collapse" model that doesn't have to give birth to a multiverse because, accounting for everything possible, all alternate possibilities cancel out to zero, just in a way so complicated and exact that it's much easier just to work with probabilities.
The Born rule is an abstract description of the absorption spectrum of the detection system. It is easier to "grok" if you consider a spectroscopy experiment. Imagine you have an atom in a large, perfectly reflecting cavity. You make a small hole into it and couple that to an optical fiber that leads to an interferometric spectrometer. That spectrometer has absorption lines that depend on the distance between the mirrors. The spectrometer can only absorb photons that have the resonance frequency of the spectrometer. So now the atom emits a photon of a certain energy. That energy bounces around the reflecting cavity and couples to the spectrometer. It either gets absorbed in the spectrometer or it doesn't. If it gets absorbed, then this energy is no longer in the atom/cavity system. If it doesn't get absorbed, then the atom gets re-excited with it and effectively stays in the original state forever. The projection operator in the Born rule projects onto states that belong to exactly the absorption energies of the spectrometer. The final system can only take on those particular states because otherwise it simply can't shed its energy.
It is unfortunate that we don't teach physical pictures like this when we teach the Born rule. It seems like a completely abstract and arbitrary thing, but in reality it is the mathematical expression of the fact that the absorption spectrum of the measurement system matters a great deal for the final state of the coupled system.
You can generalize this to the usual location/momentum/spin measurements once you realize that "a quantum" is a combination of energy, momentum, angular momentum and charges. Historically one can find these energy/momenta as indices into Heisenberg's matrices in his matrix mechanics papers. In that representation it's easier to see that what we are doing is to consider "transition probabilities" from the quantum system into the measurement's systems absorbing energy/momenta channels. In the Schroedinger equation representation the locations/momenta become test functions for linear operators and that obfuscates the physical meaning of the math quite a bit. I believe von Neumann explains this (in a somewhat hard to understand way) in his 1932 book on quantum mechanics that marks the beginning of the modern "shut up and calculate" era. It's all perfectly legit... you are just not being told what the physics behind the symbols really is in a typical undergrad course on QM.
The "randomness", by the way, is not randomness. Random numbers do not conserve energy, momentum, angular momentum and charges, i.e. quantum mechanics can not be random for trivial reasons. What happens, instead, is that the measured result is entangled with some "far away" system (the source of the energy) that in general resides in some spacelike separated region of spacetime. Nature has therefor made it impossible for us to know the state of that system, hence the local measurement looks random. It isn't, but the lack of "god mode" in a relativistic universe makes an introspection into the total state of the universe impossible.
Fascinating! Thanks for this effort
Congratulations on this beautiful video!
Great explanation, excellent graphics. Even when sending electrons though the double-slit one at a time, the interference pattern will build up - unless the electron is observed to pass through a particular slit, after which it behaves like a classical bullet. As Feynman said "No one understands quantum mechanics"! I'm an architect, but really appreciate the walk through of the equations.
Feynman was joking. Quantum mechanics is probably the most trivial theory in all of physics. We just don't teach it well and that perpetuates all of these myths. It's like the question "Why did the architect put that beam there? It's such a great artistic choice!". In reality he didn't. The structural engineer did to prevent the building from collapsing. :-)
@@schmetterling4477there is nothing trivial about the ontological questions raised by quantum mechanics, and Feynman discussed these topics at great length in person and in his writing. He was not joking and frankly your comments make you sound like someone who is aware of many concepts but has not been able to connect it with practical understanding of the physical world. If you believe you have a trivial explanation of notions like entanglement I invite you to share it here. Please go ahead and derive it from first principles.
@@howiedewitt6223 Quantum mechanics is simply the system theory of systems that go "click-click-click" like a Geiger-Mueller counter. One can derive it in like a dozen pages of slightly above high school level math from Kolmogorov's axioms and if you can overcome your intellectual laziness then you can do a literature search for papers in which it has been done. I just gave you the search terms. These papers are boring like watching paint dry. You are basically solving a partition of unity problem with Pythagoras (sin^2 + cos^2=1) instead of with the usual p+(1-p)=1 formula of ordinary probability theory.
Did Feynman know this? I don't know and I am not sure he did based on what he wrote about QM, but all of this was already known to people like von Neumann in 1932... which is a year before Kolmogorov even published his axioms.
Why is quantum mechanics the way it is? Because of special relativity. That's undergrad intuition. All you have to do is to stare at a spacetime diagram for a few minutes and then it should become clear why relativistic systems can not be deterministic in the classical 19th century sense.
This is one of the biggest things that convinced me that 'imaginary' numbers are not so imaginary. It's just a 90° turn.
Yes, that is what we teach in high school. Not sure why anybody needs a refresher on the internet. If you went to a good public school then you have seen it. It seems to be taught in Algebra II in 11th grade. That's roughly where it belongs.
Wish this had dropped before I flunked my quantum exam. I learned all this but would have been a good refresher since my notes are indecipherable.
I almost had a stroke reading that title until I figured out you’re just a math guy.
Never been so early your channel is so awesome. This is what the education system needs more of.
I am currently a student in physics at Leiden University and have learned the derivation from ground up using the ansatz for said plain wave. It is incredible just how profound the equation is and the solutions. We have not yet learned to solve for the hydrogen atom but that will come in due time in next semester as it requires much more tools.
Was wondering when you would bring those things up in the video and you did eventually do it. Great job. Sharing it with non-physics friends now so they get an idea of how to do it. Not everyone needs just 6 lines to derive it ;)
The equation isn't profound. It's not even physical. It does not conserve energy and momentum, which you will notice as soon as you learn to solve the particle in a box problem. It's a toy quantization procedure that is useful for many applications in which its fundamental flaws are of no importance and for which physically correct equations are waaaay overkill.
@@schmetterling4477what exactly do you think unitary evolution represents 😂😂😂 what could it possibly mean for the Hamiltonian to be time independent as it is in the box problem? Might it have something to do with energy conservation? 😂😂😂😂😂
@@howiedewitt6223 The problem is not the time independence. The problem is that the potential is fixed. There is no back reaction that could preserve momentum. We are simply not modelling the momentum transfer on the box. That can only happen the same way as in classical mechanics by having V(r1-r2) and at least two particles in the system. But now you are getting in trouble with the Born rule, though, because what are you measuring? The particle in the box or the box or both? The formalism is simply not self-consistent. You can squint and ignore and then the world of the Schroedinger equation is wrong but useful. If you decide not to squint then you will slide down a slippery slope that won't end before you hit quantum field theory, which, as I said, is waaaaay overkill for most applications.
So, yeah, go and learn how to do it the wrong way because you will need it for all of your applied physics classes like atomic physics, nuclear physics, solid state etc.. Then take a quantum field theory class or two or eight and learn how nature actually does this.
Ohhhh wow taking the complex conjugate to get probability makes sense now!
Excellent video! From a quantum mechanical physicist. I always feel bad about the name given to SQRT(-1) =i "IMAGINARY NUMBER", it confuses its significance often.
I remember my professor guiding us through the derivation for us. It was very complex (pun intended) to me at the time. Now it's natural to me.
Imaginary numbers are no more real than "real" ones. In particular, any equation involving imaginary numbers can be written down as two equations with only real numbers. In the case of Schrodinger equation, you can have two equations mixing amplitude and phase, both linking to other known equations. As they are equivalent, there is no telling which one is more real than the other, whether it maths the maths simpler or not.
0:50 It's actually -i h, not i; not that it matters qualitatively, but as the equation stands, it is a mashup of both the dimensionless and non-dimensionless versions.
I love this explanation, I'd love if you could do an explanation on Barande's stochastic processes for quantum mechanics. He sounds really convincing but I have no frame of reference to understand his math
Quantum mechanics is not stochastic. Nothing in physics can be stochastic. Stochastic processes do not conserve energy and momentum. Some theorists who are eternally stuck at the level of the free non-relativistic particle and some semi-classical potentials love these approximations because they can be calculate with relative ease, but I don't see any other reason for even considering them. They do not represent actual physics.
Now, it is important to understand the difference between reversible and irreversible processes, but that is almost entirely orthogonal to stochastic model.
Very clear explanation. Thanks
This topic fascinates me and has often made me wonder if using complex numbers could similarly simplify the equations for General Relativity.
Great to see the classic Cohen-Tannoudji et al. text.
There are a ton of videos discussing the surprise that complex numbers seem to be a fundamental part of reality. What is missing, and what I really would like is to see is a video explaining that this does not mean that complex numbers are "more physically real" than real numbers, but just that they are a convenient tool for describing operations on the circle, and the circle seems to be more fundamentally real than intervals. It is like saying "quaternions seem to be fundamental to reality" because they are very convenient to describe 3D rotations. Accepting that C in physics is more about the geometry and topology of the circle than the algebra of a field also helps ease the way to considering other fundamental topologies, such as SU(2) and SU(3), or even Calabi-Yau manifolds etc.
Tl;dr: C is not special. The circle is special, and C gives a coordinate system on the circle.
Yeah, but you still didn't tell us where "the circle" came from. You are halfway there, but you still lack the full understanding. Hint: What is unitarity. THAT is where the unitary groups come from. And why do we have unitarity in quantum mechanics? Your turn. ;-)
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