What is the i really doing in Schrödinger's equation?
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References
De Broglie, L. (1924). Recherches sur la théorie des quanta (Doctoral dissertation, Migration-université en cours d'affectation).
Callender, C. (2023). Quantum mechanics: Keeping it real?. *The British Journal for the Philosophy of Science*, *74*(4), 837-851.
Dyson, F. (2009). Birds and frogs. *Notices of the AMS*, *56*(2), 212-223.
Einstein, A. (2011). Letters on Wave Mechanics: Correspondence with H. A. Lorentz, Max Planck, and Erwin Schrödinger. United States: Philosophical Library/Open Road.
Fleisch, D. A. (2020). A Student's Guide to the Schrödinger Equation. India: Cambridge University Press.
Fleisch, D., Kinnaman, L. (2015). A Student's Guide to Waves. United Kingdom: Cambridge University Press.
Gamow, G. (2012). Thirty Years that Shook Physics: The Story of Quantum Theory. United States: Dover Publications.
Jammer, M. (1989). The Conceptual Development of Quantum Mechanics. United Kingdom: Tomash Publishers.
Karam, R. (2020). Schrödinger's original struggles with a complex wave function. *American Journal of Physics*, *88*(6), 433-438.
Moore, W. (2015). Schrödinger: Life and Thought. United Kingdom: Cambridge University Press.
Schrödinger: Centenary Celebration of a Polymath. (1987). United Kingdom: Cambridge University Press.
Schrödinger, E. (2003). Collected Papers on Wave Mechanics: Together with His Four Lectures on Wave Mechanics. United States: AMS Chelsea Pub..
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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.
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.
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 TH-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).
Good derivation walkthrough
In electrical engeneering, we use the same tricks, replacing trigonometric functions that represents alternate voltages and currents, with imaginary exponemtials. Ita just a way of avoiding trigonometric algebra. The i on the fornulas means that some quantities are 90 degree "displaced" in time (related to a implicit frequency oscilarion assumed). Its just "syntaxe sugar" as computer guys says. The conpile code (reality, experiments) is the same (real values oscilating in time) that needs 2 real numbers to describe (amplitude and fase), that cab be represented using a trigonometric A.cos(wt+phi) or using imaginary exponentials A.exp(i.phi) in a certain frequency w=2.pi.f (implicity assumed). People use to misunderstain for exemple imaginary electrical 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?
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.
Your videos are always a delight!. So glad you are back on TH-cam.
bro your videos are awesome, just discovered your channel, binge-watch all your content now! Please continue your excellent work
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.
there's a mistake at 5:40 onwards. E-V term is "operated" onto psi, which is missing from here on.
It re-appears at 5:57 on the second line of handwritten paper.
Awesome video, I always wanted to understand this particularity
Honorary 14th part of Imaginary Numbers Are Real
Fifteenth because of the Euler’s formula video which arguably should also be an honorary part.
I really want your book.
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
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.
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.
I am the ninth letter of the alphabet. (And you are the twenty-first.)
Bravo on the visualizations! It was a little fast for me but I can always play it over again. I remember calculating the shroedinger orbitals of the hydrogen atom towards the end of my bachelor of physics at the UofMN and just being blown away!!
i cant expres how much i love this chanel it teaches me so much
Can confirm. ^.^
Thanks! I watched your imaginary number series a few years ago!
Very well done. This is the best description of these equations and phenomena I have seen.
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.
So great, can`t wait to recive your book here in Vienna next year.. Thanks for your excellent work.
Very beautifully and clearly done. Thank you.
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.
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.
love your work man, one day I will buy your book for sure!!!!
19:52
What does peaking of the function while going through the slits represent psychically? High probability of being inside one of the slits? Because it’s constrained by the walls?
And where would that peak go in 3d, in a 4th spatial dimension?
1. Conversation of probability
2. Wherever it remains a valid solution of schrodingers equation, not a meaningful question without a specific system in mind
Incredible work, this was super fun to watch
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).
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?
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.
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.
@ exactly. I think the whole concept morphed into something that fundamentally has nothing to do with sqrt-1 but just uses the same rules.
This whole video is superb. Will order the book soon.
Outstanding video! We need to see a double slit experiment which detects those refelections.
This is the most perfect schroedinger equation video till now, I would say!
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.
My god, this finally nails quantum mechanics "weirdness" to me 😂 thank you for the fantastic video!
6:00 For German mathematician/physicist Hermann Weyl pronounced "VAH-ell", somewhat like the English word "vial." For French mathematician Andre Weil, "VAY-yuh."
6:40 Chemists creat pictures of orbitals using the real component of the wave equation; physicists use the imaginary component.
Thank you so much for making this
Ohhhh wow taking the complex conjugate to get probability makes sense now!
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.
오늘도 열심히 공부 하겠습니다. 감사합니다.
Great to see the classic Cohen-Tannoudji et al. text.
Nice explanation, thank you.
Wow! Such deep research man, respect!
Dirac would go on to factorize E^2 = (pc)^2 + (mc^2)^2, but using incredible tools like matrices in Dirac matrices. Again the imaginary unit i appears. But matrices appearing implies we're delving into a realm that not even normal (complex) numbers will work. We had to give up stuff like commutativity.
Thanks for your great work!!!!
Very well done!
".... the value of this matter wave........" What do you mean by this? The energy level? in which case, does the energy level fluctuate?
In the animation of the double slit experiment, what does the reflected wave pattern on the side of the barrier that the particles originate from indicate? Does that mean that after the particles reach the slits, there is some probability of finding the particle on the side they came from?
The explicability of nature through application of assemblages of otherwise simple mathematical and trigonometric functions still blows my mind. Why nature would allow us to find its secrets via systematic mathematical formulae seems to point more to the existence of God rather than not.
18:55 secret to double slit exp shown: it's _always_ a wave. So many yt video say "if you close one slit it becomes a particle"...and that just leads to confusion and non-existent quantum woo. It. is. always. a. wave.
It was a very good explanation video.
This topic fascinates me and has often made me wonder if using complex numbers could similarly simplify the equations for General Relativity.
10:00 Why not use a cosine?, I mean, I love the idea of using i, but why shouldnt just go for the ""simple"" way?
I've been wanting an answer to this for so long, and this video still didn't answer it for me. This focuses on the history and how we got to the current formulation. It doesn't focus on the more specific question I'm dealing with. In a more modern formulation that deals with state vectors in some hilbert space, i can easily be seen as just some antihermitian unitary operator* J that commutes with everything we care about (i.e. a complex structure).
The existence of such an operator comes mostly for free since the generators of translations (i.e. derivatives) must have some polar part, which is guaranteed to be antisymmetric and orthogonal, it perhaps it can be shown that the J that arises for each spacetime translation are all "the same-ish" (since any such J would have to commute with all of them), but it's a different story for operators that don't come from the poincare group. What's up with J and why would it commute with _everything else?_ (except maybe the antiunitary stuff, which it's supposed to anticommute with)
*or, if you prefer, antisymmetric, orthogonal operator. The inner product of this space is just the real part of what would be the complex inner product, and the complex inner product can just be expressed as (u,v) + i(u, Jv) (or with the J on the u, depending on complex inner product convention).
This video was amazing, thank you :)
5:45 small error between last and second last line on the right, you are missing phi after the brackets (not to be pendantic, doesnt change much, still a great video)
The harmonic oscilator method was due to Dirac though.
6:19 - the footnote says 'photos' but I think that is meant to be 'photons'? And there's an extra 's' at the end of the second line.
Amazing videos thank you!! Please which software do you use for animations please? I'm currently working on a math project for uni
Used Manim for the first time on this one for the 3d animations!
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!
Never been so early your channel is so awesome. This is what the education system needs more of.
suppose your double slit observation when you are at the light of light or greater, will the vector probabilities be equal to 100%?
We took math to so far it became completely unreal.
lol I just asked that myself today because of a video explaning the imaginary number 😂
From the start it’s like a cool movie! 🍿
keep up the good work
Another awesome video
18:25
When it is said electron is a wave, if you Google: "actual imagings of atoms, molecules" you will see real best possible physical detections of electron, and it compells one to deduce, that the particle/waveness of a single electron is partly to do with the electrons body being far from 1d, and thus it's 3d/4d body has blobby wiggleability;
That coupled with the fact that particles are intimately, locally surrounded and kept at certain distance via em field, may both contribute to electrons not appearing at certain locations on the detector, I geuss that is what you are saying, speed, angle, the electron body and their repulsive to one another field, seem to cause exact interactions that disallow them to land on the screen.
Possibly in the sense: billiard balls are the same size, imagine they were also not so solid hard, but a degree maybe the same degree elastic, and their degree or function of elasticity maybe reflected in variables of their emittance.
They are considered repulsive perhaps, because their bodies are elastic and do not generate local em field that compatibly maintains their bounciness in proximity;
Whereas the proton, and quark theory, is its own attractive repulsive system, that does make an external impression in its local em field, that does trap the electron in confinement.
The wonder of pong like light clocks comes to mind, in thinking of how the electron is bound to the proton at rest say, and then if you were to suddenly accelerate the system, the protons and local fields grasp on the electron is so, such that the electron does not drag or glitch or slip out, but stays sturdily remained locked in place by the power of 3-quark-activity-caused-em-field-space
@14:09 doesn't the 'i' elegantly embed orthogonality into 3D, well strictly 4d given time as well.
If so, then it shows the elegant beauty between mathematics and reality as observed and measured.
The 'i' as we now know is essential yes?
great job boss
btw, there's a mistake in the schrodinger equation that is displayed in the beginning. the coefficient of the second derivative of the wavefunction wrt x on the RHS of the equation should be the square of hbar divided by 2m.
No, both sides are divided by hbar, usually one wouldn’t divide it out but in this case it’s the SE for a free particle so it can be cleanly divided out. He definitely should not have though because it was going to inevitably lead to people incorrectly calling it a mistake.
Arosa!! That's interesting.
@11:30 : you have a note in the lower right corner (and at other places in the video). This is not helpful, esp. when they are there for a very short period as when pausing the video, the player "time bar", etc. covers over the text. Pls find a better way to do this. There is one of these texts earlier in the video that is up for about 1 or 2 seconds - not long enough to read - and if one pauses the video, it's then obscured by the player time bar.
Otherwise you make great videos! Love them all. Would be cool to see a collab with 3b1b.
I'm 15 and barely understand this stuff but from what I can understand this equation is so cool🔥🔥
Existing physics of the time absolutely would and did work for quantum mechanics; it is just wave mechanics + Hamiltonian mechanics
The real and imaginary part of wave function resembles like electric and magnetic field of electromagnetic disturbance...
Why in animation the spacing between 2 slit is less than the spatial spread of that wave function you showed as electron...Did people made slit spacing smaller than electron's spatial spread...
"The real and imaginary part of wave function resembles like electric and magnetic field of electromagnetic disturbance..."
Why do you think so? I don't see that.
really great and high quality vid^^...
❤ Beyond awesome, crazy well done ❤
Nifty AF!
Is the difference between electricity and magnesium relativistic?
Great video!
The heat equation is also a diffusion equation or an equation of Brownian motion taken as a Wiener process. Adding the *i* converts it to a pure wave equation, and nothing else.
Just looking at the equation in a mood of schoolboy whimsy, one might imagine that something like a tachyonic Wiener process is being described, but no, the equation is obstinately just a wave equation.
But a tachyonic Wiener process can be present in quantum mechanics nevertheless. There’s more than one way to travel faster than light. The Schroedinger equation describes wavelike behaviour in one of the ways. A Wiener process in the other way is a degree of freedom for an entity which is orthogonal to the Schroedinger equation. This is the origin of the well-known randomness of quantum mechanics.
when will you be able to do international shipping?
any chance you could talk about the dirac equation at some point? or is that too complicated to explain in a video like this?
Straordinario!
I find it very helpful to be able to visualize the concepts presented here, only made possible relatively recently by the wonderful computer graphics. I find myself wondering whether Schrödinger (and basically all others pre-computer-graphics era) had these visualizations in their heads when developing these concepts?
I believe he just try to combine all the fancy findings and formulas at that time in a logic way , that's algebraic way.
@@michaelzap8528 Hmmm... interesting... Thanks.
What is the music at the beginning of the video? Is it your own? :O
you are genius, man!, you are genius!
19:56 how is this different from a normal water wave? can’t you use shroedinger to represent a water surface wave?
No, water surface waves are described by a different differential equation - the "wave equation".
Every time you say matter wave I can't help but think of matter babies
I see its the same music at the begining of the video as last one, so i ask the same question 😅 what is the song ?
Great video! But where does the rightmost psi go at 5:38?
He simply forgot it.:D
The World as Fields - Dimensions and their Intersections as a Manifold
____________________________________________________________
When I first heard of manifolds I believed that they were like a cross section but in maths. Why I thought this I am not sure but I am worried that the literature does not support this in a way that would be conducive to treating math as somewhat of a field of science. I say this because I want it to be known that a cube to a 4d hyper-sphere is the same as a square to a cube or a line to a square. I choose these shapes because in a way they have a point and a counterpoint. A positive and a negative. A point is not a line but a positive and negative one are a line. While three points do not make two of the previous... a line. Though four points does, those are two different shapes. A triangle and a square. It is the completeness. Not just in the soundness of a theorem like Godel says but it is(going a bit interpretive here) incompleteness. Each one requires a pair and the first only exists because it does. Almost metaphysical. This is important to understand when I say that the world is fields because I don't know how else to define our existence other than things that interact with other things. I emphasize because you need to know that it is a key part of what I am saying.
Enough beating around the bush... what is a manifold? Well... they come in pairs of dimensions and interact with other pairs of dimensions in a manner that still allows individual expression and does not mean that they do not intersect. What I believe is that each dimension matters. That there are 14, seven pairs, & not need for further interpretation beyond the metaphysical.
~Jill P. Johnson PhD
5:43 in the last equation with the Laplacian you dropped out the wave equation function Psi that was being multiplied to the (E-V) term. Why'd you do that?
He simply forgot it.
@@bjornfeuerbacher5514 Yea after watching the rest of the video I realized he just accidentally dropped a term here or there but corrected later.
Could you list the book's that you showed at 7:00?
This is the first time in my life I've heard "de Broglie" pronounced correctly.
I disagree with dyson here. Nature does not work with complex numbers. Complex numbers happen to be how we can represent rotation operations on a 2d plane. Real numbers and all the mathematical operations on pure real numbers cannot express rotations. Nature is fundamentally rotatory on these phenomena. We simply don't have the mathematical tools to express this information in the real domain, hence the need to use the complex domain.
This honestly looks a lot like Maxwell