Intro to the Quantum Harmonic Oscillator in 9 Minutes  

That’s awesome, glad you enjoyed the video! :) And don’t worry - the equations are only scary at first 😅 After a while it all starts to make more sense.

Thanks for hosting the contest! And thanks for the honorable mention! :)

Thanks Jean, I’m glad you enjoyed the video! :)

Thanks, I’m glad you liked it! :)

Thanks Kevin, I’m glad you liked the video! :) For the next few videos I’m planning on mostly focusing on exploring the nature of gravity, but I’ll try to upload some videos on other topics too. Let me know if there are concepts in your quantum mechanics class that you’d want to see a video on! Eventually I plan on doing the particle in a box, double slit, and hydrogen atom.

Thanks for the recommendation, I’ll check it out! Graduate-level is fine. I do a lot of quantum-related stuff for work, admittedly it’s mostly X-Ray and electron diffraction which is pretty straightforward, but I’ve been meaning to get back into the fundamentals. Frankly I still have no idea why quantum mechanics is the way it is 😅 My latest attempts to approach the subject via path integrals has been a little less satisfying than I had hoped for; the intuitive starting point is nice, but evaluating path integrals is way too hard in general. But then from a matrix mechanics perspective I always get lost in the abstract nature of it all. Looking forward to checking out your recommendation.

@@RichBehiel I think you'll like it.

Thanks for watching! :)

Thanks, glad you enjoyed the video! :)

Thanks! :)

Very true! 😂 It’s all so simple at first, then the details of the situation quickly make the analysis complicated. Although, regarding statistical mechanics, it’s always cool when a complicated ensemble exhibits a new kind of simple behavior in the continuous limit. Sometimes I wonder if those kind of emergent thermodynamic laws are in some sense more fundamental than the microscale laws that give rise to them, because of their substrate independence. There might be some exotic other universe out there that’s totally unrecognizable to us, but as long as it has something resembling packets of information which can move around at various speeds and bump into each other, it’ll have something analogous to temperature. Makes you wonder about the scope of what’s possible and what’s inevitable. Anyway, thanks for watching, and I’m glad you liked it! :)

Misha Gromov says as much in one of his talks (I think powerspace and the bulk method), arguing it arises from category theory and also why it’s so difficult to model systems in the “middle” (too many particles for coupling, too few for central limit theorem (disjoint union, a sum)).

Now that’s a deep and nuanced question! I haven’t had my coffee yet so forgive any typos… First, to see the nature of electromagnetism it’s best to start with the electromagnetic four-potential, which is comprised of the vector potential A and the scalar potential phi. The four-potential is a four-vector so it’s Lorentz invariant. For example, a charged particle at rest is all scalar and no vector potential, but a moving charge picks up some vector potential by virtue of the Lorentz transform. But because motion is relative, one man’s vector potential is another man’s scalar potential. You can calculate the Faraday tensor basically by taking derivatives of the four-potential. The Faraday tensor neatly packages the electric and magnetic fields E and B, so you can see how they come from the four-potential, and you can see how E and B are really more suited for engineering than pure physics. In physics, it’s really all about A and phi. So to your question about coupling to the wave function… to that I would highly recommend the Feynman lectures volume 2, section 15-5 in which he shows how there is a change in phase of a path due to the scalar and vector potential, which you can calculate by integrating over the particle’s trajectory in time and space respectively. In those two equations you’ll find the answer to your question. Here’s a link to the Feynman text. Section 15-5 starts on page 179 of the pdf. I’d recommend buying a physical copy on Amazon if you’re into this kind of thing. joepucc.io/static_assets/projects/feynman-lectures-on-physics/vol2.pdf Also, his explanation is in the context of the path integral formulation, which is arguably a more natural/intuitive starting point for quantum mechanics, but is also a pain in the ass to do anything with, because Feynman path integrals are generally ill-defined and tortuous. On the other hand, doing QM with wavefunctions is further from intuition but is easier algebraically. The two formulations are equivalent (for now, as far as we know…), so it’s probably wise to learn them both. To conclude this wall of text, I’d like to point out that as of 2023 the human species still has no idea what electromagnetism is. Sure, we have a very nuanced and occasionally sharply predictive model in QED, but it’s fraught with philosophical difficulties vis-a-vis renormalization, and it’s a phenomenological model at best. We need a return to the old, more optimistic ways of thinking, e.g. Kelvin’s knots in the ether or some analogous thing. Whoever can draw a picture of an electron will probably understand most of what’s going on at the fundamental level of this reality. For now, we basically just have dubious statistical mechanics whose sole redeeming quality is its uncanny ability to make extremely precise predictions in a few scenarios which intuition alone can’t map out. So I guess what I’m trying to say is you should keep kindling your wonder for this kind of thing, because there’s still a whole lot of new territory that hasn’t been discovered yet!

​@@RichBehiel Renormalization does kinda feel more like a problem than a solution...

Renormalization is a problem that wasn’t solved, but was swept far enough under the rug that most physicists don’t worry about it. It greatly bothered Dirac, and to some extent Feynman. QED diverges in the ultraviolet limit, but presumably it also breaks down, so you can renormalize things based on observed charges and masses, rather than the true values they would have if not for their field interactions. Still, this is an unsatisfying sleight of hand.

Python, with matplotlib for 2D stuff and plotly for 3D.

The wavefunction asymptotically approaches zero in the limit as x -> infinity. This is because the particle is very unlikely to be very far from the point around which it oscillates. The frequency is determined by the energy. The thing that’s an integer is the number of nodes, which is closely related to the energy. Most (all?) of the integers that arise in quantum physics are due to integer numbers of nodes arising from an oscillating but continuous function.

I think the problem here is that it is really a longitudinal wave and not a transverse wave like assumed, which transverse wave would have the problem you pointed out

This video was made with python, using matplotlib.

Thank you very much!@@RichBehiel Did you start from scratch, or was there a github repo or pypi project you used as a base?

@StinkyEthien I guess I started from scratch, about a decade ago. In college I was really into programming and visualizing data, and over the years I’ve done a lot of that for work too, so that sort of evolved into these videos.

Thanks for your comment, and I’m glad you enjoyed the video! :) Quantum mechanics is deeply geometrical, in the sense that it models processes occurring in Nature, all of which have some geometric form. But to your point about the reduced Planck’s constant, in that case it’s more of a notational convention. Consider the function y = sin(k*x). This wave has a wavelength of lambda = 2*pi/k. So here we can see that the factor k, the so-called angular wave number, fits nicely into the equation sin(k*x). If we just wanted to use the reciprocal of the wavelength for the wavenumber, we could have had to write sin(k*x/(2*pi)), which is messy. Anyway, the reason we often use the reduced Planck’s constant is that hbar is easier to write than h/(2*pi). As for 4pi, that number appears a lot, in the context of spherical surface area but also just when 2pi gets multiplied by 2. But I agree that typically when you see a 4pi in physics, there’s a flux integral over a sphere hidden somewhere in the derivation, e.g. Coulomb’s law, or deriving Newtonian gravity from the statement that the laplacian of the gravitational potential is proportional to the mass density field. But I wouldn’t read too much into the numerology here. You can see from the derivations why things are what they are. As for why the wavefunction is squared, that’s a more nuanced topic that extends beyond the scope of a RU-vid comment. But you can derive the answer within the context of the Feynman path integral formulation, and the observation that in order for that formulation to work, you need probability amplitudes to be able to interfere with each other in ways that requires having a phase, so you need more than just the real numbers if you want to encode that all in one function. I’m not quite sure what you mean by your last question. And I’m also not sure if the future is probabilistic 😅 I think that’s one of the deepest philosophical problems, with major implications for free will. But I’ll end on this note: if the future is probabilistic, then may the odds be in your favor. And if it’s all predestined, then may you be a force of nature! :)

Sorry, I don’t think this was clear from my previous comment, but to clarify on the point about the wavefunction being squared: you need complex numbers to track the phase, but real numbers to represent observables, so the amplitude of the wavefunction gets squared to make that happen.

@@RichBehiel By my last question, I mean is could the wave particle duality of light and matter in the form of electrons be forming a blank canvas that we interact with forming a future relative to the energy and momentum of our actions? in such a theory light is a wave over a ‘period of time’ relative to the atoms of the periodic table with particle characteristics as an uncertain ∆×∆pᵪ≥h/4π probabilistic future comes into existence light photon ∆E=hf by light photon. This is logical because light photon ∆E=hf energy is continuously transforming potential energy into the kinetic Eₖ=½mv² energy of matter, in the form of electrons. Kinetic energy is the energy of what is actually ‘happening’ as the future unfolds.

I’m not quite sure. I know the Schrodinger equation is a way of saying that the energy landscape of a quantum system is what determines its evolution in time, and the wavefunction is both wavelike and particle-like. But I don’t know how to dig deeper into the ontology than that. I wish I did! Have you heard of Nima Arkani-Hamed’s work on the amplituhedron? Seems like something that may interest you.

@@RichBehiel No I have not come across Nima Arkani-Hamed’s. The way to have an intuitive logical explanation of what is happening is to think of the collapse of the wave-function as a new moment in time. When we make a measurement there is a somewhat random and unpredictable collapse into one of the possible measurement states and this depends on what the wave function looked like just before we made the measurement. It collapses into one state rather than continuously flowing from one state to another and then at that new moment in time just after the measurement, once again the wave-function starts following forming probability based on our new measurement state. This is totally logical because electromagnetic fields are not quantized, it is just of the energy transfer processes between field and matter, between photons and electrons that are quantized. We have electromagnetic waves, flowing out forming the carrier for potential quantized photon energy. The energy contained in a wave, spread out forming the characteristics of three-dimensional space. This energy is converted back to another form of energy in a very local and quantized event by light photon ∆E=hf energy continuously transforming potential energy into the kinetic Eₖ=½mv² energy of matter, in the form of electrons. Kinetic energy is the energy of what is actually ‘happening’. This geometrical process formed by the spontaneous absorption and emission of light photon energy is continuously forming a probabilistic uncertain ∆×∆pᵪ≥h/4π future.

You’re right about mc^2, but that’s a different observation than that the quantum harmonic oscillator’s ground state has nonzero energy. Schrodinger’s equation is nonrelativistic, so it takes the mass of the particle as just a parameter, then finds that in addition to the energy from its rest mass, it’ll have an additional nonzero energy term even in the ground state. Contrast this with classical physics: you can sit on a swing at the park, and if you’re not swinging, then your swing energy is zero. You still have E = mc^2 energy (which is an enormous amount of energy btw, like over a trillion kWh), but you don’t have swing energy if you’re not swinging. In the quantum world, in addition to the E = mc^2 energy, you also always have swing energy. It’s a subtle effect, but it can be observed. For example, it’s the reason helium doesn’t freeze solid, even at absolute zero.

I am 😂

@@RichBehiel haha great to know! that review was very eloquently written, hope you’re enjoying your state of enlightenment over us who have not felt the density of tungsten

Thanks. The joy of tungsten enlightenment is eternal! 🧘🏻‍♂️

It’s a math and physics video contest hosted by Curt Jaimungal of the Theories of Everything podcast.

Hi Henry! :)

It was a contest for educational math and physics videos.

@@RichBehiel thanks! I only knew about #some2 😅 Who is hosting it tho

Curt Jaimungal, host of the Theories of Everything podcast.

@@RichBehiel nice

I share much of your general skepticism of the theoretical physics community, and you’re right that plenty of scientists overhype their work in the pursuit of funding, but I can personally attest to the utility of the the quantum harmonic oscillator. It is useful in many contexts. Of course, a set of ideas can be taken seriously but not literally, and one of the gaping flaws of modern physics is its utter lack of an ontological framework. Many physicists have resigned themselves to the “shut up and calculate” mindset, and will regard as metaphysics any inquiry into what these equations actually *mean*. But the tide seems to be changing. I’ve seen a renewed enthusiasm for investigating the foundations of quantum mechanics in recent years. There is hope! :)

@@RichBehiel I agree, linear algebra combined with an algebraic geometry of the Lagrangian mechanics is a truly useful and beautiful framework. And the difference between that and a model of energy ("physics" in one word) is at the root of the philosophy of science. We really deserve to be fooled at this point. But it is meaningful indeed that the single word for describing physics, 'energy' is also the single expression for 'god' in quite a few religious contexts. This metaphysical quandary persists into the domain of pure maths, which will always be definitively the "metaphysics" of quantitative sciences. But I think it is hilarious that Plato describes this game by saying that the materialists are terrible fellows who are typically elderly men who love the sound of their own voices and take the tack of leading the argument into nonsense in order to avoid detection of their sophisms. But he decides for philosophical reasons to steel-man their arguments for them for the sake of examining them closely. And then he says, that is why their opponents, the idealists, defend themselves from behind mystical clouds, so as not to be forced to confine 'energy' or 'being' to mere quantification. It's a very old conversation indeed. And yes we have learned some great things, almost all of which were anticipated by Archimedes and Apollonius, millennia ago; the conic sections. How did they know? By logical necessity alone, with no recourse to an energy metric, they laid the tracks of the Standard Model train. With sticks in the sand? Hm

I’m not sure I totally follow, to be honest. Are you talking about the mystery of how our first-person qualia map onto the underlying third-person empirical world? I.e. the hard problem of consciousness? Btw to be totally honest I wouldn’t have imagined the solution to the quantum harmonic oscillator a priori 😅 It does make sense in retrospect though, in that it hangs together into a coherent story of how particles behave. Anyway, I wouldn’t take the math too literally. Most of physics strikes me as a kind of accounting over an economic substrate whose nature is unknown. We canvas the landscape with mathematical forms, but rarely ever pierce it.

The notion or feeling that (of course this is how reality works ) our ability to imagine and predict most of it even before we learn the details in school. Irony of familiarity. And the later part was just a reference to the great debate of how to categorize the universe and world around. What is or isn't physicalism? How hard it is to categorize and give defined description

Yeah for sure, it’s curious how often when you learn something, it snaps into place and there’s that moment of insight, when suddenly the idea fits into your mind and it’s almost as if you’ve seen it before. Plato was curious about this problem. How can a person go from not knowing something, to knowing it, and then knowing that they know it? Who are they to judge whether they know it, if just a moment ago they didn’t know it? That line of thinking leads to Plato’s famous quote that all knowing is remembering. I’d take a slightly different approach to the problem and say that we learn by ever-branching analogies, and then smooth down the edges of the analogies over time until our worldview coheres into a mostly unified whole. But I think there’s a danger in this, because we end up living in the world of analogy and not the world of *slaps hand on the table*, that is we live in the map and not in the territory, and our familiarity with the landscape is at least in part a familiarity with the stories we have told ourselves over the years. But Nature always has a trick up her sleeve, and there are things yet discovered which would shatter the edifice of our expectations and dissolve the remnants. We’re living on a small boat in a deep, dark sea.

I don’t think so. But the uncertainty principle implies that there can never be exactly zero waves 🤔

You bring up some interesting points there, and your intuition is in the right place. The phrase “mass on a spring” is just an analogy to introduce the quantum harmonic oscillator, because the spring also has a potential which goes with displacement from equilibrium squared, but as with all analogies its relevance is limited and it can’t be stretched too far. The quantum world is very strange. You’re right that it’s not continuous like our macroscopic world. But then again, our macroscopic world isn’t exactly continuous either, because it arises from the quantum world. We’re just in the statistical limit.

Good question! :) And I agree, if it doesn’t tie back into your intuition, then that’s something to think about! In a condensed matter physics class in grad school, I calculated the bulk modulus of a few kinds of metal, from quantum ab initio simulations over periodic lattices. What I did was to set the spacing between atoms at a variety of distances around their equilibrium value (the lattice parameter), and calculated the energy of each configuration. Lattices which were compressed or stretched relative to their equilibrium scale had higher energy, which was quadratic for small stretches and compressions (quadratic potential = spring-like effect). The bulk modulus of the material is related to the second derivative of that potential, i.e. to the spring constant of the material. My calculated values were only off by 1-2% of the values reported by experiments which pull on samples of these metals to see how stiff they are. So in my own life, I have seen how these equations make contact with reality, and if I hadn’t yet seen that, then I would be skeptical of them too. I encourage that skepticism, as long as you keep at it and test your ideas. When I tested my ideas, I found that mainstream physics provides the best framework for understanding the physics of the world around me. Nowadays I work with the diffraction of x-rays and high energy electrons at work, designing software to analyze these signals and use them for process control for the growth of superconducting wire, and I can’t imagine understanding any of this without quantum mechanics. But at the end of the day my code works and we make good product. The proof is in the pudding! I agree wholeheartedly that physicists do not do a good enough job of communicating their ideas. To be fair though, it’s not easy! The world is nothing like what we think it should be, when you get right down to it. At the bottom of things it’s an existential mystery. But the physicists have a better working map than anyone else.

@@RichBehiel And yet no answer of how a pull works. Please answer the question or admit you do not know and that any theory that uses a pull is incorrect because pulls can not exist. I guess you somewhat have admitted that you don't know because "At the bottom of things it’s an existential mystery.". But no...it really isn't a mystery. This has nothing to do with intuition or bad explanations. Further, I am not skeptical but 100% sure the theories you are using are not correct. They merely work because of measured values and derived equations that match the measurements. Further, 1% is a rather large error...which could have several causes ranging from bad theory to equipment accuracy/precision. Sounds like your job is really an engineering job based on current theory. Which I admit will work. Newton's equations worked too, to a point, but then something did not work quite right. But if you were a physicist trying to know how things actually worked I could help you.

The 1-2% error was likely due to the fact that I modeled a perfect lattice, and didn’t factor in things like grain boundaries, dislocations, stacking faults, etc. Though I think 1-2% is not too bad, since I did this for aluminum, copper, iron, and nickel and got similar accuracy across all four metals with different lattices and electron structures. To me that suggests a connection between quantum mechanics and the stiffness of those metals, so I used that as an example of how these concepts can map onto the concept of pulling on something. You are correct to point out that my response was not focused on pulling per se; it was an anecdote of how the quantum world can map to the everyday world of squishing and pulling on materials. So if you’ll give me another chance to address your question more directly, I would first start by defining what a “pull” is. I assume you’re talking about when a material thing draws another material thing towards it via direct contact. In that case, a pull arises from short-range coulomb forces. As two atoms approach each other, their electrons will begin to repel. This is why you can’t actually touch anything, but can only get close enough that static electricity kicks in (the Coulomb force) to repel your hand away from the thing. I use an atomic force microscope at work which operates on this principle, by vibrating an atomically sharp tip near a surface and converting the frequency response to a height map. Let me know if you would like to argue about whether it’s valid to think of contact forces as short-range electron repulsion, or if you agree with that part. Then, if we’re on the same page about that, you can pull something by wrapping your fingers around it, thereby pushing it toward you. So a pull is secretly a push. Do you agree? Naturally you might ask how a material can sustain the state of tension required to pull an object. There are a few ways that can happen. Molecular bonds between atoms, van der waals forces between atoms and molecules, even surface tension in some contrived scenarios. There are ways in which matter sticks to matter, otherwise there would be nothing but atomic dust. So in summary, a pull is a push mediated by some arm, biological or mechanical, which is able to grab onto something and pull it. That’s all in alignment with mainstream physics, whether you want to think in terms of coulomb interactions or go really deep and get into quantum electrodynamics. Please let me know if this has still not addressed your question, and if so in what way.

@@RichBehiel Hey thanks for the reply. Almost all other physicists do not respond. But to be more precise in my definition I am referring to a non-contact force such as gravity, magnetic fields, and the like. So the Coulomb force, therefore, is a contact force involving a magnetic field (you would say electro-magnetic, but electrons do not exist and are instead magnetic flows generated by the nucleons of an atom.) The idea of "charges" breaks down if you believe they are pulls. In the case of short-distance atomic forces, this is more obvious. So to give you an idea of math linked to physical objects I'll describe how gravity works. Not how it might work but how it actually works mechanically. Then I can support this idea with answers to why the red-shift occurs, why black holes are black, where the CMB is coming from, why gamma bursts occur as stuff is sucked into black holes, the cause of time dilation (gravitationally and via speed) and many more things. But first an example solution for gravity. Given two masses M and N the "attractive" force is from a field of particles that have previously been called Ether or Aether etc. This ether is simply very tiny specs of matter that were created by massive suns (A.K.A black holes). It creates the ether by crushing nucleons into their elementary parts. The size of these particles is unknown but can be determined by experiment. For the time being this information is not important. The ether particles fill all of space. Even between and within atomic parts and are traveling in random directions. They travel independently through space and only change direction and speed on collision. They rarely hit one another because of the large (when compared to an ether particle) empty space around each individual ether particle. Gravitation is essentially a shadowing effect within an ether field. Now math. We have masses M and N. M has m nucleons within it. N has n nucleons within it. (BTW a nucleon is a proton or neutron - some people don't use the term.) Every nucleon in M will cast a shadow pair with every nucleon in N. So we have a relationship of m * n shadow pairs. Each shadow pair removes momentum from each nucleon's side that faces its paired nucleon. The shadows being empty space offer no resistance to intruding ether particles and the force drop from the shadow is reduced as the shadow collapses. The rate of this collapse is reduced by an inverse square law. The force removed by each shadow is G. So each shadows force contribution is G/r^2. There are m * n shadows so that makes F = (m*n) * G/r^2. This is Newton's law of gravitation but with reasons behind each term. But but but....Newton's gravity is instant. Yes but this model is not. How fast doe an ether particle travel? C of course. So the shadows in space start to look like space being warped. So why is this theory better than others? For one, it will not work at very large distances and probably accounts for the differences we see at large distances. I lack the computing power to run the simulation. (I think Earth actually lacks the computing power.) If you want the answers to the other things I mention just specify which and I'll explain it. It is all connected so each explanation depends on an Ether universe. If you want to model this with software you probably already have I'd be happy for feedback...either math based or even if you just don't think that it could possibly be that way. I've had people say it can't work this way and when pressed to explain why they couldn't. Even asked them if it made sense to them...it did...still didn't believe. There is still much I'm leaving to you to understand, so if you are wondering "how does X work?" please ask.

@@buddysnackit1758 thanks for your thorough response, sincerely. The picture you are describing is effectively LeSage’s theory of gravity, right? That the force of gravity is due to shadowing of the bombardment of corpuscles? Please let me know if your picture differs substantively from the LeSage model, otherwise I’ll continue with that assumption. One problem with the LeSagian view, which I’ve personally entertained for a while, is that in addition to gravity, it would lead to a drag force in the direction opposite the body’s motion. So if we say that the earth and the moon are pushed together by some kind of particle, then we would expect a drag force from that particle to also impart some drag on earth, and cause our planet to spiral into the sun. That’s the argument that persuaded me away from the LeSagian view. If you’ve got a rebuttal against that, I would genuinely love to hear it. Now, as for gravity… Newton’s inverse square law is more elegantly cast as the statement that the Laplacian of the gravitational potential is proportional to the density distribution. This is mathematically equivalent, and not too dissimilar from Einstein’s theory of general relativity, which is in essence the statement that excess curvature of spacetime is proportional to mass/energy density. Both theories have one and the same major unresolved mystery: where the hell does gravity come from? And why is it so weak, relative to the other known forces? I assure you, my friend, this problem has not been swept under the rug. It’s known as the hierarchy problem, and it’s a major ongoing problem in physics. I wish I knew the answer to it! Every physicist dreams of solving this problem. Physicists are generally much too cocky, that’s for sure. But the community as a whole is well-grounded by the constraints of empiricism. And mysteries still abound. You seem like someone who has a passion for uncovering the unknown. I hope you will stand on the shoulders of giants, and see father than anyone has seen before. But it is a tough climb to get up there. Take it from a fellow traveler 😉