This is the first time I have really understood why the wave equation is written as it is! We did the derivation and how to solve it, but I never fully understood it. You are an epic genius at understanding and relaying physical concepts! The same for the heat and fluid equation and explaining the "flow" of probability in the Schrödinger equation. I am so grateful :)
@@ScienceAsylum orbital is created by assuming position of particle is uncertain and velocity thus energy of particle is certain but what would an orbital look like if it is created by assuming postion of particle is certain and velocity thus energy of particle(energy of orbital) is uncertain?
This exactly! First time I’ve been introduced to the idea that there are ‘standard’ heat and wave equation, shown what they look like, and shown how the Schrödinger equation fits into that, instead of just falling from the sky
@@pwinsider007 ..orbitals is a costruct of humans ... it makes it easier to work with just... in reality thers no such things as orbitals..but energy stats...the probability that a higer energy electron is further away from the nucleus is bigger...but its a it all over the place... i see iyt as a rather fundamental misconception thats been passed down in time... the real interesting q is 'Why does waves behave as particles when intersecting' and not 'why does particles behave as waves' (there is no particle...its an illusion) thankfully soundwaves behave very simillar and can be used explain it (it make it easier for us sapiens to comprehend it)...let it sink in b4 u answer...
"By the way it's actually a hundred times more complicated than that" seems to be the motto of Quantum Physics. And the collab with Arvin Ash is awesome! I've checked out a number of his videos too.
The problem is Nick Lucid will have a PhD. And yeah, how can he explain without the fancier stuff? Of course it's complicated, or Nick wouldn't have to study a doctorate to be at maximum of his career
I remember many years ago seeing an article about Josephson junctions. There was a circuit diagram and an arrow that pointed in a direction with a caption "Probability Current". I went, "WHAT?" Then the more I thought about it the cooler it seemed. Now this video completes that for me.
I'm from Austria and I'm also a Schrödinger fan. Before the country transitioned to the Euro, the currency was the "Schilling" (read 'sch' in German words as an 'sh' in English, i.e: shilling). The second last iteration of said currency (in the 80's up until the 90's) had Schrödinger and the wave equation on the 1000 bank note. Ofc I had to get one :P It contains a portrait of Schrödinger, the formula symbol of the wave equation and a stilized atom on the front and the main university of Vienna and another stilized atom on the back. For anyone who's into collecting old currencies, I can highly recommend getting one. It's an absolute beauty of a bank note. As a side note, these cannot be exchanged into Euro at the Austrian national bank anymore, but they become increasingly sought after by collectors. If you keep one for 40 years and have it remain in good condition, it may also serve as a nice investment.
@@Sanntik Aww, some people on YT/Twitter are offended by a dead person :P I am not bound by cancel culture and it's irrelevant to me whatever he did in his private life, since it's unrelated to his scientic achievements.
This is great! I learned recently (from 3Blue1Brown, actually) that the Fourier transform originated in Fourier's contribution to solving the heat equation, not the wave equation; so the role of Fourier transforms in QM now makes a lot of sense. Thank you for this!
Fourier’s contribution to the 20th and 21st century is so underrated. Even the uncertainty principle is derived from Fourier. Not surprisingly, as oscillators (everything wiggles ….Feynman) is cosine and sine dependent. And more amazingly, it is all a vector space that can also be tied to linear algebra. A soup all pointing to energy density probabilities… nature is truly amazing
Man, what a fantastic summary of some of the main relationships in physics. That free body diagram of the string section related to the equation terms was particularly clarifying.
I actually used the analogy between the Schrodinger equations and diffusion equations (which generalize the heat equation) as the key to my PhD thesis, where I applied path integration methods from QM to population genetics problems that are usually described with diffusion equations
First-year physics undergrad here and I was just studying these equations! Love your videos Nick; you're one of the reasons I'm studying Physics today ;)
Thank-you! As someone who encountered Schrödinger etc as an Undergraduate at The University of Bristol fifty--five years or so ago, I'm intrigued by the fact that this presentation weaves in modern cultural forms of the entertainment industry. All most interesting! Again my thanks!
You are brilliant, Nick! Almost every time I watch one of your videos I either learn something new or gain a new perspective. This was really helpful - thanks!
Yes the Schrodinger equation is first order in time, like the heat equation. But the i factor, by rotating the time derivative by 90 degrees, makes the result very different. In fact, if you split psi into real and imaginary parts, you can write two differential equations, then combine them to finally get a single differential equation that is second order in time. That is, if I remember correctly.
It depends what the "i"means though - there is an interpretation where time is a clock which is a cycle so anything that describes these kinds of processes at this fundamental level involves clocks diffusing.
Interesting video. It did always feel weird to me that it wasn’t a wave equation. But I am slightly confused. Wasn’t Schrödinger motivated by De Broglie’s hypothesis? Namely that electrons were standing waves around the nucleus. That all particles, not just photons, could exhibit particle wave duality. Wasn’t this also used to explain the double slit experiment? I understand that formally it resembles the heat equation, not the wave equation - but wasn’t the whole point to find waves? Even if they are “probability waves” for lack of a better term. I mean I just started studying quantum physics in my undergrad, so I really have very little idea what’s going on.
The "i" in the coefficient on the time side of the equation makes things a bit more complicated (which I glossed over). It allows for rotations in the complex plane and rotations can be viewed as waves if you take a cross-section... but that's two _extra_ levels of abstraction.
Richard Feynman, one of the greatest physicists in the last century believed that the key to truly learning and understanding a concept was to be able to explain it coherently to others in laymen terms. . And just on a general level, intelligent people don't just "have opinions" on things without at least trying to understand them inside and out. So it's likely this was a lot of his actual thought process on working through this. Also, deriving an answer through logic will be much more likely to stick in your memory than deriving an answer through memorization. . In other words, this isn't just for us. He is likely sharing his actual thought process on the matter.
Zur Heterogenität dessen, was man nomine so in der Quantenphysik vorfindet: Die Gleichung H Psi (r) = E Psi ( r ) ( das Psi für die Wellenfunktion... ) ist in dieser Form der Schrödinger-Gleichung eine Eigenwertgleichung, wobei ihre Lösungen also Eigenfunktionen zum Eigenwert E ( wer nicht weiß, was das ist, der sich sehe sich die Hauptquantenzahl < n > an... ) sind. Weil die Eigenwertsgleichung meistens nur durch diskrete Eigenwerte in Abhängigkeit der Randbedingungen erfüllt wird, heißen diese quantisiert als Antonym zu kontinuierlich. Ich finde aber das Problem ist, dass, wenn die Leute fragen < quantisiert > was'n das? ...die Antwort < es sind infinitesimale Pakete dessen, was das jeweilige Quant ist, einem zwar Ruhe bringt... ( ...die Leute hauen ab... ...sie hauen aber auch bei🖕ab ), aber keine abstraktionsorientierte Antwort sein kann, weil die Leute autosuggestiv wieder an Teilchen denken und nicht von dieser Anschauung aus dem Mesokosmos wegkommen.. ...Rettung vor dem, was die Sprache der Physik ist, bringt einem eigentlich nur der mathematische Term als Hilfsmittel der Beschreibung aus dem Reservoir, was die Mathematik abstrahiert aufgespannt hat... als letzte Zuflucht für die, die da die Schöpfung so gut wie möglich sehen wollen... ...ich will hier so schnell wie möglich wegkommen!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! Le p'tit Daniel🐕🐕🐕🏒🏒🏒
As someone with a degree in physics, it always makes me smile when I learn something from the Science Asylum. My undergraduate wave mechanics professor never explained what the Schroedinger wave equation represented. It was just, here it is, now let's do some calculations.
Same. Why don't they just say: by the way mathematical abstraction doesn't always translate so we started using probabilities.... Instead they keep it esoteric like it's unbelievably complex and of course it's complex but we already knew that didn't we! It doesn't make Maxwell's equations not true in fact Maxwell's equations account for almost all phenomena except gravity. This is why I decided not to pursue my post-graduate degree in physics seems like a waste to me...
Thanku 1000 time u have no idea how much your vedios help me. When professors explain schrödinger equation i remain only scratching my had. After watching your vedios my efficiency increase 200%
Respect for tackling this. A flow equation instead of a wave equation. Though when you take a time independent solution on a constant potential energy term , you'll get a 'standing (co)sine wave' Which is probably where the confusion originates. As these simpler solutions are always the first ones used in a classroom environment
I think it all comes down to semantics and something you mentioned on your earlier videos. Namely that quantum particles don't sometimes behave as particles and sometimes as waves, they always behave as particle-waves 100% of the time, i.e. they're something different entirely. It's a shame concepts in physics get mislabeled and the label sticks, but it's a learning opportunity and that's why I appreciate these videos.
As Feynman once said in one of his lectures: "They do not behave like particles! They do not behave like waves! They behave in their own inimitable way!"
Excelent video about the wave equation. Would have helped me a lot when I was taking differential equations lectures in undergrad school.
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The best explanation of the wave equation ever seen. This intuitive approach to the meaning of the second derivatives with respect to time (acceleration) and space (curvature) is needed to a full understanding of the equation, but almost never explained. You are the best in clarifying physics equations!!
Thanks! Honestly, I (partly) regret putting it inside a video about Schrodinger's equation. It should have been its own 5 minute video.
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@@ScienceAsylum But you can make another video about waves and repeat it with other things. I suppose that is hard to find a different subject of physics every week. Doesn't matter to make several videos about the same subject. It is usually made with special relativity, but it can be made with everything.
OMG... If I had instructors in my engineering program who had a quantum bit of the talent to explain things so clearly... I would've saved so much head scratching and confusion. Brilliant overview!! Thanks...
If the Schrodinger equation is a diffusion equation, then it has an imaginary diffusion coefficient, which in turn makes it weird to call it a "diffusion equation". I mean, the "i" in the Schrodinger equation makes a big difference... Because of the "i", the solutions must be complex, and the equation has U(1) phase invariance, which leads to conservation of probability in QM. None of that is true for the heat equation with a real diffusion coefficient, because there's no phase invariance and no conserved quantity. As similar as these two equations may seem, the "i" introduces some properties to the Schrodinger equation that are just not present in the diffusion equation.
Don’t be ashamed of plugging your book, it’s a damn good one. I’ve been working through it for fun in my free time which is a lot more than I can say about my copy of Jackson’s e&m book.
Thanks for the clarification! I found the term ‚wave equation’ always very confusing: in High School I thought an electromagnetic wave propagates like a water wave through space.
at 1:50 , *in general* the horizontal component of the tensions does NOT remain the same after applying the downward force. To keep it the same, you need a very very *soft* string (I am assuming linear-elastic spring behavior) and a very shallow dip (so that cos of the angle is very close to 1)
To me the videos you make with humor and/or describing the equations (math) with simple words, instead of leaving them out completely, are your best ones.👍 (I think most people just want to know what the equations come to anyway, without studying mathematics.)
I love the description of the difference between the broad equation types. Personally, I think of the heat equation in terms of diffusion (mass transfer, but conduction in the heat model). It represents movement across a gradient compared to the beautifully explained wave function's restoration from concavity.
The Schrödinger equation (SE) is technically a parabolic equation (PE), also known as the parabolic wave equation. This is an excellent approximation to the full wave equation when the motion is primarily in one direction. The approximation in the SE is that particles are only moving forward in time, in other words: non-relativistic. IMO it is rather silly to claim PEs are not wave equations (in their applicable domain) since they give solutions that are extremely close to those of the full wave equation. For example a PE equation is routinely used to calculate how sound waves propagate thousands of miles in the ocean. The "waviness" in the SE is in time, not necessarily in space. The eigenfunctions typically have a factor of e^-iEt which contains the waviness. It is no surprise that when you ignore these factors, the waviness goes away. Stanley Flatté published an excellent pedagogical article that shows the relationship between the SE and the full wave equation (Klein-Gorden equation): The Schrödinger equation in classical physics American Journal of Physics, Volume 54, Issue 12, pp. 1088-1092 (1986).
This video was excellent. I immediately went to look up the Black-Scholes equation as I had remembered it had been derived from the Brownian heat equations. In that equation (at least for European options) we see a first order term for time and a second order term for motion. This video has revealed a relationship that I had never connected before. It is like a curtain has been lifted. Thank you!
This vid completely ignores the fact we're dealing with complex numbers here, and the way curviness in complex phase influences amplitude makes it very different from ordinary heat equation where all numbers are real, not complex.
This is a great intro to understanding the Schrodinger Equation! And this is also what my research professor taught me on how to view the equation: you can recast the Schrodinger Equation such that it can be explicitly shown that it is indeed a continuity (or a flow) equation in which Probability Density is conserved and a so called "Probability Current Density" carries the flow. Think of it as if there is no current, then the probability won't change; similar to when there is no flow of fluid, there will be no change in the accumulation of fluid. After all, the Schrodinger Equation enshrines the Unitarity Principle. Anyway, my comment I guess is about the notion that Schrodinger Equation is a derived equation. Here, it is said that it was derived from Conservation of Energy. But my Prof always says (I am in his Quantum Foundations theory group btw) that Schrodinger Equation cannot be derived. It is a conclusion of a concoction of several experiments and postulates. It just so happen that the Schrodinger Equation fits snugly in our current view of Physics such that one could say that it could be derived from say the Conservation of Energy. But with the same logic, Schrodinger Equation could also be "derived" from a proper Lagrangian Density and from the Hamilton-Jacobi Equation, among others, which stems from the fact that there is a symmetric transformation that gives rise to a conservative quantity. Hence, you cannot pin-point where it really came from. He always corrects me: "You cannot derive the Schrodinger Equation: it is a result of many experiments trying to describe our weird nature." Still this is a great video and I learned a lot! Thank you!
Pedantic me says it's a diffusion equation of which the heat equation is a special case. Whether it is also a kind of wave equation depends who you ask but diffusion waves are definitely a thing. Fun video from Nick as always.
I've always thought the "wave" aspect of the Schrödinger was contained inside its use of imaginary and complex numbers, as these are pretty useful for describe some "cyclic" behaviors like the rotational and harmonic phenomena 🤔
Only rarely do I learn something that blows up my mind and lets me emerge with a new, completely altered understanding of our world. This video is one those rare moments and is amazingly the second time you've done that for me. Keep up the amazing job educating us about science!
I like during these explanations when mathematical explanations are used and explained. However, what is enjoyable is the split-screen or caption box that shows visually what is described with an arrow or another box encapsulating the specific item of the math equations. This way it does not really matter if it is just an equation that is not yet understood completely, I would think? ...Nice guitar, Alvarez.
Oh, Nick, I once thought of the very same question - "Is this a dissipation equation?" - and asked a physicist about it. The physicist answered that it is not "quite" heat equation because of the imaginary one, the "i"! ("The Imaginary One", now that's something right out from a fantasy book... I mean "imaginary unit", sorry, English isn't my native language). This changes it from dissipation equation to something quite different because every derivative "turns it 90 degrees" in this weird imaginary space. Could you elaborate on this a little? Does it make big enough a difference for it NOT to be a dissipation equation? Or does this only mean that dissipation itself is weird?
This is awesome! The only thing swept under the rug is that Ψ isn't a pdf, but a complex number, and |Ψ|^2 is a pdf. So it's not probability that is flowing but something very closely related. An intuitive explanation of the meaning of Ψ would be incredible
@@ScienceAsylum Are there any other links between quantum mechanics and thermodynamics? They both seem to be describing something statistical - lots of little tiny things, running around and interacting with each other on a scale that's too small to follow the details, so we are only really aware of a summary (averages and distributions) of what happens, not so much the details of individual interactions.
"Quantum Mechanics is the dreams that stuff is made of." That saying makes even more sense after learning that the Schrodinger Equation is about the flow of probability. Probability, as a quantifiable value, is about as non-tangible as you can get.
You know, I actually got a lot from this video. Thank you Nick for clarifying the perspective of the *wave* and *flow* itself. Arvin also has some great videos and I've watched nearly all of those too. Thank you both very much for making these videos! ♥️🙏 I look forward to reading your *Fine Structure* education!! Thank you so much!
I struggled with Science at school finding it extremely difficult to grasp. However 40 odd years later I’m finding your vids really fascinating - I won’t pretend I understand the detail but the way you put things across it certainly makes me grasp the overall concepts so cheers for that 😊 One final thought and it might be a daft question but I will ask it anyway: How do we know a particle can be in more than one place at once until we measure it - as the act of measurement will fix it to a particular place, so thefore we can never observe these multiple states? 🤔💭
I loved it, thinking about Schrodinger's equation as heat dissipation equation was fun! 👌👌👍👍 Well I think the comment about three types of equations in classical physics is correct if we change the "type" with 'category"! Most of physics equations are of order 2 and linear with constant coefficients mostly, when working with second order differential equations, we only have hyperbolic, parabolic and elliptic variations (discarding the degenerate cases).
As an Austrian fellow, like Schrödinger ;-), I have to disagree here. You forgot the imaginary unit "i" in the Schrödinger equation. This is a very important difference to the heat equation and leads to a different behavior. The time-dependent Schrödinger equation can describe (standing) waves rotating in the complex plane, having the same solution as two coupled wave equations - one real and one imaginary. The heat equation cannot describe such behavior. Considering also the imaginary unit, in my opinion the Schrödinger equation is more similiar to the wave equation than to the heat equation.
I love your videos...even if I only get half of what you're talking about 🤣 Yet...at the end of your videos I feel like "I got this". Feel smart for about 20 minutes 😂 Seriously...I love physics and even though I could never be a physicist there's just something about the science that tweaks my brain. Thanks for making a tough (for me) subject enjoyable.
Man...that's really a perfect timing...I have engineering physics exam Tomorrow and I have searched many videos....but your video made me a clear mind of seeing equations 🙂🙂
But it does relate "curviness" (in complex phase) to oscillation! The stronger WF spirals in complex plane, the faster its amplitude changes, the faster it "moves" through space. And for free particles this function does look like a wave with all the wavy properties. Nice visualisation here: th-cam.com/video/LZie2QC5Jbc/w-d-xo.html
Nice description. Identifying the second derivative as the curvature and distinguishing its behavior in one equation from the other is particularly insightful. This intuition beautifully agrees with the solution of the 1-D diffusion equation, where the two curvatures of opposite signs (one at the head and two at the tails) cause the head of the gaussian function to come down with time, while its tails grow in height. When I first understood the diffusion equation during my early days of learning, it gave me a better physical insight into the dynamics of the Schrödinger equation. I feel it's always nicer to view the Schrödinger equation as a form of diffusion equation describing a complex function, which interestingly gets called as the 'wave function'.
Great video. I am fascinated by wave & heat equations for a long time but haven't look at it from this angle yet. They are super easy to understand using grid based simulations. One grid for the wave height (concavity in the video) and one for velocity. All the scary looking derivatives just turn into simple subtraction. Totally worth a look into it.
I haven't thought this through, but an argument could be made that the Schrödinger equation is a wave equation in disguise. There are complex numbers, not real numbers, and the argument of the wave function is indeed "waving" all the time (you only address the dissipation of the square). One can reformulate every second order differential equation as a first order equation in more variables, which might be exactly what happens here using the complex numbers (which consist of two independent numbers). What if you rewrite the Schrödinger equation as a system of two coupled equations in two variables, perhaps its waviness becomes more apparent? If you think of the solution of the Schrödinger equation for the harmonic oscillator, with a wave package moving back and forth, it does not look like dissipation, and dissipating heat cannot do this periodic motion without a changing potential. Lastly, you got the Dirac equation which hides even more of the derivatives by introducing even more components. This is a linear equation. But noone would claim it behaves like the usual linear systems we study for ODEs, because these are pretty boring. The upshot is: what lacks in derivatives can be compensated by using more components, potentially in the form of complex numbers.
Since I can't give two thumbs up for your second order content, I leave this comment in its place. Two thumbs up man! Keep making this amazing content! You are one of my favorite science communicators, and easily one of the most entertaining while also maintaining an incredible fidelity! (Ps: I only said "one of" because I don't want the other kids to get jealous, but between you and me? We know who my favorite actually is, right? Wink wink...)
Very good. I wish my teachers way back then had talked about electron orbitals in terms of probability. It would have helped my understanding a lot. I suspect that they didn't understand enough themselves.
In Geophysics we use what we call “one-way wave equations” that have the same form as Schroedinger’s equation. We use them as approximations of the wave equation that only need one boundary condition instead of two because you only allow the waves to go one way in depth…
This is a bit misleading. To call the schrodinger equation a heat equation is really missing the point. The time derivative of psi comes with a factor of i, so really the schrodinger equation is in fact two coupled differential equations for the real and imaginary parts of psi. What the schrodinger equation really says, is the change of the imaginary part over time is due to the spatial derivatives in the real part and vice versa, where as the heat equation would not have this coupling. The flow in the schrodinger equation describes motion about the complex circle. It also has a conserved current (probability current), which the heat equation does not have (it describes flow to a stationary state). In other words, motion in schrodinger's equation can carry on indefinitely (plane waves for example). The heat equation describes dissipation of energy until a stationary state (with no spatial curvature) is achieved. Another why to put it, Schrodinger has wave solutions, the heat equation does not. Congratulations for making it to the end of this rant.
I agree with this, and wrote a long comment to the extent elsewhere. There is an interesting formal similarity though and if you treat complex solutions to both equations the only sing that differs is one of the coefficients. So there would be mathematical connections one could exploit for sure. But that doesn’t mean the two should be classified as behaving at all similarly as they stand. If you ignore factors of i, there would be no real difference between Laplace’s equation and the wave equation which is kind of one of the motivating distinctions to begin with.
Didn't you kind of gloss over the i on the right-hand side? Since the solution is a complex power of e, multiplying by i is kinda-sorta taking another derivative. So, the right-hand side behaves second-derivativy.. So, that's still kinda wavy?
Once you get complex amplitudes involved, you can have first order or mixed order equations that admit wave solutions. The heat equation is basically a diffusion equation, and while it's interesting to look at Schrödinger's equation in that lens (diffusion and waves both relate to moving things around, after all) it's not really the same thing, because the heat equation is restricted to real amplitudes, cutting out a huge number of solutions. It's fascinating that you need two derivatives in both time and space to get real wave solutions, but basically any non-trivial combination of derivatives in time and space will work when you allow complex amplitudes. I think that shows that the classical wave equation is really just a special case for real valued quantities. The other cases are still wave equations, they just don't look like the classical one because they allow complex wave functions. You can argue that by adding potential functions and boundary conditions you can create "static" solutions, but I don't think that's fair. The classical wave equation can produce standing waves, which are just as static as the hydrogen orbitals you mentioned. Does that mean the classical wave equation is not a wave equation because it allows solutions that don't vary in time? Also, what we call a "wave" can be made precise. If you mean a spatial shape moving along in time (some of the illustrations used here suggest pulses), then any function of the form f(kx - wt) will produce that behavior. Not all of those are physically realizable (i.e. solutions to physical wave equations), but do we want to call a travelling parabola a wave, or do we want to only call solutions to wave equations waves? Is that definition too circular? I dunno.
Wow, great video. I got more insight into Schrodinger from that video in 15 minutes than in some full courses. Professors always bury those kinds of insights or scatter them weeks apart so they no longer fit together. With this new found clarity, I have one question... You said the equation expresses how probably (density) flows over time. That makes sense to me, but how do you connect that to the idea of fixed quantum states in an atom and the discrete quantum jumps when changing energy levels. What is "flowing" in a fixed state? and how does it characterize the state transitions? Aren't they discontinuous? Keep up the great work. I'm going to go track down a copy of your book. I did an engineering physics undergrad in the late 80's. Channels like yours have helped my rekindle my love for the subject any refresh many underused brain cells. I'm sure the book will be as good. Joe
11:33 Fun fact: Schrödinger actually came up with the Klein-Gordon equation before Klein & Gordon; but it gave the wrong answers, which is why he rejected it.
Don't forget to check out Arvin Ash's video on the ubiquitous harmonic oscillator: th-cam.com/video/BZRv8Nko9XQ/w-d-xo.html 🤓
So it's not any coincidence that Arvin's video an yours popped up in my notifications almost simultaneously. I knew it: coincidences do not exist! 🤔
It's okay to be a little self-promotional for time to time, Nic :)
If I made videos and mentioned Arvin then just for that part of the video I'd be wearing an Arvin-type hat.
Thanks for the collab Nick! It was fun. Your video is not only funny and creative, as usual, but also important!
@@clmasse Fair point. It would have been more accurate to say "seemingly ubiquitous."
This is the first time I have really understood why the wave equation is written as it is! We did the derivation and how to solve it, but I never fully understood it. You are an epic genius at understanding and relaying physical concepts! The same for the heat and fluid equation and explaining the "flow" of probability in the Schrödinger equation. I am so grateful :)
Glad it helped! 🤓
@@ScienceAsylum orbital is created by assuming position of particle is uncertain and velocity thus energy of particle is certain but what would an orbital look like if it is created by assuming postion of particle is certain and velocity thus energy of particle(energy of orbital) is uncertain?
Exactly, I always wonder if I'm not listening in classes or the teachers don't teach well.
This exactly! First time I’ve been introduced to the idea that there are ‘standard’ heat and wave equation, shown what they look like, and shown how the Schrödinger equation fits into that, instead of just falling from the sky
@@pwinsider007 ..orbitals is a costruct of humans ... it makes it easier to work with just... in reality thers no such things as orbitals..but energy stats...the probability that a higer energy electron is further away from the nucleus is bigger...but its a it all over the place... i see iyt as a rather fundamental misconception thats been passed down in time... the real interesting q is 'Why does waves behave as particles when intersecting' and not 'why does particles behave as waves' (there is no particle...its an illusion)
thankfully soundwaves behave very simillar and can be used explain it (it make it easier for us sapiens to comprehend it)...let it sink in b4 u answer...
"By the way it's actually a hundred times more complicated than that" seems to be the motto of Quantum Physics. And the collab with Arvin Ash is awesome! I've checked out a number of his videos too.
Yeah, I think "By the way it's actually a hundred times more complicated than that" pretty much sums up every explanation of quantum physics.
The problem is Nick Lucid will have a PhD. And yeah, how can he explain without the fancier stuff?
Of course it's complicated, or Nick wouldn't have to study a doctorate to be at maximum of his career
Dang, a flow of probability makes a lot more intuitive sense to me than a wave of probability. Thanks for the awesome video!
Glad I could help 🤓
It only makes intuitive sense when you ||psi^2|| it...I don't think intuition covers a flow of complex probability amplitude. That gives me an idea.
So, what, probability particles are moving?
@@bozo5632 no, that's (psi*)grad(psi) -(psi)grad(psi*).
I remember many years ago seeing an article about Josephson junctions. There was a circuit diagram and an arrow that pointed in a direction with a caption "Probability Current". I went, "WHAT?" Then the more I thought about it the cooler it seemed. Now this video completes that for me.
This is the equation we all can understand without understanding it.
It's just P.E + K.E for subatomic particles. The Science Asylum is just complicating the notion..
@@Pleasing_view it isn't "just" that. It's what happens when you replace the ideas of classical energy with quantum operators.
Trust me... You dont. There are whole research branches dedicated to push this equation to its limits...
That moment when you understand that QM can be done but not understood
Actually, you can be in a state of understanding it and in a state of not understanding it at the same time... and you will collapse upon examination.
I'm from Austria and I'm also a Schrödinger fan. Before the country transitioned to the Euro, the currency was the "Schilling" (read 'sch' in German words as an 'sh' in English, i.e: shilling). The second last iteration of said currency (in the 80's up until the 90's) had Schrödinger and the wave equation on the 1000 bank note. Ofc I had to get one :P
It contains a portrait of Schrödinger, the formula symbol of the wave equation and a stilized atom on the front and the main university of Vienna and another stilized atom on the back. For anyone who's into collecting old currencies, I can highly recommend getting one. It's an absolute beauty of a bank note.
As a side note, these cannot be exchanged into Euro at the Austrian national bank anymore, but they become increasingly sought after by collectors. If you keep one for 40 years and have it remain in good condition, it may also serve as a nice investment.
My brother still has a 10 DM bank note, with Gauss on it, for similar reasons. Unlike you, he could exchange it into Euros if he liked.
Underrated comment. 👍
@@mal2ksc It's in a superposition of these two states, how fitting :D
I’m sorry to say that “I’m a schrodinger fan” is no longer something you can say, given the controversy around him ^^”
@@Sanntik Aww, some people on YT/Twitter are offended by a dead person :P
I am not bound by cancel culture and it's irrelevant to me whatever he did in his private life, since it's unrelated to his scientic achievements.
This is great! I learned recently (from 3Blue1Brown, actually) that the Fourier transform originated in Fourier's contribution to solving the heat equation, not the wave equation; so the role of Fourier transforms in QM now makes a lot of sense. Thank you for this!
Glad I could help 🙂
Fourier’s contribution to the 20th and 21st century is so underrated. Even the uncertainty principle is derived from Fourier. Not surprisingly, as oscillators (everything wiggles ….Feynman) is cosine and sine dependent. And more amazingly, it is all a vector space that can also be tied to linear algebra. A soup all pointing to energy density probabilities… nature is truly amazing
Thanks Nick! This semester I’m taking my first proper rigorous QM class! Perfect timing!
Good luck!
gl hf
Man, what a fantastic summary of some of the main relationships in physics. That free body diagram of the string section related to the equation terms was particularly clarifying.
Thanks. Glad you liked it. 🤓
“The wave equation isn’t about wave shapes, it’s about wave motion” 10:05 great line
Thanks 🤓
But the geometrics of wave motion is also critically important, it can't be overlooked
I actually used the analogy between the Schrodinger equations and diffusion equations (which generalize the heat equation) as the key to my PhD thesis, where I applied path integration methods from QM to population genetics problems that are usually described with diffusion equations
Ooh, has it been published? I'd be interested in reading that
First-year physics undergrad here and I was just studying these equations! Love your videos Nick; you're one of the reasons I'm studying Physics today ;)
Thanks for sharing. I'm glad I could inspire you 🙂
After 4 months I will be studying the same equation.
Thank-you!
As someone who encountered Schrödinger etc as an Undergraduate at The University of Bristol fifty--five years or so ago, I'm intrigued by the fact that this presentation weaves in modern cultural forms of the entertainment industry.
All most interesting!
Again my thanks!
You are brilliant, Nick! Almost every time I watch one of your videos I either learn something new or gain a new perspective. This was really helpful - thanks!
Thanks! Glad you liked it 🤓
I LOVE your videos, Nick! I'm a physics teacher in high school here in Brazil and always learn and have fun with you! Thanks a lot!! Hugs
Yes the Schrodinger equation is first order in time, like the heat equation.
But the i factor, by rotating the time derivative by 90 degrees, makes the result very different.
In fact, if you split psi into real and imaginary parts, you can write two differential equations, then combine them to finally get a single differential equation that is second order in time.
That is, if I remember correctly.
Indeed, you can remove time dependance in schrodinger equation and get the time independent version of schrodinger!
Yes, I recall the complex diffusion factor is why "probability flow" doesn't have the same broad characteristics as temp flow and can look "wave-y".
This. I recall reconfiguring the Schrodinger equation to be second order in time, and the i factor was important
It depends what the "i"means though - there is an interpretation where time is a clock which is a cycle so anything that describes these kinds of processes at this fundamental level involves clocks diffusing.
Well, Dirac’s version with second derivative of time (relativistic) got him the Nobel Prize.
Awesome Video as Always Bro!
You Make Science so Interesting
Thanks a Lot
Appreciate it a LOT🔥❤️
Glad you enjoy it! 🤓
Interesting video. It did always feel weird to me that it wasn’t a wave equation. But I am slightly confused. Wasn’t Schrödinger motivated by De Broglie’s hypothesis? Namely that electrons were standing waves around the nucleus. That all particles, not just photons, could exhibit particle wave duality. Wasn’t this also used to explain the double slit experiment? I understand that formally it resembles the heat equation, not the wave equation - but wasn’t the whole point to find waves? Even if they are “probability waves” for lack of a better term.
I mean I just started studying quantum physics in my undergrad, so I really have very little idea what’s going on.
The "i" in the coefficient on the time side of the equation makes things a bit more complicated (which I glossed over). It allows for rotations in the complex plane and rotations can be viewed as waves if you take a cross-section... but that's two _extra_ levels of abstraction.
“Where was I”? My thoughts all through this, and loved every second of it!
Nice cameo 😎 I have been following both channels for years 👍🏻 excellent as always Nick ❤
I like how he is questioning everything just to explain stuff to us.
Richard Feynman, one of the greatest physicists in the last century believed that the key to truly learning and understanding a concept was to be able to explain it coherently to others in laymen terms.
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And just on a general level, intelligent people don't just "have opinions" on things without at least trying to understand them inside and out. So it's likely this was a lot of his actual thought process on working through this. Also, deriving an answer through logic will be much more likely to stick in your memory than deriving an answer through memorization.
.
In other words, this isn't just for us. He is likely sharing his actual thought process on the matter.
Nick always manages to make these gnarly topics extremely lucid! 😆
Thanks! 🤓
Actually, I watched Arvin last night, February 2, 2023. Thank you for putting it all together.
"Curviness determines acceleration"
"Probability flows through space"
Love these
Just ask this question.. "is probability a physical object? If no then how and where is it flowing? In our heads or in physical cosmos? 😀
@@brigittelars5564 Love this. Probability maybe the first step towards a definition of consciousness? Thank you!
Zur Heterogenität dessen, was man nomine so in der Quantenphysik vorfindet:
Die Gleichung H Psi (r) = E Psi ( r ) ( das Psi für die Wellenfunktion... ) ist in dieser Form der Schrödinger-Gleichung eine Eigenwertgleichung, wobei ihre Lösungen also Eigenfunktionen zum Eigenwert E ( wer nicht weiß, was das ist, der sich sehe sich die Hauptquantenzahl < n > an... ) sind. Weil die Eigenwertsgleichung meistens nur durch diskrete Eigenwerte in Abhängigkeit der Randbedingungen erfüllt wird, heißen diese quantisiert als Antonym zu kontinuierlich.
Ich finde aber das Problem ist, dass, wenn die Leute fragen < quantisiert > was'n das? ...die Antwort < es sind infinitesimale Pakete dessen, was das jeweilige Quant ist, einem zwar Ruhe bringt... ( ...die Leute hauen ab... ...sie hauen aber auch bei🖕ab ), aber keine abstraktionsorientierte Antwort sein kann, weil die Leute autosuggestiv wieder an Teilchen denken und nicht von dieser Anschauung aus dem Mesokosmos wegkommen.. ...Rettung vor dem, was die Sprache der Physik ist, bringt einem eigentlich nur der mathematische Term als Hilfsmittel der Beschreibung aus dem Reservoir, was die Mathematik abstrahiert aufgespannt hat... als letzte Zuflucht für die, die da die Schöpfung so gut wie möglich sehen wollen... ...ich will hier so schnell wie möglich wegkommen!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Le p'tit Daniel🐕🐕🐕🏒🏒🏒
As someone with a degree in physics, it always makes me smile when I learn something from the Science Asylum. My undergraduate wave mechanics professor never explained what the Schroedinger wave equation represented. It was just, here it is, now let's do some calculations.
*"It was just, here it is, now let's do some calculations."*
Yeah, that's pretty common.
Same. Why don't they just say: by the way mathematical abstraction doesn't always translate so we started using probabilities.... Instead they keep it esoteric like it's unbelievably complex and of course it's complex but we already knew that didn't we! It doesn't make Maxwell's equations not true in fact Maxwell's equations account for almost all phenomena except gravity. This is why I decided not to pursue my post-graduate degree in physics seems like a waste to me...
@@ScienceAsylum H sigh = E sigh and then exercise.
Thanku 1000 time u have no idea how much your vedios help me. When professors explain schrödinger equation i remain only scratching my had. After watching your vedios my efficiency increase 200%
Thanks for your commitment to lucidity in science.
Thanks for the support!
Respect for tackling this.
A flow equation instead of a wave equation.
Though when you take a time independent solution on a constant potential energy term ,
you'll get a 'standing (co)sine wave'
Which is probably where the confusion originates.
As these simpler solutions are always the first ones used in a classroom environment
I think it all comes down to semantics and something you mentioned on your earlier videos. Namely that quantum particles don't sometimes behave as particles and sometimes as waves, they always behave as particle-waves 100% of the time, i.e. they're something different entirely. It's a shame concepts in physics get mislabeled and the label sticks, but it's a learning opportunity and that's why I appreciate these videos.
As Feynman once said in one of his lectures:
"They do not behave like particles! They do not behave like waves! They behave in their own inimitable way!"
@@nybble Yes, I saw that video of Feynmann, it was really cool.
Cannot tell you how delighted I am to see the two of you collaborate like this. I took me by surprise. You guys made my day. LOVE YA BOTH! ❤
Our pleasure! 🤓
Saw you on Arvin's site before seeing your latest upload! Was a nice surprise to see you appear!
It's been so long it seems every time you post a video. Great work as always sir
Came here from a reference from Arvin... already subscribed long long ago for both. Great teamwork!
Excelent video about the wave equation. Would have helped me a lot when I was taking differential equations lectures in undergrad school.
The best explanation of the wave equation ever seen. This intuitive approach to the meaning of the second derivatives with respect to time (acceleration) and space (curvature) is needed to a full understanding of the equation, but almost never explained.
You are the best in clarifying physics equations!!
Thanks! Honestly, I (partly) regret putting it inside a video about Schrodinger's equation. It should have been its own 5 minute video.
@@ScienceAsylum But you can make another video about waves and repeat it with other things. I suppose that is hard to find a different subject of physics every week. Doesn't matter to make several videos about the same subject. It is usually made with special relativity, but it can be made with everything.
I just took a course on schrodingers equation so this was a treat to watch.
Nice! 👍 I hope this was some good reinforcement.
OMG... If I had instructors in my engineering program who had a quantum bit of the talent to explain things so clearly... I would've saved so much head scratching and confusion. Brilliant overview!! Thanks...
I'm so glad you liked it 🙂
If the Schrodinger equation is a diffusion equation, then it has an imaginary diffusion coefficient, which in turn makes it weird to call it a "diffusion equation". I mean, the "i" in the Schrodinger equation makes a big difference... Because of the "i", the solutions must be complex, and the equation has U(1) phase invariance, which leads to conservation of probability in QM. None of that is true for the heat equation with a real diffusion coefficient, because there's no phase invariance and no conserved quantity. As similar as these two equations may seem, the "i" introduces some properties to the Schrodinger equation that are just not present in the diffusion equation.
Sure, it might be better just to label it it's own unique thing 🤷♂️
Don’t be ashamed of plugging your book, it’s a damn good one. I’ve been working through it for fun in my free time which is a lot more than I can say about my copy of Jackson’s e&m book.
Thanks for the clarification! I found the term ‚wave equation’ always very confusing: in High School I thought an electromagnetic wave propagates like a water wave through space.
Ah, education. They don't care what you know/think as long as you can regurgitate the exact list of things they fed to you.
I mean.... kinda? It does propagate. Through space. And something is waving. Just happens to be the EM field instead of material water
PDE's a favorite topic of mine. Very well done!
Thanks! 🤓
at 1:50 , *in general* the horizontal component of the tensions does NOT remain the same after applying the downward force. To keep it the same, you need a very very *soft* string (I am assuming linear-elastic spring behavior) and a very shallow dip (so that cos of the angle is very close to 1)
I just thought of this too
I've been watching Arvin's and Nick's videos back and forth until Nick's video helped me reach a steady-state.
Love your videos, Nick! Keep it up.
Thanks, will do! 🙂
To me the videos you make with humor and/or describing the equations (math) with simple words, instead of leaving them out completely, are your best ones.👍
(I think most people just want to know what the equations come to anyway, without studying mathematics.)
Thanks Tommy!
I love the description of the difference between the broad equation types. Personally, I think of the heat equation in terms of diffusion (mass transfer, but conduction in the heat model). It represents movement across a gradient compared to the beautifully explained wave function's restoration from concavity.
The Schrödinger equation (SE) is technically a parabolic equation (PE), also known as the parabolic wave equation. This is an excellent approximation to the full wave equation when the motion is primarily in one direction. The approximation in the SE is that particles are only moving forward in time, in other words: non-relativistic. IMO it is rather silly to claim PEs are not wave equations (in their applicable domain) since they give solutions that are extremely close to those of the full wave equation. For example a PE equation is routinely used to calculate how sound waves propagate thousands of miles in the ocean.
The "waviness" in the SE is in time, not necessarily in space. The eigenfunctions typically have a factor of e^-iEt which contains the waviness. It is no surprise that when you ignore these factors, the waviness goes away.
Stanley Flatté published an excellent pedagogical article that shows the relationship between the SE and the full wave equation (Klein-Gorden equation):
The Schrödinger equation in classical physics
American Journal of Physics, Volume 54, Issue 12, pp. 1088-1092 (1986).
An episode about the History of the Schrödinger Equation would be great!
This video was excellent. I immediately went to look up the Black-Scholes equation as I had remembered it had been derived from the Brownian heat equations. In that equation (at least for European options) we see a first order term for time and a second order term for motion.
This video has revealed a relationship that I had never connected before. It is like a curtain has been lifted. Thank you!
This vid completely ignores the fact we're dealing with complex numbers here, and the way curviness in complex phase influences amplitude makes it very different from ordinary heat equation where all numbers are real, not complex.
Keep up the good work, I love your videos
Thanks, will do! 🤓
Props for having Arvin on. I'm a hard guy to make laugh, but this time your law of conversation bit broke me.
You're welcome 😉
This is a great intro to understanding the Schrodinger Equation! And this is also what my research professor taught me on how to view the equation: you can recast the Schrodinger Equation such that it can be explicitly shown that it is indeed a continuity (or a flow) equation in which Probability Density is conserved and a so called "Probability Current Density" carries the flow. Think of it as if there is no current, then the probability won't change; similar to when there is no flow of fluid, there will be no change in the accumulation of fluid. After all, the Schrodinger Equation enshrines the Unitarity Principle.
Anyway, my comment I guess is about the notion that Schrodinger Equation is a derived equation. Here, it is said that it was derived from Conservation of Energy. But my Prof always says (I am in his Quantum Foundations theory group btw) that Schrodinger Equation cannot be derived. It is a conclusion of a concoction of several experiments and postulates. It just so happen that the Schrodinger Equation fits snugly in our current view of Physics such that one could say that it could be derived from say the Conservation of Energy. But with the same logic, Schrodinger Equation could also be "derived" from a proper Lagrangian Density and from the Hamilton-Jacobi Equation, among others, which stems from the fact that there is a symmetric transformation that gives rise to a conservative quantity. Hence, you cannot pin-point where it really came from. He always corrects me: "You cannot derive the Schrodinger Equation: it is a result of many experiments trying to describe our weird nature."
Still this is a great video and I learned a lot! Thank you!
Every wave equation implies a continuity equation.
Pedantic me says it's a diffusion equation of which the heat equation is a special case. Whether it is also a kind of wave equation depends who you ask but diffusion waves are definitely a thing. Fun video from Nick as always.
I've always thought the "wave" aspect of the Schrödinger was contained inside its use of imaginary and complex numbers, as these are pretty useful for describe some "cyclic" behaviors like the rotational and harmonic phenomena 🤔
Only rarely do I learn something that blows up my mind and lets me emerge with a new, completely altered understanding of our world. This video is one those rare moments and is amazingly the second time you've done that for me. Keep up the amazing job educating us about science!
I like during these explanations when mathematical explanations are used and explained. However, what is enjoyable is the split-screen or caption box that shows visually what is described with an arrow or another box encapsulating the specific item of the math equations. This way it does not really matter if it is just an equation that is not yet understood completely, I would think? ...Nice guitar, Alvarez.
Oh, Nick, I once thought of the very same question - "Is this a dissipation equation?" - and asked a physicist about it. The physicist answered that it is not "quite" heat equation because of the imaginary one, the "i"! ("The Imaginary One", now that's something right out from a fantasy book... I mean "imaginary unit", sorry, English isn't my native language). This changes it from dissipation equation to something quite different because every derivative "turns it 90 degrees" in this weird imaginary space. Could you elaborate on this a little? Does it make big enough a difference for it NOT to be a dissipation equation? Or does this only mean that dissipation itself is weird?
I'll admit the "i" in the coefficient muddles things. You could argue that Schrodinger's equation is really it's own thing.
It took rewatching some parts but the explanations here were great.
This is awesome! The only thing swept under the rug is that Ψ isn't a pdf, but a complex number, and |Ψ|^2 is a pdf. So it's not probability that is flowing but something very closely related. An intuitive explanation of the meaning of Ψ would be incredible
Unfortunately, we don't have a physical interpretation of Ψ because we've never observed it directly.
@@ScienceAsylum Are there any other links between quantum mechanics and thermodynamics?
They both seem to be describing something statistical - lots of little tiny things, running around and interacting with each other on a scale that's too small to follow the details, so we are only really aware of a summary (averages and distributions) of what happens, not so much the details of individual interactions.
@@TooSlowTube look into statistical mechanics and quantum decoherence. Have fun going down the rabbit hole, see you on the other side!
@@ScienceAsylum so in place of the real physical thing, humans just plug in probability which is not also a physical thing, right?
"Quantum Mechanics is the dreams that stuff is made of."
That saying makes even more sense after learning that the Schrodinger Equation is about the flow of probability. Probability, as a quantifiable value, is about as non-tangible as you can get.
You know, I actually got a lot from this video. Thank you Nick for clarifying the perspective of the *wave* and *flow* itself.
Arvin also has some great videos and I've watched nearly all of those too. Thank you both very much for making these videos! ♥️🙏
I look forward to reading your *Fine Structure* education!! Thank you so much!
Brilliant! Such a lucid explanation. Love it! 💗
Glad you liked it! 🤓
@@ScienceAsylum Between your channel, Arvin Ash, and Sabine Hassenfelder we amateurs actually have a chance to correctly understand physics! 😁
3 equations within the first minute! Love it, this is turning into PBS Spacetime without the beard.
😆
6:18
Putting Boxxy in the square wave is a very old school internet reference that I am all for!
I struggled with Science at school finding it extremely difficult to grasp.
However 40 odd years later I’m finding your vids really fascinating - I won’t pretend I understand the detail but the way you put things across it certainly makes me grasp the overall concepts so cheers for that 😊
One final thought and it might be a daft question but I will ask it anyway:
How do we know a particle can be in more than one place at once until we measure it - as the act of measurement will fix it to a particular place, so thefore we can never observe these multiple states? 🤔💭
I thought I understood this stuff pretty well, but your videos consistently blow my mind!
Thanks!
Very awsm bro
Love from India :'))
Excellent video... As usual. Thank You!
I loved it, thinking about Schrodinger's equation as heat dissipation equation was fun! 👌👌👍👍
Well I think the comment about three types of equations in classical physics is correct if we change the "type" with 'category"! Most of physics equations are of order 2 and linear with constant coefficients mostly, when working with second order differential equations, we only have hyperbolic, parabolic and elliptic variations (discarding the degenerate cases).
The classification "elliptic, parabolic, hyperbolic" is complete only in 2 spatial dimensions.
Loved your way of explaining this! Made something very complex seem shockingly simple.
Great! This video leads nicely into a discussion of the epistemic vs. ontic nature of the wavefunction.
Okay. This is like the best explanation I've seen.
As an Austrian fellow, like Schrödinger ;-), I have to disagree here. You forgot the imaginary unit "i" in the Schrödinger equation. This is a very important difference to the heat equation and leads to a different behavior. The time-dependent Schrödinger equation can describe (standing) waves rotating in the complex plane, having the same solution as two coupled wave equations - one real and one imaginary. The heat equation cannot describe such behavior. Considering also the imaginary unit, in my opinion the Schrödinger equation is more similiar to the wave equation than to the heat equation.
Another important difference: The Schrödinger equation can describe interference, the heat equation cannot!
7:32 Every 100 videos on a subject you will find something truly enlightening that you will remember forever
I love your videos...even if I only get half of what you're talking about 🤣 Yet...at the end of your videos I feel like "I got this". Feel smart for about 20 minutes 😂 Seriously...I love physics and even though I could never be a physicist there's just something about the science that tweaks my brain. Thanks for making a tough (for me) subject enjoyable.
Man...that's really a perfect timing...I have engineering physics exam Tomorrow and I have searched many videos....but your video made me a clear mind of seeing equations 🙂🙂
But it does relate "curviness" (in complex phase) to oscillation! The stronger WF spirals in complex plane, the faster its amplitude changes, the faster it "moves" through space. And for free particles this function does look like a wave with all the wavy properties. Nice visualisation here: th-cam.com/video/LZie2QC5Jbc/w-d-xo.html
Sweet Lord this was great and extremely helpful Nick. You and AA are connecting the disparate dots into a cohesive whole.
Thanks! 🤓
That boxy wave radiates some old school vibes. A signal from a distant past.
Here's the comment I was looking for.
Nice description. Identifying the second derivative as the curvature and distinguishing its behavior in one equation from the other is particularly insightful. This intuition beautifully agrees with the solution of the 1-D diffusion equation, where the two curvatures of opposite signs (one at the head and two at the tails) cause the head of the gaussian function to come down with time, while its tails grow in height. When I first understood the diffusion equation during my early days of learning, it gave me a better physical insight into the dynamics of the Schrödinger equation. I feel it's always nicer to view the Schrödinger equation as a form of diffusion equation describing a complex function, which interestingly gets called as the 'wave function'.
Everyone remember that it’s ok to be a little crazy
Great video. I am fascinated by wave & heat equations for a long time but haven't look at it from this angle yet. They are super easy to understand using grid based simulations. One grid for the wave height (concavity in the video) and one for velocity. All the scary looking derivatives just turn into simple subtraction. Totally worth a look into it.
That was really very exciting and eye-opening. Thanks Nick 😃
I haven't thought this through, but an argument could be made that the Schrödinger equation is a wave equation in disguise. There are complex numbers, not real numbers, and the argument of the wave function is indeed "waving" all the time (you only address the dissipation of the square). One can reformulate every second order differential equation as a first order equation in more variables, which might be exactly what happens here using the complex numbers (which consist of two independent numbers). What if you rewrite the Schrödinger equation as a system of two coupled equations in two variables, perhaps its waviness becomes more apparent? If you think of the solution of the Schrödinger equation for the harmonic oscillator, with a wave package moving back and forth, it does not look like dissipation, and dissipating heat cannot do this periodic motion without a changing potential. Lastly, you got the Dirac equation which hides even more of the derivatives by introducing even more components. This is a linear equation. But noone would claim it behaves like the usual linear systems we study for ODEs, because these are pretty boring. The upshot is: what lacks in derivatives can be compensated by using more components, potentially in the form of complex numbers.
Since I can't give two thumbs up for your second order content, I leave this comment in its place.
Two thumbs up man! Keep making this amazing content!
You are one of my favorite science communicators, and easily one of the most entertaining while also maintaining an incredible fidelity!
(Ps: I only said "one of" because I don't want the other kids to get jealous, but between you and me? We know who my favorite actually is, right? Wink wink...)
That may have been your best yet thank you!
in my opinion, the term of "wave-particle duality" is wrong. The right term must be "vortex-particle duality"..
It's huge! The book that is. Looking forward to having time to read it! 😊
Turn out seeing Arvin and Science Asylum both uploaded at the same time is not a coincidence! 😭😭😂
Nope! Not a coincidence.
Very good. I wish my teachers way back then had talked about electron orbitals in terms of probability. It would have helped my understanding a lot. I suspect that they didn't understand enough themselves.
Somehow my highly paid professors half a century ago failed to make this as clear as you have…🤓
In Geophysics we use what we call “one-way wave equations” that have the same form as Schroedinger’s equation.
We use them as approximations of the wave equation that only need one boundary condition instead of two because you only allow the waves to go one way in depth…
Cool!
This is a bit misleading. To call the schrodinger equation a heat equation is really missing the point. The time derivative of psi comes with a factor of i, so really the schrodinger equation is in fact two coupled differential equations for the real and imaginary parts of psi. What the schrodinger equation really says, is the change of the imaginary part over time is due to the spatial derivatives in the real part and vice versa, where as the heat equation would not have this coupling. The flow in the schrodinger equation describes motion about the complex circle. It also has a conserved current (probability current), which the heat equation does not have (it describes flow to a stationary state). In other words, motion in schrodinger's equation can carry on indefinitely (plane waves for example). The heat equation describes dissipation of energy until a stationary state (with no spatial curvature) is achieved. Another why to put it, Schrodinger has wave solutions, the heat equation does not.
Congratulations for making it to the end of this rant.
I agree with this, and wrote a long comment to the extent elsewhere.
There is an interesting formal similarity though and if you treat complex solutions to both equations the only sing that differs is one of the coefficients. So there would be mathematical connections one could exploit for sure. But that doesn’t mean the two should be classified as behaving at all similarly as they stand. If you ignore factors of i, there would be no real difference between Laplace’s equation and the wave equation which is kind of one of the motivating distinctions to begin with.
Thanks Ordon for the prompt. I had the exact same thought. It'd be better called the "quantum _diffusion_ function".
Didn't you kind of gloss over the i on the right-hand side? Since the solution is a complex power of e, multiplying by i is kinda-sorta taking another derivative. So, the right-hand side behaves second-derivativy.. So, that's still kinda wavy?
But it's an _abstract_ kind of wavy, and more of a rotation than a wave.
Man.. I love these kinds of videos ♥︎
Once you get complex amplitudes involved, you can have first order or mixed order equations that admit wave solutions. The heat equation is basically a diffusion equation, and while it's interesting to look at Schrödinger's equation in that lens (diffusion and waves both relate to moving things around, after all) it's not really the same thing, because the heat equation is restricted to real amplitudes, cutting out a huge number of solutions. It's fascinating that you need two derivatives in both time and space to get real wave solutions, but basically any non-trivial combination of derivatives in time and space will work when you allow complex amplitudes. I think that shows that the classical wave equation is really just a special case for real valued quantities. The other cases are still wave equations, they just don't look like the classical one because they allow complex wave functions. You can argue that by adding potential functions and boundary conditions you can create "static" solutions, but I don't think that's fair. The classical wave equation can produce standing waves, which are just as static as the hydrogen orbitals you mentioned. Does that mean the classical wave equation is not a wave equation because it allows solutions that don't vary in time?
Also, what we call a "wave" can be made precise. If you mean a spatial shape moving along in time (some of the illustrations used here suggest pulses), then any function of the form f(kx - wt) will produce that behavior. Not all of those are physically realizable (i.e. solutions to physical wave equations), but do we want to call a travelling parabola a wave, or do we want to only call solutions to wave equations waves? Is that definition too circular? I dunno.
Wow, great video. I got more insight into Schrodinger from that video in 15 minutes than in some full courses. Professors always bury those kinds of insights or scatter them weeks apart so they no longer fit together.
With this new found clarity, I have one question... You said the equation expresses how probably (density) flows over time. That makes sense to me, but how do you connect that to the idea of fixed quantum states in an atom and the discrete quantum jumps when changing energy levels. What is "flowing" in a fixed state? and how does it characterize the state transitions? Aren't they discontinuous?
Keep up the great work. I'm going to go track down a copy of your book. I did an engineering physics undergrad in the late 80's. Channels like yours have helped my rekindle my love for the subject any refresh many underused brain cells. I'm sure the book will be as good.
Joe
11:33 Fun fact: Schrödinger actually came up with the Klein-Gordon equation before Klein & Gordon; but it gave the wrong answers, which is why he rejected it.