What is quantum mechanics? Explain in simple terms.

Quantum mechanics is a branch of physics that studies the behavior of very small particles, such as atoms and subatomic particles . It is a theory that describes the nature of particles and their interactions. In simple terms, quantum mechanics introduces concepts that are different from classical physics, where objects have definite positions and velocities .

One key aspect of quantum mechanics is the uncertainty principle, which states that certain pairs of properties, such as position and momentum, cannot both be precisely known at the same time . This means that the more accurately we try to measure one property, the less accurately we can know the other property .

Quantum mechanics also introduces the concept of wave-particle duality, where particles can exhibit both wave-like and particle-like behavior . This means that particles can exist in multiple states simultaneously, known as superposition, and can also undergo interference, similar to waves .

Quantum mechanics has been very successful in explaining the behavior of microscopic systems and is used in various technological applications, such as in the development of quantum computers . However, quantum mechanics can be challenging to understand fully because it involves complex mathematical formalisms and concepts that differ from our everyday classical experience .

If you are interested in learning more about quantum mechanics, there are various resources available, including books and online materials. Sean Carroll's book "Something Deeply Hidden: Quantum Worlds and the Emergence of Spacetime" provides an in-depth exploration of quantum mechanics . Additionally, listening to podcasts or watching videos by physicists can help to gain a better understanding of the subject .

Please note that the explanation provided here is a simplified overview of quantum mechanics, and there is much more to explore and learn about this fascinating field.

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Sean Carroll: Yeah, there are absolutely experimental implications for measuring the same particle and that in fact that's where the implications are most obvious, most dramatic. If you take literally the idea that you're measuring the position of a particle and you measure it very accurately, then after the measurement, you know strictly nothing about its momentum because a position, a localized quantum state, sorry, let me say it this way, a quantum state that is localized in position is completely delocalized in momentum and vice versa. The states that have small variations in position and momenta simultaneously are a different kind of state. You cannot have a small variation in one Um... very small variation in one and also very small variation in the other. So the Heisenberg Uncertainty Principle is a statement about how quantum states relate to observables. It's actually not a statement about measurement or experimental results at all. It is a statement about how quantum states relate to observables. And we can make the statement very very crisp and clear if you think about spins instead of position and momentum. So just like position and momentum are complementary variables, in the Heisenberg Uncertainty Principle, so are the spin in the z-direction and the spin in the x-direction. So it's kind of hard to visualize sometimes or to grasp, but in Hilbert space, in the space of all possible quantum states, a spin in one direction, let's say spin up, is perpendicular as a vector in Hilbert space to spin in the other direction, which is spin down.

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Sean Carroll: That's just not true anymore in quantum mechanics. And like physicists always do, we use the same word for a rather different concept. So in quantum mechanics, of course you can have a circumstance where you literally have a bunch of atoms moving around with different velocities and then you could attribute a temperature to it, you're just describing it quantum mechanically, that's fine, but it's a special case of a more general thing that can happen. The more general thing that can happen is that you have a, like I said, what we call a density matrix or a mixed state. What does that mean? Ordinarily in quantum mechanics, when you hear a little bit about quantum mechanics, you're talking about wave functions, right? The electron is not just a point particle with a location, it's a wave function all spread out. Well, you also probably have heard that in statistical mechanics, even though we know that there really are, well, we would have known if classical mechanics had been true, that there really were particles with definite positions in momenta, in statistical mechanics we recognize that we don't know. the exact position and momentum of everything, and therefore we have a probability distribution over all those things. So you might think that in quantum mechanics you could have a probability distribution over wave functions, right? Maybe the analogy is you don't know what the wave function of the electrons is. And that's almost true, but there's a technical complication because wave functions are vectors. You can add them and subtract them. You can't add and subtract positions and momenta in classical mechanics.

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Sean Carroll: So that's the Newtonian classical universe. How is quantum mechanics different? You might think that quantum mechanics adds a certain fuzziness to the classical picture, right? Quantum mechanics says that when we predict the outcome of an observation, we don't know exactly what we're going to see. Everyone agrees on that. That is definitely part of the quantum mechanical story. So unlike classical mechanics, if I knew exactly the position and momentum of a particle, I can predict what's going to happen next, and I could tell you what measurement outcome I would get. In quantum mechanics, I cannot, even in principle, make absolutely reliable predictions for any observation I want to make. So if you started your brain With a classical intuitive perspective, there are particles, they're moving in some way. You might think about quantum mechanics as just adding some uncertainty, some fuzziness to that. And that's a lot, that's a big part of the popular picture of what quantum mechanics really says. But it's actually deeper than that. And the many worlds perspective, this is where it comes in. In quantum mechanics, the position and velocity of a particle, it's not just that you don't know what they are, or you can't predict what you will measure them to be. It's that they don't exist. Positions and velocities are not what quantify the state of a particle in quantum mechanics.

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(someone): Oh boy. Um, quantum mechanics is a phenomenally successful physical theory. Uh, it is, uh, it is the best theory we have to account for the behavior and predict the behavior of very small objects, but also things that are constituted of very small objects, which is basically everything. And so, so quantum mechanics. So one, one definition of quantum mechanics is it is, well, hold on quantum physics, right?

Sean Carroll: I don't care.

(someone): Quantum theory, quantum physics, quantum mechanics, those mean the same thing. Quantum mechanics. We usually use that for the non relativistic theory, but, but, but anyway, yeah. Um, quantum physics, um, because it's this theory for tiny things and things made of tiny things, which is everything. Um, it's basically our best physical theory of everything with a little asterisk on it. And the asterisk is general relativity, which is a whole other thing. But basically anything that where gravity doesn't matter that much, Uh, where that much means matters less than it does, I don't know, inside of the heart of a dense star or near a black hole. Um, quantum mechanics is, is fantastic. Okay.

Sean Carroll: But what does it actually say? Let's imagine someone is listening who just doesn't know about uncertainty or wave functions or anything like that. Yeah.

(someone): Um, Well, that's the question, right?

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Sean Carroll: You know, F equals ma, there's positions and velocities, and on the basis of the forces acting on a system, you can predict what's going to happen next. Newtonian mechanics is a really good approximation to the world we see. It only fails to be a good approximation when we look at microscopic systems like individual elementary particles. And the point is that in classical mechanics, in Newtonian mechanics, if you had a particle, what is that? That's an object that is point-like, right? It just has a certain location in space, and Isaac Newton would say it also has a velocity through space. And if you tell me the position and the velocity of a particle, you're telling me the state. You're telling me all the information I need to know to predict what will happen next. If you know the other forces acting on the particle caused by other particles out there or fields or whatever, you can predict the entire future of the universe. This is called the clockwork universe. Laplace's demon, if you've ever heard of that concept. Laplace, Pierre-Simon Laplace, the French mathematician and physicist, said that if there were a demon that knew everything about the current state of the universe, he could predict the future and retrodict the past with perfect accuracy. He didn't actually say demon, he said vast intelligence, but you know what we mean. So that's the Newtonian classical universe. How is quantum mechanics different?

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Sean Carroll: In classical mechanics, the fundamental equations of motion are often nonlinear. In quantum mechanics, the Schrodinger equation itself is linear. Okay the Schrodinger equation has a wave function as its fundamental variable and the wave function appears to the power one on both sides of the Schrodinger equation. One side is the Hamiltonian which is asking how much energy is there and it says h operating on one power of psi, the wave function. The other side is the time derivative. How fast is the wave function changing? Acting on one power of the wave function. So the kind of evolution that is chaotic in classical systems naively doesn't appear at all in quantum mechanical systems. The wave function itself does not evolve chaotically, full stop. But of course in quantum mechanics we have a classical limit, right? And this is where it becomes a subject that requires a study and papers written about it and why there's a whole thing called quantum chaos. Because even though the wave function evolves linearly and non-chaotically, the classical observables, the classical limit of that wave function, you know, gives rise to classical mechanics. The quantum, the triple pendulum that is a chaotic system is a classical limit of some quantum system, right? So there has to be some classical limit of this quantum system that does behave non-linearly and chaotically, okay? So there is an interesting thing to talk about. which is this relationship between the underlying behavior of the wave function and the emergent classical world, and the emergence of nonlinearities in that classical world.

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(someone): this thing that's called the Copenhagen interpretation, but also goes by other names like the Orthodox interpretation or whatever. Um, I am not a fan to put it mildly. Uh, the other thing though that I think makes my book somewhat unusual as a history of quantum mechanics is, you know, there are a lot of other popular histories of quantum mechanics out there and most of them sort of start in say 1900 with Max Planck and then they say, you know, Max Planck found the black body radiation law and that was the beginning of the quantum revolution. And then they sort of move forward from that. with Einstein and the photoelectric effect in bore and his model of the atom. And then, you know, they, they end with, and then, uh, Heisenberg and Schrodinger developed, you know, fully fledged modern quantum mechanics and everybody lived happily ever after. And then maybe, you know, Einstein and bore got into a fight, but bore one. And then, you know, 30 years later, John Bell did a thing and that's the end of the book. Right. My book, kind of inverts that structure. I only have a chapter or two at the beginning about the very early days of quantum mechanics. And most of the book focuses on stuff that happened, uh, from say 1945 onward, which is fascinating.

Sean Carroll: So just, just so there's, if there's any people out there in the audience who, I think quantum mechanics is fun, but history is not.

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(someone): And what then happens is the GeoD6, essentially in branchial space, the shortest paths in this branchial space, are also deflected by the presence of energy and momentum. Okay? So this is where it gets... it's really fun, because the... the sort of... the... you know, the sort of a good core formulation of quantum mechanics is the Feynman path integral. And the Feynman path integral says, you look at all these possible paths of history, and you say the phase, the quantum phase associated with the path of history is, you know, e to the is over h bar, where s is this quantity, the action, which is an integral of Lagrangian density, which is essentially the aversion of the amount of energy momentum in a particular region of something. Now, in our world, that's a particular region of branchial space. Effectively, the density of energy momentum in branchial space leads to the deflection of Jd6 in branchial space. And so what does that mean? If the position in branchial space is the phase of the quantum amplitude, that means the deflection of a Jd6 is a change of phase. And the Feynman path integral is precisely telling you that the presence of energy momentum leads to a change of phase in the quantum amplitude. So what it's saying is, the bigger picture here is that the same mathematics that leads to the Einstein equations in space-time leads to the Feynman path integral in multi-way graphs and branchial space and so on.

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Sean Carroll: This is quantum mechanics we're talking about. It's not classical mechanics. Quantum mechanics, in its basic formulation with the Schrodinger equation, etc., is not discrete, okay? It is just wrong to think that the fact that the world is quantum somehow implies some discreteness to it. I ramble on about this in the upcoming book, the upcoming volume two of Biggest Ideas in the Universe on Quanta and Fields, because in something like an atom, where you have discrete energy levels for the electron, that's not because the laws of physics are discrete. The laws of physics are smooth. It's a wave equation, the Schrodinger equation. It's that the set of solutions to that equation in that particular setup is discrete. So quantum mechanics itself, if you just do regular quantum mechanics, nature is not discrete. Full stop. I recently wrote a paper, very recently, pointing out that in certain special cases, if things line up just right, you can take quantum mechanics and you can discretize it. You can discretize the evolution of the wave function in Hilbert space so that it is truly discrete. There are some cosmological issues that come up there, but otherwise it's fairly plausible that that could be the real world. But we have no reason to think it's the real world, it was just sort of an intellectual exercise in pointing out this is a possibility. that we can consider.

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Sean Carroll: I do think they should say true things about the foundations of quantum mechanics. They should be correct when talking about the measurement problem or Bell's inequality or things like that. And they're often not. Okay, so I think I can try to get that better. But more importantly, very often quantum textbooks are just excuses to solve the Schrodinger equation over and over and over again. Most often for single particle systems. And the notion that particles can be entangled is mentioned and then slid by, right? So that's terrible. That to me is just doing a disservice to what makes quantum mechanics quantum mechanics. So I think that without getting philosophical or anything like that about it, I definitely want to do a lot more than the typical quantum mechanics books does to explain both the phenomenon of entanglement and the technical apparatus that goes along with thinking about it, decoherence and density matrices and things like that. And this is not just because I think it's interesting and true, although there's that, it's also relevant to how we use quantum mechanics in the modern world, whether we are grown-up professors, that are now doing research in quantum mechanics, or whether you go into industry to build a quantum computer. Quantum computers rely on entanglement, and it turns out that it's not that hard and enormous fun to teach people the basics of quantum information and quantum computing. That's 100% appropriate for an intro quantum course. So ideally, if it all works out, there will be stuff like that in my quantum textbook, but it will be optional.

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Sean Carroll: But still, there's no quest to unify classical mechanics and quantum mechanics. Classical mechanics is a proper subset of quantum mechanics. So – but one other thing to say about that is that people sometimes say quantum mechanics begins to be relevant when systems are small or something like that. And that's not exactly precisely rigorously right. The right thing to say is quantum mechanics is always true. But there is also a subset of circumstances under which classical mechanics is a good approximation. And there's a subset of circumstances under which classical mechanics is not a good approximation. And you need to invoke those parts of quantum mechanics that are not present in the classical limit. That's really what we mean when we say that quantum mechanics becomes important in the small scale microscopic world. Brent Meeker says, in the solo episode on finding gravity within quantum mechanics, Mindscape 63, you speculated that there are only finite many degrees of freedom in a given volume. Wouldn't this imply that there are also only finitely many possible states, and so there would be a smallest non-zero probability of any possible event? So no, that's not quite right. So yes to the first half of that sentence, but not to the second half. So finitely many possible states, right, that is in fact what is implied by imagining only finite number of degrees of freedom. But think about how quantum mechanics works. States are vectors, okay?

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Sean Carroll: just by Pythagoras's theorem, right? But you want it to evolve without changing the fact that its length is one. If you think of a wave, an individual wave oscillating up and down, it can oscillate up and down, but that means it has to go through zero. If you have a real valued number that goes between minus one and plus one, it goes through zero on the way. But a complex number can oscillate from minus 1 to plus 1 without going through 0. Its magnitude can stay equal to 1 by going e to the i omega t, where omega is the frequency and t is time. So that's basically saying you go from minus 1 to i to plus 1 to minus i to minus 1, et cetera. And that turns out to be extremely useful for what actually happens in quantum mechanics for interference effects that maintain the probability rule, all that stuff. So the fact that the wave function is complex is very important in quantum mechanics, or that it's conveniently represented using complex numbers is very important, but I'm happy to just take it as true and go from there. If you prefer something else that you want to use to derive it, that's also okay. One of the things about axiomatic systems is there's often different ways to write down the fundamental axioms and derive them from each other. I'm going to group two questions together. One is from John Morgan, who says, in a prior AMA, you mentioned that should human lifetimes be extended indefinitely, you feel like you've got a good several thousand years of things to keep you occupied and interested in staying alive.

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Sean Carroll: I've been reading, listening to many of your books and books by other fellow physicists, in addition to YouTubing and Googling the crap out of every piece of info I don't understand, in an effort to really give it an honest try, because it is truly fascinating stuff. Do you have any techniques or suggestions for helping the average Jane get a firm understanding of quantum physics in her spare time? I mean this is both a super important question and a super difficult question because I feel I can say things but they're nothing other than platitudes. That's like the obvious things. There's no – I don't have any secrets in other words, okay? I don't have any like, oh, if you do this, suddenly you will understand quantum mechanics really well. I think that the only thing I will say is that There are levels of understanding, and it's important to appreciate that. So if you read a popular book on quantum mechanics, like my book, for example, Something Deeply Hidden, you will be challenged because the ideas and the concepts are very, very hard. But even if you really understand it, even if it's very, very clear and you really feel you're getting something, you're never getting the full thing because you don't have the equations, okay? That's why I'm currently working on this series of books, The Biggest Ideas in the Universe, where I do the equations. I mean, I should be I'm finishing book two right now, but I'm pretty close. I'm getting there. Don't worry.

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(someone): Okay, so, you know, the easily checkable problems and the problems that are easily solvable by a quantum computer. Now, you know, in some sense, the biggest mistake that popular writers make when they write about quantum computers you know, and this has been true for like 25 years by now, okay, is that they conflate NP with BQP, right? They write as if a quantum computer would just be a magical machine for solving NP problems. They say, well, look, a quantum computer just has to try each possible answer in a different parallel universe, you know, a different branch of the wave function, And in some sense, it is actually true that a quantum computer can do that. That's not even a hard thing to do. to program your quantum computer to create a superposition over every possible answer. The hard part is reading out the answer that you want, okay? So, you know, if I just, you know, I mean, you know this, Sean, but, you know, for the benefit of our listeners, right? If I just create an equal superposition over every possible answer to some, you know, super hard problem, like, you know, all possible keys for, you know, breaking a cryptographic code, let's say. and then I just measure it, not having done anything else, the rules of quantum mechanics tell me that what I will see will just be a random key, a random answer. Well, if I just wanted a random key, I could have picked one myself with a lot less trouble, right? I didn't need to build a quantum computer for that, okay? But then that makes it sound like

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Sean Carroll: So again, for those of you who don't know, The nice thing about a Lagrangian, a Lagrangian is an expression from which you can derive many different equations of motion at once. So you might have a theory that has different kinds of particles and fields and whatever. In good old-fashioned Newtonian mechanics, classically, you would have to separately have equations of motion. For all those pieces of your theory, they might involve interactions with each other, but they would have separate equations. The Lagrangian point of view lets you combine all of the things in your theory into one expression, the Lagrangian, and you integrate that over all of spacetime, and you minimize that, and those are your equations of motion. And the great thing about this single expression is that it not only has equations of motion for everything, but that it's very easy to implement the symmetries. You just look at the Lagrangian as a whole, rather than all the separate equations of motion, okay? So there's various conveniences involved in doing things the Lagrangian way. But at the classical level, there's no deep difference between the Lagrangian and the equations of motion. You just use one to get the other. quantum mechanically there's something extra going on which is not mentioned in this question which is that you're not classical anymore you're quantum mechanical and the neither the Lagrangian nor the equations of motion are the whole story. One nice way of thinking about that is to think about the path integral You know, when you have a regular integral of a function f of x, you write integral f of x dx, right?

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(someone): He said, physics is about describing what's in the world. Uh, and now you're telling me, no, physics is about predicting the outcomes of experiments. what's an experiment, what, you know, if it's really only about predicting the outcomes of experiments, then how are we supposed to use it to explain things that happen in the natural world when we're not around, which people were doing already at the time, you know, quantum mechanics has always been used to do more than just predict the outcomes of experiments. It has also been used to explain all kinds of very, you know, interesting natural phenomena ranging from stuff as basic as, you know, why does the sun shine to, you know, why is it that, uh, silicon behaves in this funny way where it's not really an insulator and it's not really a conductor. It's a semiconductor, right? Yeah. and that lets us build computers. So everything from basic fundamental facts about the world around us, like why I can't pass my hand through the table, to all sorts of strange and wonderful modern technology like LEDs, or most of this recording equipment.

Sean Carroll: And I thought, you know, you mentioned this very quickly, but I want to dwell on this philosophical leap made by the Copenhagen interpretation. So if you really say that the job of your physical theory in this case is to simply predict the outcomes of experiments. And they said this, the advocates of Copenhagen, as far as I can tell, were sometimes more explicit about this than others. They were never, completely on the same page, but okay, there's a version of it which really says that.

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Sean Carroll: There's a whole subject called quantum chaos, and here's the reason why it's a non-trivial question. Because in classical chaos, classical systems are often chaotic. Not always, there are non-chaotic classical systems, but you don't need to work very hard to get a classical system that is chaotic. Like you put a couple of pendulums and hang them from each other and you get a chaotic motion for the triple pendulum as it is called. But quantum systems are a little bit different because the reason why classical systems can be chaotic has to do with the non-linearity of the interactions between them. What I mean by that is linear is when everything just depends on the variable you're looking at to the power one, right? X to the first power. As soon as something depends on higher powers, you know, x squared, etc., that's nonlinear. And the difference is that when you have a linear system, if you perturb it a little bit, it moves a little bit, and that's it. A small perturbation leads to a small difference in what is happening. But when you have a nonlinear system, a small perturbation can grow very rapidly, and that's where you get chaos. That's where you get sensitive dependence on initial conditions, because the The tiny perturbation you give it can feed on itself and grow very, very quickly. In classical mechanics, the fundamental equations of motion are often nonlinear. In quantum mechanics, the Schrodinger equation itself is linear.

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(someone): I mean, they're either true or they're false. What's this hard or easy? What does it mean? My teacher, my high school teacher, told me that the four-column map theorem is thought to be very, very hard. And I said, what do you mean hard? It's either true or it's not true. How do you make sense out of this idea of hard or easy? And I didn't know. Then I later learned out that there were theorems that were so hard that they were true, but they were so hard that they couldn't be proved, period. That was Mr. Gödel's invention. And I got very curious. What does it mean to say that a theorem is harder than another theorem? And it turns out what it means is that it's more complex. What does more complex mean? that the minimal number of steps to be able to prove it, not the number of steps that some mathematician might have used, but the absolute minimal number of steps, logical operations, starting with the postulates and using logical operations, and, or, and so forth, that the complexity of a theorem is what the minimal number of steps to prove that theorem is. And it's a hard number. Given a theorem, you can say what the complexity of it is. Hard theorems are more complex than easy theorems.

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Sean Carroll: That phenomenon is why particles arise in quantum mechanics and in quantum field theory. So in particular in The original word quantum came to be because we were seeing discrete packets of energy released by atoms. So I'm not going to get the history exactly right here, but you know that electrons can move to higher and lower energy levels in atoms, and they emit or absorb photons while doing it, and only certain photons are emitted or absorb certain energies of those photons. And the reason why is because there's a discrete set of orbitals that the electrons can be in. If the electron were not in the orbit of an atom, it could have any energy at all. But once it falls into the potential field of the electromagnetic field of the nucleus of the atom, then it has a boundary condition. Namely, its wave function is near the nucleus and goes to zero far away, right? And that boundary condition is enough to say there's a discrete set of possibilities exactly with the vibrating violin string or whatever. In quantum field theory, it's a little bit different, but the concepts are exactly analogous. So in quantum field theory, you do what we did before. You consider a certain wavelength of a mode of a quantum field, and that mode, you know, that wavelength of the field can vibrate up and down, right? Its amplitude goes up and down. And you say, well, I don't see any boundary conditions there. It can vibrate up and down as much as it wants, right?

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Sean Carroll: Well, that's a difficult question. Part of it is easy, which is that we don't only experience the world in the classical sense. If you've ever heard a Geiger counter, much less done a quantum experiment of any sort, then you have absolutely experienced the world in a quantum mechanical sense. But I get what the question is asking about. There's absolutely a classical regime in which classical mechanics is a very good approximation to the underlying quantum dynamics. Why is that true? I think that we actually don't give the right or at least a complete answer to this when we teach quantum mechanics, when we talk about quantum mechanics. There are two things that we talk about. One is what is called Ehrenfest's theorem. If you have a situation where the pushes and pulls on a system, the forces acting on it, very slowly over space, right? Like if you're the Earth and you're orbiting the Sun, the Sun's gravitational field does not change dramatically from place to place. So if that's true and you are heavy like the Earth is, for example, then your quantum mechanical observables, your position and your momentum, do a very, very, very, very good approximation of obeying the classical equations of motion. So there's a very well-defined sense in which quantum mechanics gives you back classical mechanics in a well-defined regime.

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Sean Carroll: And then Hamilton figured out a way to do classical mechanics based on the energy. So the whole idea of Hamiltonian mechanics is just a rewrite of Newtonian mechanics. It's not a new physical theory, but it's a new language in which to talk about Newtonian mechanics, where the energy of the system is the primary thing. So Hamiltonian mechanics starts with an equation for energy as a function of position and momentum. And from that one equation, it gives you a recipe for deriving all of the equations of motion in Newtonian mechanics, okay? And we call that Hamiltonian mechanics, even though it's physically equivalent to Newtonian mechanics. By the way, I have a book coming out that will explain all of this. This is all talked about, including Hamiltonian mechanics and Lagrangian mechanics in The Biggest Ideas in the Universe, coming out in September. But in quantum mechanics, the point is that we have the Schrodinger equation, yes, and the Schrodinger equation certainly features the Hamiltonian, which is that thing invented by Hamilton, the way to express the energy of the system in terms of its velocities and its momentum. Schrodinger turns that into an operator on wave functions, but it's the same basic idea. So I would say that energy doesn't play any more important role in quantum mechanics than it did in classical mechanics. It's just that you're using a useful accounting trick to write a theory in a way that uses energy in a very straightforward, important, central way.

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Sean Carroll: I know that the Uncertainty Principle is much more general than that, and that it is fundamental in quantum physics, so is the above explanation wrong, or is it correct but just an illustration? you Well, it is correct, it is not wrong, but it is not an explanation of the uncertainty principle. It is one way of thinking about one of the consequences of the uncertainty principle. But the uncertainty principle itself does not make any reference to measurements or observations, certainly doesn't make any reference at all to the mechanism by which we measure things, okay? The uncertainty principle is just a statement about what possible quantum states exist. not a statement about what happens when we measure them. As a consequence of the uncertainty principle and the nature of quantum mechanics, you can derive things like, if you were to measure the variables like position and momentum, there is no quantum state for which you could measure them exactly or you could predict exactly what you're going to get for both such variables, position and momentum. But even if you chose never to measure it, it would still be true that expressed as a superposition of positions or momenta, there are no quantum states that are exactly localized in position and momentum simultaneously. And in fact, the more localized any one particular state is with respect to position, the less localized it is with respect to momentum, and vice versa. That's the uncertainty principle. And it's really just a matter of choosing different coordinates, right? I mean, the explanation that I use in Something Deeply Hidden, and probably elsewhere, is just think about a two-dimensional plane, right, with X and Y coordinates.

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