Tuesday, 6 October 2020

definition - Quantum made easy: so what *is* quantum mechanics all about?




Being a physics grad student, I got used to the weird concepts behind quantum mechanics (used to doesn't mean I fully understand it though). What I mean is that I'm not surprised anymore by the fact that a quantum system might be in a superposition state, that a particle in a potential well has a discrete spectrum of energy, that some pairs of observables cannot both be measured with arbitrary precision, etc.


However, the problem arises when I'm talking about my work to my (non-physicists) friends and family. They are not used to think "quantum mechanically", so when I'm saying that Schrödinger's cat is really, simultaneously, both "alive" and "dead", they don't get my point as it just seems crazy. So I have been thinking for a while of what would be the easiest way to introduce them to the core principles of QM. (EDIT: This part seems to have upset people; I don't want to go into the details of distinguishing "simultaneously alive and dead until you measure" VS "superposition of alive and dead states", I mainly want them to understand that the outcome of the measure is indeterminate)




When they ask me "So what is QM all about?", I'd like to give them a short answer, a simple postulate/principle that, once accepted, makes most key features of QM seem more natural.



In my opinion, some of the "weird" ideas people find most difficult to accept (due to some popular pseudoscience) and/or that are central to QM are:



  • Quantum superposition

  • Complex amplitude/phase (as it leads to interference)

  • Uncertainty principle

  • Entanglement

  • Quantization of physical quantities/particles



My problem is that I can't explain how these ideas are related. If my friends accept that particles can be in two states at once, why couldn't they know both its position and momentum? Or if they accept that an electron has a super abstract property called a "complex phase", why should it lead to a discrete spectrum of energy?


Of course, I could just tell them that they need to accept it all at once, but then the theory as a whole becomes hard to swallow. In my opinion, QM sometimes suffers from a bad reputation exactly because it requires to put all our classical intuitions to scrap. While actually, I think it could make perfect sense even for a non-physicist given only 1 or 2 "postulates" above. I tried looking back at my own learning of QM with little success since undergrad students are often precisely bombarded with poorly justified postulates until they just stop questioning them.


A few more points:



  1. I don't want to get started on mathematical definitions, so nothing like "Quantum systems are in an infinite-dimensional Hilbert space" as it doesn't mean anything to my relatives. That includes pretty much all of the actual postulates of QM. I want my explanation to be more intuitive.

  2. I'd like to avoid any mumbo jumbo related to "wave-particle duality". I think (and I know I'm not the only one) that most explanations regarding this are inacurrate and actually inefficient at giving a good description of quantum behaviour, since most non-physicists aren't familiar with the properties of a classical wave anyway. And it's mostly used when talking about the position of a quantum particle; I've never heard someone say that spin "behaves like a wave" or "like a particle". I'm looking for something more fundamental, more universal.

  3. That being said, I think it's very important to include a discussion about the relative phase between two superpositioned states as it leads to quantum interference (perhaps the only "wave-like" feature that applies throughout QM). If there were only quantum superposition, quantum computing would be pretty much useless: even if we perform one billion calculations simultaneously, when measuring the system we would get a single random value. Our goal is to manipulate the system, to create destructive interference between the unwanted results in order to increase the probability of measuring the desired value. And that requires the notion of quantum phase.

  4. Uncertainty principle might be the toughest challenge here, since we can get the basic idea without invoking QM at all. First, even in classical mechanics, it's impossible to measure something with an arbitrarily big precision simply because of instrumental limitations. Second of all, it makes sense to get different results if we measure the position before or after the momentum ($[x,p]\neq 0$) since we interact with the system when making the first observation, leaving it in a different state afterward. How could I explain that there are more fundamental implications from QM?

  5. I wasn't sure about entanglement and quantization. The weirdness of entanglement doesn't come from how we achieve it, but from how we interpret it, which is still an unresolved issue as far as I know (Many-world? Copenhagen interpretation? etc). As for the quantization of particle properties, it doesn't seem like an intrinsic feature of QM, more of a consequence of boundary conditions. A free electron in a vaccum has a continuous energy spectrum, it's only when we put it in a potential well that some states are "forbidden" (not exactly forbidden of course; even Bohr explained the hydrogen atom in a semi-classical model using destructive interference between the state of the electron). Then, there is also the quantization of matter itself (photons, electrons, quarks, etc.) which is another story. But I think most people are familiar with the idea that matter is made of atoms, so it isn't a far stretch to include light as well. My point is that these two principles don't appear quite as essential as the other ones to understand QM.



I haven't found a satisfying answer in the similar posts of this forum (here, here or here for instance) since I'm more interested in the overall description of QM - not just of wavefunctions or superposition - and I don't want to get into the motivations behind the theory. Also, I'd like to give an accurate, meaningful answer, not one that leads directly to pseudoscientific interpretations of QM ("Particles are waves, and vice versa", "Everything not forbidden is mandatory", "Consciousness affects reality"; that kind of thing).




EDIT: Just to clarify my question perhaps, here what I would say concerning special relativity (SR).


The basic assumption of SR is that everyone always sees light traveling at the same speed, no matter what circumstances. So far so good.


Now imagine that I'm in a train going at 50 km/h and that I throw a ball in front of me at 10 km/h. From my perspective, the ball is going at 10 km/h. But for you, standing on the ground, the ball is going at 50+10 = 60 km/h with respect to you. It all makes perfect sense since you have to consider the velocity of the train. Good.


Now, let's replace the ball with a photon. Oh oh, according to our previous assumption, we will both see the photon travelling at the same speed, even though I am moving at 50 km/h. We cannot add my speed to your measurement as we did earlier. This implies that we both perceive time and space differently. Boom. And from this you can get to pretty much all of SR (of course, one should also mention that $c$ is the maximum speed).


This simple assumption implies the core of the theory. What would be the equivalent in QM, the little assumption that, when accepted, implies most of the theory?


@Emilio Pisanty I saw the post in your comment (it's one of the links at the end of mine). As I said, I'm not interested in why we need QM, I want to know/explain what fundamental assumption is necessary to build the theory. However it's true both questions are linked in some ways I guess.




EDIT 2: Again, I'm not interested in WHY we need QM. It's pretty obvious we'll turn to the theory that best describes our world. I want to explain WHAT is QM, basically its weird main features. But QM has a lot of weirdness, so if I simply describe the above phenomena one by one, it doesn't reflect the essence of the theory as a whole, it just looks like a "patchwork" theory where everyone added its magical part.



And I KNOW all about the QM postulates, but those are more a convention of how to do QM, a mathematical formalism, (we'll use hermitian matrix for observables, we'll use wavefunction on a complex Hilbert space to represent a particle, and so on) rather than a description of what QM is about, what happens in the "quantum world", what phenomena it correctly predicts, no matter how counterintuitive to our classical minds.




EDIT 3: Thank you all for your contributions. I haven't chosen an answer yet, I'm curious to see if someone else will give it a try.


There's a little point I want to stress: many answers below give a good description of the weird concepts of QM, but my goal is to wrap it all up as much as possible, to give an explanation where most of these are connected with as little assumptions as possible. So far @knzhou's answer is the closest to what I'm looking for, but I'm not entirely satisfied (even though there's probably no answer good enough). Most answers are very close to the wave-particle duality - which I'm not a fan of because people understand this as "Electrons are particles on week days and waves during the weekend". However perhaps it's unavoidable as it truly does include most of the phenomena listed above. Then I thought perhaps if we were talking about quantum fields rather than "waves of matter", the discussion would be less likely to lead to misinterpretation?


Also, for all those who said "Well QM is just incomprehensible for lay people" I think that's giving up too easily and it feels a little dishonest. QM is not a secret society for a special elite. I think it's important that researchers explain what is their goal and how they intend to achieve it to the general public. I think the 2nd question physicists are asked most often is "Why should my tax money pay for your work?". Do you really think an answer such as "Well... you couldn't understand it. Just trust me." is satisfying?


QM is counterintuitive, but well explained it can make sense, and I'm just looking for the easiest way to make people realize this without simply saying "Well forget anything you know". Also I think it is a theory that would be much easier to accept if people were more often exposed to these ideas. After all, children and high school students learn that the world is made of atoms and they accept it even though they cannot see atoms. Why then couldn't QM be part of the common scientific culture? It's true it may sound ridiculous at first to hear about superposition and indeterminate states, but it isn't by giving up and by saying "Well you need a degree to understand it" that we are going to change it and make it more appreciable.



Answer



You're asking a tough question! I try to explain QM to nonphysicists all the time, and it's really hard; here are a few of the accessible explanations I've found.


What I'll say below is not logically airtight and is even circular. This is actually a good thing: motivation for a physical theory must be circular, because the real justification is from experiment. If we could prove a theory right by just talking about it, we wouldn't have to do experiments. Similarly, you can never prove to a QM skeptic that QM must be right, you can only show them how QM is the simplest way to explain the data.


Quantization $\rightarrow$ Superposition



If I had to pick one thing that led to as much of QM as possible, I'd say quantum particles can exist in a superposition of the configurations that classical particles can have. But this is a profoundly unintuitive statement.


It's easier to start with photons, because they correspond classically to a field. It's intuitive that fields can superpose; for example the sound waves of two musicians playing at once superpose, adding onto each other. When a light wave hits a partially transparent mirror, it turns into a superposition of a transmitted and reflected wave, each with half the energy.


We know experimentally that light comes in little bullets called photons. So what is the state of a photon after hitting a partially transparent mirror? There are a few possibilities.



  1. The photon is split in half, with one half transmitted. This is wrong, because we observe all photons to have energy $E = hf$, never $E = hf/2$.

  2. The photon goes one way or the other. This is wrong, because we can recombine the two beams into one with a second partially transparent mirror (the would-be second beam destructively interferes). This could not happen if the photon merely bounced probabilistically; you would always get two beams.

  3. The photon goes one way or the other, but it interferes with other photons somehow. This is wrong because the effect above persists even with one photon in the apparatus at a time.

  4. The photon is something else entirely: in a superposition of the two states. It's like how you can have a superposition of waves, and it is not a logical "AND" or "OR".


Possibility (4) is the simplest that explains the data. That is, experiment tells us that we should allow particles to be in superpositions of states which are classically incompatible. Then you can extend the same reasoning to "matter" particles like electrons, by the logic that everything in the universe should play by the same rules. (Of course such arguments are much easier in hindsight and the real justification is decades of experimental confirmation.)



Superposition $\rightarrow$ Probability


Allowing superposition quickly brings in probability. Suppose we measure the position of the photon after it hits one of these partially transparent mirrors. Its state is a superposition of the two possibilities, but you only ever see one or the other -- so which you see must be probabilistically determined.


This is not a proof. It just shows that probabilistic measurement is the simplest way to explain what's going on. You can get rid of the probability with alternative formulations of QM, where the photon has an extra tag on it called a hidden variable telling it where it "really" is, but you really have to work at it. Such formulations are universally more complicated.


Superposition $\rightarrow$ Entanglement


This is an easy one. Just consider two particles, which can each have either spin up $|\uparrow \rangle$ or spin down $|\downarrow\rangle$ classically. Then the joint state of the particles, classically, is $|\uparrow \uparrow \rangle$, $|\uparrow \downarrow \rangle$, $|\downarrow \uparrow \rangle$, or $|\downarrow \downarrow \rangle$. By the superposition principle, the quantum state may be a superposition of these four states. But this immediately allows entanglement; for instance the state $$|\uparrow \uparrow \rangle + |\downarrow \downarrow \rangle$$ is entangled. The state of each individual particle is not defined, yet measurements of the two are correlated.


Complex Numbers


This is easier if you're speaking to an engineer. When we deal with classical waves, it's very convenient to "complexify", turning $\cos(\omega t)$ to $e^{i \omega t}$, and using tools like the complex Fourier transform. Both here and in QM, the complex phase is just a "clock" that keeps track of the wave's phase. It's trivial to rewrite QM without complex numbers, by just expanding all of them as two real numbers, as argued here. Interference does not require complex numbers, but it's most conveniently expressed with them.


Wave Mechanics


The second crucial fact about quantum mechanics is momentum is the gradient of phase, which by special relativity means that energy is the rate of change of phase. This can be motivated by classical mechanics, appearing explicitly in the Hamilton-Jacobi equation, but I don't know how to motivate it without any math. In any case, these give you the de Broglie relations $$E = \hbar \omega, \quad p = \hbar k$$ which are all we'll need below. This postulate plus the superposition principle gives all of basic QM.


Superposition $\rightarrow$ Uncertainty



The uncertainty principle arises because some sets of questions can't have definite answers at once. This arises even classically. For instance, the questions "are you moving north or east" and "are you moving northeast or southeast" do not have definite answers. If you were moving northeast, then your velocity is a superposition of north and east, so the first question doesn't have a definite answer. I go into more detail about this here.


The Heisenberg uncertainty principle is the specific application of this to position and momentum, and it follows because states with definite position (i.e. those that answer the question "are you here or there") are not states of definite momentum ("are you moving left or right"). As we said above, momentum is associated with the gradient of the wavefunction's phase; when a particle moves, it "corkscrews" along the direction of its phase like a rotating barber pole. So the position-definite states look like spikes, while the momentum-definite states look like infinite corkscrews. You simply can't be both.


Long ago, this was understood using the idea that "measurement disturbs the system". The point is that in classical mechanics, you can always make harmless measurements by measuring more gently. But in quantum mechanics, there really is a minimum scale; you can't measure with light "more gently" because you can't get light less intense than single photons. Then you can show that a position measurement with a photon inevitably changes the momentum by enough to preserve the Heisenberg uncertainty principle. However, I don't like this argument because the uncertainty is really inherent to the states themselves, not to the way we measure them. As long as QM holds, it cannot be improved by better measurement technology.


Superposition $\rightarrow$ Quantization


Quantization is easy to understand for classical sound waves: a plucked string can only make discrete frequencies. It is not a property of QM, but rather a property of all confined waves.


If you accept that the quantum state is a superposition of classical position states and is hence described by a wavefunction $\psi(\mathbf{x})$, and that this wavefunction obeys a reasonable equation, then it's inevitable that you get discrete "frequencies" for the same reason: you need to fit an integer number of oscillations on your string (or around the atom, etc.). This yields discrete energies by $E = \hbar \omega$.


From here follows the quantization of atomic energy levels, the lack of quantization of energy levels for a free particle, as well as the quantization of particle number in quantum field theory that allow us to talk about particles at all. That takes us full circle.


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