What is the i really doing in Schrödinger's equation?

[@WelchLabsVideo]

Thanks to PIA for helping make this video possible! Go to piavpn.com/WelchLabs to get 83% off Private Internet Access with 4 months free!

[@Player_is_I]

The way you combined the wave phase perspective to the interference was very well done. Great video ❤

[@pkul9583]

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 = !!

[@youerny]

Would it be possible to have a part II of the video to show how Schrödinger eq was dual to matrix mechanics by Heisenberg? It would complete the analysis of that shared Nobel

[@RockBrentwood]

I have a simpler explanation and I can (mostly) get rid of the "i" in a *much* easier way. Check it out: 1. -iħ∂Ψ/∂u = mΨ (Eigenvalue equation for mass-phase), 2. ∂²Ψ/∂x² + 2 ∂²Ψ/∂t∂u = 0 (2+1 dimensional wave equation), 3. ψ(x,t) = Ψ(x,t,u=0): ψ is Ψ on the u=0 plane. There you go. You're welcome. Like Qui-Gonn (almost) said: there's always a bigger dimension; and the extra coordinate, u, is conjugate to the mass, m. Oh, and lest I forget: Ψ(x,t,u) = ψ(x,t) exp(imu/ħ).

[@yahyasheikhnejad]

Since the imaginary part is not going to participate to the final solution, why cos(omega.t) was substituted by exp(i.omega.t)? what was the reason?

[@3blue1brown]

Nice animations 😉

[@oishiine6781]

The man himself. Manim is amazing and I'm glad to see it being adopted by other math and science creators!

[@dr.shrimppuertorico2749]

@3blue1brown Nice animations

[@pnaarav.]

You and other science guys are really crazy about knowledge of science and other fields.❤

[@WelchLabsVideo]

lol thanks 😁

[@clusteringmiu]

@@dr.shrimppuertorico2749 Grant: you bet

[@kirksneckchop7873]

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!

[@schmetterling4477]

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.

[@kellymoses8566]

@@schmetterling4477 "It turns out that it was the wrong equation and he never understood where it came from." What?

[@schmetterling4477]

@@kellymoses8566 If you read Schroedinger then it is highly doubtful that he understood that quantum mechanics is a relativistic ensemble theory.

[@MrTit-ey8cx]

loser nerd

[@AnshumanBaruah1609]

Exactly my feelings!

[@agargamer6759]

The animation and explanation of interference being caused by the phase of the wave function was so well done!

[@scroipt]

I especially like the interference in the reflected wave off the single slit.

[@danlock1]

@@scroipt I'm happy that your reply to the reply came through without any interference. That didn't open my eyes (they were open), but it probably helped enhance certain neuronal connections in my brain.

[@Paul-ty1bv]

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.

[@danlock1]

If you get a book with included animation, it might come with audio for the the text. Don't let any mispronunciations of names or other terminology throw you off. 😜 If it does, use the text for visual reference of those words and names from other languages or sciences. Come to think of it, the mathematics will require visual representations (or a way to see them) anyway unless the vocabulary is predefined and your memory is superb (unless you're transcribing them as they are related).

[@civwar64bob77]

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.

[@schmetterling4477]

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.

[@hyperduality2838]

The Schrodinger representation is dual to the Heisenberg representation -- quantum mechanics is dual. Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Real is dual to imaginary -- complex numbers are dual or photons are dual -- pure energy is dual. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Syntax is dual to semantics -- languages, communication or information. If mathematics is a language then it is dual. Electro is dual to magnetic -- Maxwell's equations. Alive is dual to not alive -- Schrodinger's cat. Structure (syntax) is dual to function (semantics) -- protein folding in biology. Protein shape or structure determines its function or purpose -- protein folding is dual. "Always two there are" -- Yoda. Waves are dual to particles -- quantum duality. Certainty (predictability, syntropy) is dual to uncertainty (unpredictability, entropy) -- the Heisenberg certainty/uncertainty principle. Syntropy (knowledge, prediction) is dual to increasing entropy (lack of knowledge) -- the 4th law of thermodynamics! Lacking is dual to non lacking. Synthetic a priori knowledge -- Immanuel Kant or knowledge is dual. Two quantum states differing by a global phase are said to be equivalent or dual -- indistinguishable.

[@cpt.yoinks5981]

@@hyperduality2838☯️

[@kapilchhabria1727]

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.

[@T3sl4]

Still my favorite homework problem of all time, found in this textbook: model the Earth-Sun system as a hydrogenic atom.

[@howiedewitt6223]

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.

[@hyperduality2838]

The Schrodinger representation is dual to the Heisenberg representation -- quantum mechanics is dual. Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Real is dual to imaginary -- complex numbers are dual or photons are dual -- pure energy is dual. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Syntax is dual to semantics -- languages, communication or information. If mathematics is a language then it is dual. Electro is dual to magnetic -- Maxwell's equations. Alive is dual to not alive -- Schrodinger's cat. Structure (syntax) is dual to function (semantics) -- protein folding in biology. Protein shape or structure determines its function or purpose -- protein folding is dual. "Always two there are" -- Yoda. Waves are dual to particles -- quantum duality. Certainty (predictability, syntropy) is dual to uncertainty (unpredictability, entropy) -- the Heisenberg certainty/uncertainty principle. Syntropy (knowledge, prediction) is dual to increasing entropy (lack of knowledge) -- the 4th law of thermodynamics! Lacking is dual to non lacking. Synthetic a priori knowledge -- Immanuel Kant or knowledge is dual. Two quantum states differing by a global phase are said to be equivalent or dual -- indistinguishable.

[@kapilchhabria1727]

@@hyperduality2838 time to switch to a different kind of gummy bro.

[@ricomajestic]

@@howiedewitt6223 So what QM textbook do you recommend instead that explains things in a clearer way?

[@Rubrickety]

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.

[@elijahngolofwana]

I've watched tons of videos on Schrodinger's equation.... You are first one I've seen explain it so well. Love your work. ~EN

[@rv706]

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.

[@hyperduality2838]

The Schrodinger representation is dual to the Heisenberg representation -- quantum mechanics is dual. Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Real is dual to imaginary -- complex numbers are dual or photons are dual -- pure energy is dual. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Syntax is dual to semantics -- languages, communication or information. If mathematics is a language then it is dual. Electro is dual to magnetic -- Maxwell's equations. Alive is dual to not alive -- Schrodinger's cat. Structure (syntax) is dual to function (semantics) -- protein folding in biology. Protein shape or structure determines its function or purpose -- protein folding is dual. "Always two there are" -- Yoda. Waves are dual to particles -- quantum duality. Certainty (predictability, syntropy) is dual to uncertainty (unpredictability, entropy) -- the Heisenberg certainty/uncertainty principle. Syntropy (knowledge, prediction) is dual to increasing entropy (lack of knowledge) -- the 4th law of thermodynamics! Lacking is dual to non lacking. Synthetic a priori knowledge -- Immanuel Kant or knowledge is dual.

[@piergiorgio919]

​@@hyperduality2838 anytime a scientist talks theres the philosopher with his bullshit

[@hyperduality2838]

@@piergiorgio919 Space is dual to time -- Einstein. Energy is dual to mass -- Einstein. Dark energy is dual to dark matter. AdS is dual to CFT. Sense is dual to nonsense (bullshit). You are using duality to claim that duality does not exist -- your comment is asinine. The master is dual to apprentice -- the rule of two, Darth Bane, Sith lord or the Hegelian dialectic. "Philosophy is dead" -- Stephen Hawking. Stephen Hawking accepted the metaphysics of Schrodinger's cat -- philosophy. Fear is dual to anger, anger is dual to hate, hate is dual to suffering -- the Yoda dualities.

[@solconcordia4315]

The 'i' should go with the t because the canonical quantization for the three space-related p_sub_x, p_sub_y, p_sub_z, and the one time-related E operators can put them onto equal footing to achieve symmetry. In geometric algebra, i = e1e4 and -i = e4e1 are bivectors so time has an intrinsic direction because 'i' represents an *ORIENTED* area. The quaternions i, j, k also include j = e2e4, -j = e4e2, and k = e3e4, -k = e4e3. Time is *FLUX* , an oriented area orthogonal to any spatial direction. For example, 3 seconds in the e1 direction is 3-seconds e1 i = 3-seconds e1 e1e4 = 3-seconds (e1 e1) e4 = 3-seconds 1 e4 = 3-seconds e4. Similarly for 1 second in e2 direction, it gives 1-second e4. e4 is the unit vector in the time dimension. So if a particle has traveled 3 seconds in e1 direction and 1 second in e2 direction, it has traveled a total of 3-seconds e4 + 1-second e4 = 4-seconds e4.

[@solconcordia4315]

Note that if a particle has traveled 4-seconds e4 from spatial location A to spatial location B, there is an *INHERENT* ambiguity of which path the particle has traveled by. So we say that the particle has a probability of having traveled by a particular path depending upon the number of the path's microstates.

[@sonofasalesman]

I wish I had this visualization so many years ago when I was in physics...

[@potatoonastick2239]

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.

[@stopthephilosophicalzombie9017]

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.

[@IndranilBiswas_]

EXACTLY

[@ricomajestic]

@@stopthephilosophicalzombie9017 Haha! In QM, you can solve problems without truly understanding what is going on at a deeper level.

[@stopthephilosophicalzombie9017]

@@ricomajestic Absolutely.

[@gyurbanvikrenc8267]

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.

[@ayushagrawal8198]

Perfect timing! I have a Basic Quantum Mechanics Course test coming in a few days, this video gave me so much insight of the topic. We derived the particle in a box equation, loved to see it action. Thank you for creating such a beautiful video, it really tied everything together so neatly.

[@nerophon]

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!

[@yocats9974]

_Unfortunately, the name stuck_

[@lyapunovnchill3028]

Gauss hated the word "imaginary", and suggested they should be called lateral numbers.

[@OmDeLaTara]

negative numbers are also imaginary

[@QuantumBell-xp1qg]

Real Numbers are not Real. Nature uses Complex Numbers.

[@christianadler1297]

@@OmDeLaTara Not according to my bank. ;))

[@Adowrath]

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.

[@alexradac589]

Good derivation walkthrough

[@hyperduality2838]

Yin is dual to yang. The Schrodinger representation is dual to the Heisenberg representation -- quantum mechanics is dual. Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Real is dual to imaginary -- complex numbers are dual or photons are dual -- pure energy is dual. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Syntax is dual to semantics -- languages, communication or information. If mathematics is a language then it is dual. Electro is dual to magnetic -- Maxwell's equations. Alive is dual to not alive -- Schrodinger's cat. Structure (syntax) is dual to function (semantics) -- protein folding in biology. Protein shape or structure determines its function or purpose -- protein folding is dual. "Always two there are" -- Yoda. Waves are dual to particles -- quantum duality. Certainty (predictability, syntropy) is dual to uncertainty (unpredictability, entropy) -- the Heisenberg certainty/uncertainty principle. Syntropy (knowledge, prediction) is dual to increasing entropy (lack of knowledge) -- the 4th law of thermodynamics! Lacking is dual to non lacking. Synthetic a priori knowledge -- Immanuel Kant or knowledge is dual. Two quantum states differing by a global phase are said to be equivalent or dual -- indistinguishable.

[@sandmaster4444]

If you are wondering what is too fast, it's the flashed footnotes that are impossible to pause on!!!!

[@declandougan7243]

Sounds like a you problem.

[@twopie6911]

skill issue

[@gwenaellepage4450]

Why is De Broglie an "obscure" Frenchman ?

[@jamesmnguyen]

If you're on the desktop version of KZbin, you can press the period and comma buttons to move forward or backward one frame.

[@jfverboom7973]

You could set the settng of the video display speed from 1 (normal) to 0.25 (lowest).

[@hu5116]

Marvelous! This is the most concise actual derivation of the Schoedinger equation I’ve ever seen. I had a modern physics professor who was a really smart guy that took us through a rather convoluted “derivation” of the equation for almost half a semester. You did it in 10 minutes, and it’s actually understandable! This is one of the reasons I think it’s so important to look at the history of theories, because when you do you find out that although brilliant, our heroes also struggled with coming up with this stuff, and they really thought it through usually much more simply and with straight forward analogies than how we tend to complicate it in the 21st century. Bravo!

[@schmetterling4477]

It's just too bad that the equation isn't correct, so it can not be derived from anything. ;-)

[@ricomajestic]

@@schmetterling4477 A lot of equations aren't correct. Newton's laws are incorrect and so is Bohr's theory of the atom but that doesn't mean that we should discard them or forget about all the reasoning that was done in order to derive them. The essence of understanding and thinking lies in trying to follow what these scientists of yester year were trying to do. Schrodinger had to combine ideas from Wave Physics, Mechanics, E&M and new experimental observations to try to come up with something. It was quite the feat and it made a lot of accurate predictions.

[@JuanDavidGamboaSaenz]

Wholesome comment

[@gooblepls3985]

@@schmetterling4477 sure bud, please provide your 'corrected' formula or criticisms of Schrödinger's equation then

[@schmetterling4477]

@@gooblepls3985 You can find the real theory in hundreds of textbooks on relativistic quantum field theory, kid. Happy reading. ;-)

[@jacobwilcox289]

This is an extremely well stitched story to explain the origins of the Schrödinger equation, and it provides great physical intuition. Well done!

[@yesvee7377]

bro your videos are awesome, just discovered your channel, binge-watch all your content now! Please continue your excellent work

[@AshishVaidya3]

Your videos are always a delight!. So glad you are back on KZbin.

[@KiraandtheSurvivalPod]

What the heck, your production quality - video and book are top notch! Who the heck are you? what's your background?

[@spartaleonidas540]

Math prof iirc

[@tka3]

this guy is basically Veritasium but WAYYY more in-depth

[@claudiocosta4535]

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

[@Achrononmaster]

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).

[@pyropulseIXXI]

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

[@CH3LS3A]

I immediately thought of wave polarization when I saw the spatial curvature of the Schroedinger equation.

[@CH3LS3A]

@@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.

[@claudiocosta4535]

@@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?

[@dila2990]

i cant expres how much i love this chanel it teaches me so much

[@TheMemesofDestruction]

Can confirm. ^.^

[@rickcarroll]

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

[@coooseee]

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

[@schmetterling4477]

There is nothing to derive. Schroedinger simply guessed it and it's not even the correct equation. It works for a small number of cases that happen to be important in applications, beyond that it is fundamentally false.

[@jamesmcadory1322]

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!

[@hyperduality2838]

The Schrodinger representation is dual to the Heisenberg representation -- quantum mechanics is dual. Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Real is dual to imaginary -- complex numbers are dual or photons are dual -- pure energy is dual. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Syntax is dual to semantics -- languages, communication or information. If mathematics is a language then it is dual. Electro is dual to magnetic -- Maxwell's equations. Alive is dual to not alive -- Schrodinger's cat. Structure (syntax) is dual to function (semantics) -- protein folding in biology. Protein shape or structure determines its function or purpose -- protein folding is dual. "Always two there are" -- Yoda. Waves are dual to particles -- quantum duality. Certainty (predictability, syntropy) is dual to uncertainty (unpredictability, entropy) -- the Heisenberg certainty/uncertainty principle. Syntropy (knowledge, prediction) is dual to increasing entropy (lack of knowledge) -- the 4th law of thermodynamics! Lacking is dual to non lacking. Synthetic a priori knowledge -- Immanuel Kant or knowledge is dual. Two quantum states differing by a global phase are said to be equivalent or dual -- indistinguishable.

[@abdesamedbenbedra5665]

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.

[@Govstuff137]

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.

[@swyveu]

I really appreciate the build up of logical steps. Helps to truly grasp the underlying logic.

[@kitarakirja]

This was just really amazing. Truly great work! Thank you!

[@ablobofgarbage]

Honorary 14th part of Imaginary Numbers Are Real

[@Autumn_username]

Fifteenth because of the Euler’s formula video which arguably should also be an honorary part.

[@Cpt_John_Price]

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.

[@Robert-p7b]

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.

[@das_it_mane]

Quite possibly one of the best videos I've ever seen on this topic

[@altube73]

Thanks. Always wanted a simple and intuitive explanation. Finally, the one I always wanted.

[@AlanTheBeast100]

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).

[@hyperduality2838]

The Schrodinger representation is dual to the Heisenberg representation -- quantum mechanics is dual. Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Real is dual to imaginary -- complex numbers are dual or photons are dual -- pure energy is dual. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Syntax is dual to semantics -- languages, communication or information. If mathematics is a language then it is dual. Electro is dual to magnetic -- Maxwell's equations. Alive is dual to not alive -- Schrodinger's cat. Structure (syntax) is dual to function (semantics) -- protein folding in biology. Protein shape or structure determines its function or purpose -- protein folding is dual. "Always two there are" -- Yoda. Waves are dual to particles -- quantum duality. Certainty (predictability, syntropy) is dual to uncertainty (unpredictability, entropy) -- the Heisenberg certainty/uncertainty principle. Syntropy (knowledge, prediction) is dual to increasing entropy (lack of knowledge) -- the 4th law of thermodynamics! Lacking is dual to non lacking. Synthetic a priori knowledge -- Immanuel Kant or knowledge is dual. Two quantum states differing by a global phase are said to be equivalent or dual -- indistinguishable.

[@TheBooker66]

Great video! Been watching you since your old "imaginary numbers aren't imaginary" series. Love the new developments.

[@hyperduality2838]

The Schrodinger representation is dual to the Heisenberg representation -- quantum mechanics is dual. Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Real is dual to imaginary -- complex numbers are dual or photons are dual -- pure energy is dual. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Syntax is dual to semantics -- languages, communication or information. If mathematics is a language then it is dual. Electro is dual to magnetic -- Maxwell's equations. Alive is dual to not alive -- Schrodinger's cat. Structure (syntax) is dual to function (semantics) -- protein folding in biology. Protein shape or structure determines its function or purpose -- protein folding is dual. "Always two there are" -- Yoda. Waves are dual to particles -- quantum duality. Certainty (predictability, syntropy) is dual to uncertainty (unpredictability, entropy) -- the Heisenberg certainty/uncertainty principle. Syntropy (knowledge, prediction) is dual to increasing entropy (lack of knowledge) -- the 4th law of thermodynamics! Lacking is dual to non lacking. Synthetic a priori knowledge -- Immanuel Kant or knowledge is dual. Two quantum states differing by a global phase are said to be equivalent or dual -- indistinguishable.

[@danielkerr5583]

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.

[@howiedewitt6223]

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.

[@danielkerr5583]

@@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!

[@WelchLabsVideo]

Wow great points! I did the numerical simulation with QMSolve - they have other interesting simulations as well!

[@hyperduality2838]

The Schrodinger representation is dual to the Heisenberg representation -- quantum mechanics is dual. Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Real is dual to imaginary -- complex numbers are dual or photons are dual -- pure energy is dual. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Syntax is dual to semantics -- languages, communication or information. If mathematics is a language then it is dual. Electro is dual to magnetic -- Maxwell's equations. Alive is dual to not alive -- Schrodinger's cat. Structure (syntax) is dual to function (semantics) -- protein folding in biology. Protein shape or structure determines its function or purpose -- protein folding is dual. "Always two there are" -- Yoda. Waves are dual to particles -- quantum duality. Certainty (predictability, syntropy) is dual to uncertainty (unpredictability, entropy) -- the Heisenberg certainty/uncertainty principle. Syntropy (knowledge, prediction) is dual to increasing entropy (lack of knowledge) -- the 4th law of thermodynamics! Lacking is dual to non lacking. Synthetic a priori knowledge -- Immanuel Kant or knowledge is dual. Two quantum states differing by a global phase are said to be equivalent or dual -- indistinguishable.

[@atlantasailor1]

Incredible animations. Thanks a lot!! Brilliant explanation

[@YandiBanyu]

My god, this finally nails quantum mechanics "weirdness" to me 😂 thank you for the fantastic video!

[@nazim851]

One of the most amazing videos in my life on this topic

[@Radianx001]

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

[@hyperduality2838]

The Schrodinger representation is dual to the Heisenberg representation -- quantum mechanics is dual. Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Real is dual to imaginary -- complex numbers are dual or photons are dual -- pure energy is dual. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Syntax is dual to semantics -- languages, communication or information. If mathematics is a language then it is dual. Electro is dual to magnetic -- Maxwell's equations. Alive is dual to not alive -- Schrodinger's cat. Structure (syntax) is dual to function (semantics) -- protein folding in biology. Protein shape or structure determines its function or purpose -- protein folding is dual. "Always two there are" -- Yoda. Waves are dual to particles -- quantum duality. Certainty (predictability, syntropy) is dual to uncertainty (unpredictability, entropy) -- the Heisenberg certainty/uncertainty principle. Syntropy (knowledge, prediction) is dual to increasing entropy (lack of knowledge) -- the 4th law of thermodynamics! Lacking is dual to non lacking. Synthetic a priori knowledge -- Immanuel Kant or knowledge is dual. Two quantum states differing by a global phase are said to be equivalent or dual -- indistinguishable.

[@joshuatorres3342]

Thank you so much for this! This is by far the best video I have watched on the Schrodinger equation

[@algoboi]

I really want your book.

[@shukla96]

Thank you for the physical interpretation of i and Quantum Mechanics. Really appreciable. 🎉

[@schmetterling4477]

You didn't get the physical interpretation.

[@MarcoAurelioRosaJimenes]

You really should adress the measurement problem and the question of non-locality in the EPR paper. That would be a delight to see.

[@schmetterling4477]

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.

[@ivocanevo]

​@@schmetterling4477 do we have an Everettian here?

[@schmetterling4477]

@@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. ;-)

[@thomasjamesbullock]

This was great. One of the best on this topic. Well done.

[@HaveANceDay]

Awesome video, I always wanted to understand this particularity

[@dabeet757]

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!

[@binbots]

Does the i in these equations have anything to do with sqrt-1? From what I can gather the complex plane started out for sqrt-1 and showed us how to combine these numbers but eventually it just turned into 2 dimensional numbers or alternate numbers which combine in the same way.

[@killianobrien2007]

I find it helps to think of them as rotation, and i²=-1 means i is a rotation you have to do twice to reverse direction, i.e. 90 degrees.

[@binbots]

@ exactly. I think the whole concept morphed into something that fundamentally has nothing to do with sqrt-1 but just uses the same rules.

[@howiedewitt6223]

@@binbots quantum mechanics requires the algebra of complex numbers. You can use matrices with only real values or describe the group action of rotation but the truth is that is all representations of the same thing. Whatever extent to which you believe the theory describes reality (and there is legitimate debate), the theory must include the complex algebra and therefore complex numbers under a certain representation. IF it turned out to be the case that an interpretation such as the quantum-stochastic correspondence was real, then you could actually show that the complex numbers are not required, but would use them anyway because the extra structure makes the problems easier.

[@JohnUrbanic-m3q]

Your intuition is correct - we have eventually figured out that it really isn't about i as much as geometry. All of this is best summarized with Geometric Algebra, or more specifically Space Time Algebra here. David Hestenes has been fantastic at explaining this, and the text by Doran is a pretty good intro if you care. In 50 years this will be the standard language used for QM and QFT. It is a shame all the confusion that will persist in the interim.

[@binbots]

@@JohnUrbanic-m3q yes it seems every lecture or video on this subject doesn’t like to clarify this point. My guess is because it makes it all sound much more mysterious than it really is and therefore it more interesting.

[@BiswajitBhattacharjee-up8vv]

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.

[@willyou2199]

there's a mistake at 5:40 onwards. E-V term is "operated" onto psi, which is missing from here on.

[@JivanPal]

It re-appears at 5:57 on the second line of handwritten paper.

[@Schraiber]

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

[@no-one-in-particular]

"Nature works with complex numbers" - no, it is just as valid to say nature works with a pair of real numbers. Using complex numbers is just a mathematical convenience that allows phases and reduces two coupled equations to one equation.

[@hyperduality2838]

The Schrodinger representation is dual to the Heisenberg representation -- quantum mechanics is dual. Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Real is dual to imaginary -- complex numbers are dual or photons are dual -- pure energy is dual. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Syntax is dual to semantics -- languages, communication or information. If mathematics is a language then it is dual. Electro is dual to magnetic -- Maxwell's equations. Alive is dual to not alive -- Schrodinger's cat. Structure (syntax) is dual to function (semantics) -- protein folding in biology. Protein shape or structure determines its function or purpose -- protein folding is dual. "Always two there are" -- Yoda. Waves are dual to particles -- quantum duality. Certainty (predictability, syntropy) is dual to uncertainty (unpredictability, entropy) -- the Heisenberg certainty/uncertainty principle. Syntropy (knowledge, prediction) is dual to increasing entropy (lack of knowledge) -- the 4th law of thermodynamics! Lacking is dual to non lacking. Synthetic a priori knowledge -- Immanuel Kant or knowledge is dual. Two quantum states differing by a global phase are said to be equivalent or dual -- indistinguishable.

[@0FAS1]

Literally so beautiful and clear that it brings tears to my eyes thinking about how Schrödinger himself would have reacted to having his ideas communicated so clearly and how these perspectives, avaliable to so few then, are avaliable for hundreds of thousands of people for free. Actually mind blowing when you really think about it. The ideas that made the internet possible nestling inside it. Thanks for this!

[@0FAS1]

Also: I realized that you are the one who blew my mind regarding complex numbers in the first place a couple of years ago when I watched the "imaginary numbers are real!" video. Cant even begin to explain the impact it has had on my life and thinking so thank you so much for communicating the otherwise invisible! I cant wait to get my hands on your book!

[@hyperduality2838]

The Schrodinger representation is dual to the Heisenberg representation -- quantum mechanics is dual. Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Real is dual to imaginary -- complex numbers are dual or photons are dual -- pure energy is dual. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Syntax is dual to semantics -- languages, communication or information. If mathematics is a language then it is dual. Electro is dual to magnetic -- Maxwell's equations. Alive is dual to not alive -- Schrodinger's cat. Structure (syntax) is dual to function (semantics) -- protein folding in biology. Protein shape or structure determines its function or purpose -- protein folding is dual. "Always two there are" -- Yoda. Waves are dual to particles -- quantum duality. Certainty (predictability, syntropy) is dual to uncertainty (unpredictability, entropy) -- the Heisenberg certainty/uncertainty principle. Syntropy (knowledge, prediction) is dual to increasing entropy (lack of knowledge) -- the 4th law of thermodynamics! Lacking is dual to non lacking. Synthetic a priori knowledge -- Immanuel Kant or knowledge is dual. Two quantum states differing by a global phase are said to be equivalent or dual -- indistinguishable.

[@schmetterling4477]

Curious that it didn't have any impact on your life when we taught you about complex numbers in school. You were obviously not listening back then. ;-)

[@0FAS1]

@@schmetterling4477 Lol are you my terrible math teacher that told me to simply accept everything as a given the moment I posed a question beyond the curriculum?

[@schmetterling4477]

@@0FAS1 Teachers don't react well to students in 11th grade who are still struggling at the kindergarten level. Neither do I. ;-)

[@NotBroihon]

4:14 plays a G but shows the frequency and harmonics of an A. Literally unwatchable 😡💢

[@davidwright8432]

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.

[@KilgoreTroutAsf]

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.

[@susmitislam1910]

It IS just i. The Hamiltonian equals ih/(2π) * ∂/∂t, and it also equals P²/2m for the free particle. The momentum operator P is the same as -ih/(2π) ∂/∂x, so this tells us that -(h/2π)²/(2m) ∂²ψ/∂x² = i (h/2π) ∂ψ/∂t.

[@nwnanwna-md4cj]

Incredible didactic work, first time I have the feeling of an intuitive grasp of the S. equation Sadly the book is out of stock already. Hope there will be a reprint In the meantime I look forward to read your book on imaginary numbers Thanks

[@MarshmallowRadiation]

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.

[@schmetterling4477]

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.

[@victor_anik]

" Where does that "randomness" come from?" Why you reject many-world or more ortogonal(parallel) time dimenstions? Randomness seems to hint that there are many attempts. Not to mention the direct hint in Feyman integrals over all possible trajectories

[@schmetterling4477]

@@victor_anik There is no randomness in nature. Random numbers don't satisfy any of the conservation laws. A local measurement is necessarily entangled with another system. Special relativity guarantees that that system is outside of our causal spacetime region. It either resides on the future light cone (which we can't reach) or in a spacelike separated region. Either way that physical state is not accessible to the local observer, hence the local state appears "random". The Feynman propagator doesn't propagate a single system. It propagates the probability of the infinite quantum mechanical ensemble. The infinitely many paths in it are merely a reordering of the possible classical paths of each individual system in the ensemble. The "magic" of quantum mechanics emerges simply from the ignorance of the average user of quantum mechanics who doesn't realize that it is an ensemble theory.

[@victor_anik]

@@schmetterling4477 "There is no randomness in nature. Random numbers don't satisfy any of the conservation laws" Well, when you are a point on a circle, the sum of the points is always a circle, so we don’t break the “circle-energy” law :) However, you don’t know which point you are on and it is random for you. Quite abstract, yeah.

[@schmetterling4477]

@@victor_anik What upsets you about that? If you would live in a world without that "randomness", then you wouldn't be alive. It's not much of a trade. ;-)

[@TheEngineeringHub]

This is a beautiful video! Telling a story of the development of a mathematical/physical principle just beats any other story out there. Loved every second of it!

[@pelasgeuspelasgeus4634]

You mean the development of arbitrary invented math. Right?

[@TheEngineeringHub]

@pelasgeuspelasgeus4634 hm I wouldn't call math arbitraryly invented. Math follows very strict rules, which, when obeyed, tend to describe the inner workings of the university. But I agree with you that some derivations do seem arbitrary.

[@pelasgeuspelasgeus4634]

@@TheEngineeringHub Indeed math have rules which are violated by imaginary numbers.

[@TheEngineeringHub]

@pelasgeuspelasgeus4634 Which rule do you have in mind? I work with quantifying loads on floating platforms, which is mostly done in the Laplace domain, so all functions are complex (real+imaginary), but I find the mathematics very consistent and indeed when tested against experimental results we get great matches.

[@pelasgeuspelasgeus4634]

@TheEngineeringHub I'm an aircraft engineer, so I'm not familiar with what you described. I searched it but I don't get where you need complex numbers. Measuring loads on xyz axes seems to not involve what you mentioned. Anyway, complex numbers are based on the arbitrary assumption there's a number i which when squared gives -1. That's not possible in math because all squared numbers should be positive. Also, the symbol i=sqrt(-1) is meaningless since it is uncountable, non measurable.

[@sb-hf7tw]

Mam - give me an example of a sentence that contains "is" after "i", not "am". Student - "i" is really doing something in Schrodinger's equation.

[@columbus8myhw]

I am the ninth letter of the alphabet. (And you are the twenty-first.)

[@sb-hf7tw]

@columbus8myhw nice ..loved it 😅🥰🙂‍↕️

[@snowdrop9810]

i is used before am when using the verb "be"

[@sb-hf7tw]

@@snowdrop9810 innovative...using passive voice 👍👍

[@janlang8605]

Absolutely splendid!

[@sb-hf7tw]

"i"th comment again 😅

[@WelchLabsVideo]

lol

[@PranavKumar-m6j]

perfect explanation!! keep up the work bro!! 🙂😊🙌

[@melwinjohnthomas4177]

Incredible documentary..go ahead...a nice work gifted to humanity

[@tomgraupner171]

What a wonderful video ! Thanks a lot for your work. The result is amazing.

[@plantnt489]

This channel is truly one of the best in youtube

[@JohnSmith-zy1ur]

You smashed it with this video. Superb.

[@cerezabay]

I love this video so much, it's so elegant. This is my third watch. I am a first-year PhD in chemical engineering. Great job!

[@victor_anik]

This is brilliant! It's all about rotations and spirals. This should be emphasized in every textbook on quantum mechanics.

[@schmetterling4477]

It is. That is what the word "unitary" means. ;-)

[@jupa7166]

Beautifull. You have gone well beyond just earning a new subscriber. Watching this was a pleasure.

[@brucerosner3547]

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.

[@schmetterling4477]

That's because it came from Schroedinger's mind. There is no rational derivation for an equation that is fundamentally unphysical and incorrect.

[@vinzent1992]

Your videos are always such a pleasure to watch :), perfect balance in the level of detail and very aesthetically pleasing, well done!

[@Celesteciel.__.6]

There so many books you read for the video😮i appreciate that bro,thank you so much

[@cdkw2]

love your work man, one day I will buy your book for sure!!!!

[@LebesgueIntegrator]

This lecture was a work of art! Thank you for a truly phenomenal presentation. I will likely ["almost certainly"] use it as a reference for my students, who by that point will have no problem understanding the complex PDEs and will love the tactile & historical perspective you have wonderfully provided! For interested students & Physicists who frequently say "Oh, QM really isn't my field..." because of specialization in other branches, I highly recommend "A Student's Guide to the Schrödinger Equation," by Daniel Fleisch. His presentation of the material complements this video quite well, although readers of his excellent book will benefit greatly from your visuals and careful explanations. Great work! Keep it up!

[@WelchLabsVideo]

Thanks! Fleisch's book was one of my main sources!!

[@schmetterling4477]

Does Daniel Fleisch mention relativity and Kolmogorov's axioms? If not, then he is useless for a deeper understanding.

[@infinite3574]

It's beautiful ❤️ both the video and the schrödinger wave equation ❤❤

[@Kaenguruu]

Incredible work, this was super fun to watch

[@ShlokParab]

Watching this in hopes of understanding it later in life, I just want to say right now that the animations are just rignt; not too much, just enough to expertly convey the message.

[@pelasgeuspelasgeus4634]

And how do you know the animations are accurate?

[@rubhern8187]

great visualisation and explanation, on a complex field, next dirac equation

[@michaelsatkevich]

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!

[@RealEverythingComputers]

Great video. Very well explained and simplified. Honestly, you should author a book on the mathematics behind quantum mechanics 😀😀

[@ChetanG-fh3bc]

you got my respect, when you perfectly showed double slits experiment,

[@utiogul]

"what is the i really doing in Schrödinger's equation"? Makes things COMPLEX

[@CoreyCanuck]

My greatest discovery of a KZbin channel ever. Kudos

[@semmering1]

So great, can`t wait to recive your book here in Vienna next year.. Thanks for your excellent work.

[@d.v.faller9251]

Very beautifully and clearly done. Thank you.

[@kbojdfotboajnbob9898]

Great Video, really learned a lot! I think that the exponent in the third equation at 11:16 is missing a minus sign but I might also just be missing something 😅. Keep up the great work :)

[@geoffb1418]

Wonderful animations. Brilliant

[@morfmitMeinung]

thats cool, back in my quantum mechanics lecture semester, we had no time for a derivation of the equation. professor just mapped the operators with the usual method and said thats it. it was totally unsatisfying. now, 15 yrs later i watch a youtube video with a coffee and a chill out moment in the morning. thanks for your work cheers from germany

[@schmetterling4477]

The Schroedinger equation can not be derived from first principles because it is not a physically correct equation. What one can derive from first principles easily is the structure of the Hilbert space and (with a lot more effort) the equations used in the standard model. If you want you can then introduce a few not so trivial approximations and then arrive at the SE from the QFT end, but that's rather mind numbing and not useful for an introductory class. A well taught QM 101 class should, IMHO, sketch out the derivation of unitary dynamics, then introduce the SE as an ad-hoc quantization procedure, go through the usual exercises and maybe end on a discussion of the derivation of the Born rule from first principles. That's a reasonable program for a second year undergrad class.

[@drako3659]

> And his mistress, lol This is why Welch Labs is the GOAT of math explanation. Human interest is the best hook.

[@LukeScherschel-r4h]

wow. just a great breakdown and explanation.

[@amguardia]

A side effect of the i being present in this equation is that it makes it invariant by time reversal (reversing time and taking the complex conjugate), as we expect from a elementary theory. The closely related but purely real diffusion equation famously breaks this symmetry. It is noteworthy that it is the only way one can have time reversal with a 1rst order differential equation.

[@pelasgeuspelasgeus4634]

Time reversal? 😂😂😂 In which marvel universe do you live?

[@AlKordesch]

@@pelasgeuspelasgeus4634 I find it hard to imagine the simulated wave moving backwards in time through the double slits.

[@jochemvanderspek5168]

to me the pace of this video was perfect!

[@ChrisSeltzer]

This helped me understand so much. Thank you for making this video.

[@AeroconX]

Exceptional video. Thank you

[@hussanulmaab872]

Literally beautiful, Thank you !!!

[@jjdavidian]

You are a really gifted person

[@richardcasey4439]

This whole video is superb. Will order the book soon.