The previous collection of things everyone should know about quantum physics is a little meta– it’s mostly talking up the importance and relevance of the theory, and not so much about the specifics of the theory. Here’s a list of essential elements of quantum physics that everyone ought to know, at least in broad outlines:
1) Particles are waves, and vice versa. Quantum physics tells us that every object in the universe has both particle-like and wave-like properties. It’s not that everything is really waves, and just sometimes looks like particles, or that everything is made of particles that sometimes fool us into thinking they’re waves. Every object in the universe is a new kind of object– call it a “quantum particle” that has some characteristics of both particles and waves, but isn’t really either.
Quantum particles behave like particles, in that they are discrete and (in principle) countable. Matter and energy come in discrete chunks, and whether you’re trying to locate an atom or detect a photon of light, you will find it in one place, and one place only.
Quantum particles also behave like waves, in that they show effects like diffraction and interference. If you send a beam of electrons or a beam of photons through a narrow slit, they will spread out on the far side. If you send the beam at two closely spaced slits, they will produce a pattern of alternating bright and dark spots on the far side of the slits, as if they were water waves passing through both slits at once and interfering on the other side. This is true even though each individual particle is detected at a single location, as a particle.
2) Quantum states are discrete. The “quantum” in quantum physics refers to the fact that everything in quantum physics comes in discrete amounts. A beam of light can only contain integer numbers of photons– 1, 2, 3, 137, but never 1.5 or 22.7. An electron in an atom can only have certain discrete energy values– -13.6 electron volts, or -3.4 electron volts in hydrogen, but never -7.5 electron volts. No matter what you do, you will only ever detect a quantum system in one of these special allowed states.
3) Probability is all we ever know. When physicists use quantum mechanics to predict the results of an experiment, the only thing they can predict is the probability of detecting each of the possible outcomes. Given an experiment in which an electron will end up in one of two places, we can say that there is a 17% probability of finding it at point A and an 83% probability of finding it at point B, but we can never say for sure that a single given electron will definitely end up at A or definitely end up at B. No matter how careful we are to prepare each electron in exactly the same way, we can never say for definitiviely what the outcome of the experiment will be. Each new electron is a completely new experiment, and the final outcome is random.
4) Measurement determines reality. Until the moment that the exact state of a quantum particle is measured, that state is indeterminate, and in fact can be thought of as spread out over all the possible outcomes. After a measurement is made, the state of the particle is absolutely determined, and all subsequent measurements on that particle will return produce exactly the same outcome.
This seems impossible to believe– it’s the problem that inspired Erwin Schrödinger’s (in)famous thought experiment regarding a cat that is both alive and dead– but it is worth reiterating that this is absolutely confirmed by experiment. The double-slit experiment mentioned above can be thought of as confirmation of this indeterminacy– until it is finally measured at a single position on the far side of the slits, an electron exists in a superposition of both possible paths. The interference pattern observed when many electrons are recorded one after another is a direct consequence of the superposition of multiple states.
The Quantum Zeno Effect is another example of the effects of quantum measurement: making repeated measurements of a quantum system can prevent it from changing its state. Between measurements, the system exists in a superposition of two possible states, with the probability of one increasing and the other decreasing. Each measurements puts the system back into a single definite state, and the evolution has to start over.
The effects of measurement can be interpreted in a number of different ways– as the physical “collapse” of a wavefunction, as the splitting of the universe into many parallel worlds, etc.– but the end result is the same in all of them. A quantum particle can and will occupy multiple states right up until the instant that it is measured; after the measurement it is in one and only one state.
5) Quantum correlations are non-local. One of the strangest and most important consequences of quantum mechanics is the idea of “entanglement.” When two quantum particles interact in the right way, their states will depend on one another, no matter how far apart they are. You can hold one particle in Princeton and send the other to Paris, and measure them simultaneously, and the outcome of the measurement in Princeton will absolutely and unequivocally determine the outcome of the measurement in Paris, and vice versa.
The correlation between these states cannot possibly be described by any local theory, in which the particles have definite states. These states are indeterminate until the instant that one is measured, at which time the states of both are absolutely determined, no matter how far apart they are. This has been experimentally confirmed dozens of times over the last thirty years or so, with light and even atoms, and every new experiment has absolutely agreed with the quantum prediction.
It must be noted that this does not provide a means of sending signals faster than light– a measurement in Paris will determine the state of a particle in Princeton, but the outcome of each measurement is completely random. There is no way to manipulate the Parisian particle to produce a specifc result in Princeton. The correlation between measurements will only be apparent after the fact, when the two sets of results are compared, and that process has to take place at speeds slower than that of light.
6) Everything not forbidden is mandatory. A quantum particle moving from point A to point B will take absolutely every possible path from A to B, at the same time. This includes paths that involve highly improbable events like electron-positron pairs appearing out of nowhere, and disappearing again. The full theory of quantum electro-dynamics (QED) involves contributions from every possible process, even the ridiculously unlikely ones.
It’s worth emphasizing that this is not some speculative mumbo-jumbo with no real applicability. A QED prediction of the interaction between an electron and a magnetic field correctly describes the interaction to 14 decimal places. As weird as the idea seems, it is one of the best-tested theories in the history of science.
7) Quantum physics is not magic. Yeah, this was on the other list as well, but it’s so important that it needs repeating. As strange as quantum physics is– and don’t get me wrong, it’s plenty weird– it does not suspend all the rules of common sense. The bedrock principles of physics are still intact: energy is still conserved, entropy still increases, nothing can move faster than the speed of light. You cannot exploit quantum effects to build a perpetual motion machine, or to create telepathy or clairvoyance.
Quantum mechanics has lots of features that defy our classical intuition– indeterminate states, probabilitistic measurements, non-local effects– but it is still subject to the most important rule at all: If something sounds too good to be true, it probably is. Anybody trying to peddle a perpetual motion machine or a mystic cure using quantum buzzwords is deluded at best, or a scam artist at worst.
2) — NOOOO!!!!! You need to talk about measurement operators, not about states, if you want to say “discrete”.
Perhaps: Measurement operators that have discrete spectra are used to represent measurement apparatus/procedures that produce discrete measurement results. Measurement operators that have continuous spectra are idealizations that do not correspond to real experimental data that is written in lab books or in computer memory.
The state space is usually taken to be vectors in a Hilbert space over the complex field, or density operators (arguably always one of these, by quantum physicists?), which are pretty much continuous linear spaces.
4) is problematic, I think in that a specific interpretation interweaves with mathematics that might be considered interpretation independent.
6) takes the path integral formulation of quantum theory very, very seriously. Quantum theory in other formulations wouldn’t make this claim. Crudely put, the empirical consequences that can be presented as caused by interference effects between trajectories can also be presented without introducing trajectories. The introduction of superpositions of valuations associated with continuous trajectories is a specific metaphysics that is not warranted by discrete measurement results.
Sorry, Chad, 2) took my breath away. I believe my alternative is approximately conventional, but it’s sometimes hard to differentiate my own significantly nonstandard views on QM and QFT from the canonical. Feel free to rip it apart (or someone else on your behalf).
8) “How to Teach Physics to Your Dog is a very nice introduction to all this. At least, for the relatively naive reader like myself.
@1
regarding number 2:
I think Chad’s statement here is fine. Specifically: “no matter what you do, you will only ever detect a quantum system in one of these special allowed states” is exactly correct and requires no revision. Yes, it happens because operators corresponding to observables have some discrete or continuous spectrum, but that doesn’t add anything to the discussion he’s making at this level.
Regarding number 6: whether or not you want to do nonrelativistic QM with the path integral, when you do field theory you find the result that every single allowed way a process can happen contributes to the amplitude. This is independent of whether you start with functional integrals or canonical quantization.
“The introduction of superpositions of valuations associated with continuous trajectories is a specific metaphysics that is not warranted by discrete measurement results.”
How so? The propagation of the wavefunction through space is equivalent to assigning an amplitude to each classical trajectory. If the mathematics doesn’t distinguish the ideas, how can you act like one is natural and one is superfluous?
is it possible to use simultaneous measurement of entangled particles to damage the fabric of reality and/or destroy the universe?
Bohmian mechanics is a counter-example to 4. It also conflicts with 3 to some extent, although if the Bohmian “quantum equilibrium” condition holds then it is correct in practice, but probably less fundamental than you intended it to be.
Perhaps you are not counting BM as a viable interpretation, but it is taken seriously in the foundations community so I always check my assertions about QM against it as a rule of thumb.
I was also a bit taken aback by “quantum states are discrete”. Not as much as Peter Morgan at #1, but it still strikes me as a bit misleading. Most of what we encounter in the real world (no matter whether your religion says the world is a density matrix or a wavefunction) are not states of definite energy, definite photon number, etc.
The nit-picker in me would replace that with something like “most measurements have discrete outcomes”: when you measure your coherent state you always find an integer number of photons, when you measure the energy of your electron in the H atom you always find certain discrete values. But I’ll admit that my version lacks the punch of your simpler, catchier version.
The goal here was to produce a list of core elements of the theory that wouldn’t make people run screaming for the exits. Given that constraint, I stand by my phrasing of #2 as being both essential to the theory– the limited numbe rof possible states is what puts the “quantum” in “quantum mechanics,” after all– and clear enough to get the idea across without making anyone wrestle with the mathematical apparatus. As it is, nobody who isn’t an expert is commenting on this– if I started talking about operator algebra, nobody would read past the first few words of that sentence.
My knowledge of Bohmian mechanics is extremely limited, and as a result, I tend not to think of it when I’m banging this sort of list out. If pressed, I would say that having a definite but unknown state between measurements that is determined by a non-local quantum potential is not a whole lot less weird than indeterminate states, and might be weirder. If there’s a better way to make a concise statement that gets the core of Bell’s theorem while also encompassing Bohmian mechanics, I’m not sure what it would be.
Finally:
is it possible to use simultaneous measurement of entangled particles to damage the fabric of reality and/or destroy the universe?
I hope not, because if you could rip the fabric of reality with only tens of thousands of dollars of optical hardware, the LHC people would feel pretty silly about spending billions on an accelerator that won’t even produce a planet-killing black hole…
So Chad,
You sound much too positive and cheery for my tastes.
I canât believe there arenât a few alligators out there hiding behind the cheese bunnies.
What about the problems of quantum mechanics? (Are there any?)
In particular what about the measurement problem (MP)?
Maybe Iâll phrase some of the possible answers as a poll:
1) Iâve solved the MP.
(please explain how in the comments)
2) The MP is a non problem and can be ignored.
(please explain why in the comments)
3) Itâs obvious to every good experimentalist
whatâs a measurement and whatâs not.
(let me in on the secret)
4) It all collapses in your brain.
5) Shut up and compute!!
6) I object to this poll because I believe in MWI
Iâm looking hard for enlightenment.
Maybe I should start looking for cheese bunnies instead.
Best,
Jim Graber
Re #8 You are talking about problems with interpretations and not with QM. Please state what exactly the problem is one has to solve in the measurement problem and how any possible experiment would choose one answer over another.
Or give an example of a real experiment we cannot predict the the outcome of because of this measurement problem.
Or is it a problem with the consistency of QM such that one could get contradictory predictions, if so how about an example?
So it is # 2 on you list for me.
Chad,
I was not picking at your discussion of nonlocality when I brought up Bohmian Mechanics, which is point 5, but rather at the “measurement determines reality” section, which is point 4. I largely agree with your discussion of nonlocality, but since you have inadvertently invited me to pick at that as well I will just pedantically point out that entanglement is not the same thing as nonlocality because there are plenty of (mixed) entangled states that do not violate any Bell inequality, so you should say that the weird feature is nonlocality rather than entanglement.
Anyway, returning to point 4, I must still strongly object to any “measurement determines reality” type of statement. Sure, it is the case that a lot of people believe this (just as a lot of people believe that science is compatible with religion), but people believing something does not prove that it is true. To be rigorous, one would have to prove some sort of “measurement determines reality” theorem, just as we have Bell’s theorem for nonlocality, rather than relying on the proclamations of the Copenhagen school, which we now know were a bit over-zealous. Things like the Kochen-Specker theorem, the “free will” theorem and the recent tests of Leggett’s inequalities come close, but not without assumptions that are easily challenged — the main challenge being that there are reasonable interpretations of QM that do not satisfy said assumptions. In fact, the main point of most realist interpretations of quantum theory is precisely to negate your point 4, i.e. to remove measurement as any kind of fundamental primitive. Therefore, I would say that not only Bohmian mechanics, but also spontaneous collapse theories and almost all modal interpretations do not satisfy point 4. I would also be prepared to defend the idea that many-worlds does not satisfy 4 either, but I admit that this is more ambiguous than the other cases. In any case, all these theories are perfectly compatible with current QM, and only spontaneous collapse could potentially be ruled out by experiment.
Given that I had to reread 4 in order to write this comment, I just also noticed:
“After a measurement is made, the state of the particle is absolutely determined, and all subsequent measurements on that particle will return produce exactly the same outcome.”
I don’t think this is actually part of QM, but is just an additional assumption that is valid for some measurements and invalid for others. In particular, it is never valid for POVM measurements, but more seriously, it is not valid for some extremely common projective measurements, such as detecting the position of a photon on a photographic plate after which there is no longer a photon to repeat the measurement on.
As a master quantum optician, I am sure that you know all of this and were referring to some notion of “ideal” measurement. Just to let you know, there are a fair few people (including myself) that argue about whether the projection postulate is really the best state update rule to call “ideal”, since there is no update-rule that can have all the properties of the classical Bayes’ rule and there are different rules that inherit different properties. It is also one of the reasons why I don’t regard the quantum Zeno effect as being particularly fundamental and why there was so much controversy about it in the 1990s. Sometimes it happens and sometimes it doesn’t — it depends on how you do the measurement.
I’d take some issue with a couple of statements. They are slightly niggling points, but I think this is a great list for undergraduate physics majors trying to internalize QM, so it seems worth rephrasing. For a more general audience the nitpicking below is probably largely below the radar.
(My comments are not intended as suggested rephrasings!)
“No matter how careful we are to prepare each electron in exactly the same way, we can never say for definitiviely what the outcome of the experiment will be. Each new electron is a completely new experiment, and the final outcome is random.”
Umm…the headline, that probabilities are all we will ever know, does not conflict with the possibility that the quantum probability of some measurement yielding a particular value is 100% (or 0%). For instance, what if my “experiment” is to measure the z spin component of an electron, right after prepared the state by having already measured it once?
Getting 100% probabilities is possible even if it’s not an eigenstate of the Hamiltonian. An example is the Rabi formula for spin 1/2, where, at particular times, probabilities for finding the spin up or spin down do become 100% certain.
“Until the moment that the exact state of a quantum particle is measured, that state is indeterminate, and in fact can be thought of as spread out over all the possible outcomes. After a measurement is made, the state of the particle is absolutely determined, and all subsequent measurements on that particle will return produce exactly the same outcome.”
Again, I think the headline is OK. But the final clause is actually in conflict with the first for a careful reader. More generally, the first clause is only true if the prepared state is not an eigenstate of the Hamiltonian; the final clause only if the measurement operator does commute with the Hamiltonian. If you measure a particle’s position, it doesn’t stay there for all time.
I think I know what my trouble is, Chad. You write about experimental apparatus beautifully, including lots of gory details that could really make someone run away, but it’s infinitely absorbing. It may not be you, but the theory deserves the same.
When you write in a theory mode that when you measure the energy of an electron it comes out at 13.7 eV, you have to know that if you really put together an experimental apparatus to measure the energy of an electron, it would probably take months, and you would relish telling us all the gory details. You would tell us how you make sure that the apparatus really measures exactly the energy of an electron and not something else, how you overcome the finite line width that’s inevitable with a thermal source, how a passing bus wrecked the experimental apparatus. You’d tell us how the first measurement run obtained the value 13.65 eV, and how you fixed it. You’re glorious on this stuff because you put into it stuff that isn’t put into papers in Physical Review explicitly.
My position is that such details matter as much when understanding theory.
Many things in quantum mechanics are debatable, especially “interpretations.” I have my own opinions and have wrangled over it here, maybe even too fervently. But this time, I propose an practical experiment to actually test an issue involving decoherence. It involves recovering the sort of information that is usually thought lost to decoherence. Read about it at name link. If done and results are as predicted, it’s worth more than tons of ink.
Re #4. Talking about measurement is always confusing and leads people to believe in mystical nonsense. Think of the CMB — the photons we measure today, are they only “real” once we “measure” them? Do we, during our measurement today, determine what scattering happened 14 By ago? What if we humans were not here — would the universe still exist?
VERY interesting article and commentary, all. Very illuminating.
Me. I’ve always wanted to be an Applied Metaphysician.
When I first learned about QM as a teenager I was fascinated by it’s weirdness and though it represents a completely new paradigm. As my knowledge about it increased however it became more and more obvious that all this weirdness is due to nothing more then our own ignorance of the underlying reality.
By now I am pretty certain that once a proper TOE is developed (and I am sure it will be eventually) most of this weirdness will be gone with the exception of things originating from relativity which imho IS fundamental.
Here is my personal take on this list:
1) Particles are waves, and vice versa.
I would say that there are no particles at all, only waves and *quasi* particles. The mechanics of those waves is not properly understood yet obscuring some of their properties and their fundamental nature.
2) Quantum states are discrete.
That is true by definition, but if by “quantum states” one means any states QM deals with then it’s not true as position, momentum, time and energy – the four most fundamental observables imho – are continuous in general.
Discretization is the artifact of certain kinds of interactions.
3) Probability is all we ever know.
Yes, but not all we can ever know, this is simply the limitation of QM not the limitation of underlying reality. Probabilities cannot be fundamental or to put it another way Nature does not play dice 😉
4) Measurement determines reality.
Should be measurement *affects* reality. It does not determine reality in that reality is independent of measurement though by doing measurement we do force changes in reality. There is nothing weird about it and it’s also true for classical measurements though there the perturbation can be negligible while in QM it is always severe.
5) Quantum correlations are non-local. One of the strangest and most important consequences of quantum mechanics is the idea of “entanglement.”
One has to remember that locality as such does not have to be based on the speed of light. It is possible that entanglement is local in the sense that there is still some physical speed limit of the interaction though higher then c. I am not saying this has to be the case only that it is a possibility which has to be kept in mind.
I am also pretty sure entanglement will be much less mysterious when the nature of the underlying physical process – and there certainly is one – is determined.
6) Everything not forbidden is mandatory. A quantum particle moving from point A to point B will take absolutely every possible path from A to B, at the same time.
Not necessarily. This is simply a consequence of the probabilistic nature of predictions. QM can only make statements about ensembles of events so it is natural that all the possible ways in which an event can be realized influence it’s probability. The actual single events might very well use only a small subset of all possible paths.
7) Quantum physics is not magic.
Hard to argue with this statement 😛
Another point I’d like to comment on is the tradition of using magnetic moment calculation to prove the awesomeness of QM:
“A QED prediction of the interaction between an electron and a magnetic field correctly describes the interaction to 14 decimal places.”
Yes, that’s great but the fact that we can predict one thing with such a high accuracy doesn’t prove much about theory as a whole. If you tried you could probably find an example where Newtonian mechanics gives the good answer to some quantum problem by accident or because quantum effects cancel out but will it prove anything? The same may be true for QM and magnetic moment, one should instead talk about the accuracy of QM predictions in general.
Don’t get me wrong, QM is very useful, but it’s certainly not fundamental or complete as far as I am concerned.
The collapse of the superposition basically equates to common logic about choosing from alternatives. Contrary to what the article states, the fact that it can turn out one way or another should be basically considered to be the operation of “magic”. Any statements about what is doing this job of making it turn out the one way instead of the other is subjective, a personal belief. So in commenting on such a collapse, it is perfectly alright for a scientist to opinionate that it was “love” that made it turn out the way it did. This is not unscientific, it is just so that subjective personal opinion applies here, but to assert any obective fact about what did the job, that is actually unscientific.
INVITATION:
Change the worldil become âpossible only by changing the mechanicistic ideas that are the intellectual cause to devast the world during the industrial epoch.
Developments on cognitive change as are necessary to promote the framework for renewal of green economy will be discussed during the workshop to be held in the old A. Volta classroom at the University of Pavia on 24 SETT/10.
You are invited to attend.
View: http://www.edscuola.it/archivio/lre/3rd_Quantumbionet_Workshop.pdf
see also the explanatory memorandum in: http://www.wbabin.net/science/manzelli85.pdf
Paolo Manzelli: lre@unifi.it ; http://www.edscuola.it/lre.html
On Quantum Mechanics And Entanglement
A.
Essence Of Quantum Mechanics
Life and the universe are not conglomerations of mechanisms. Their mechanisms are routes of evolution. The mechanisms do not set/determine the classical physics end-target/states. They are routes of evolution between classical physics states. Quantum mechanics are mechanisms, probable, possible and actual mechanisms of getting from one to other classical physics states WITHIN the expanse from cosmic singularity to the maximum expanded universe and back to singularity states.
B.
Entanglement loophole closed
A long-distance experiment rejects a challenge to quantum physics.
http://www.sciencenews.org/view/generic/id/64935/title/Entanglement_loophole_closed
An old USSR joke:
Question: Is it true that the USSR-made car “Volga” makes a 90-degree turn at 100 km/hr?
Answer: Yes…. but only once.
– Is entanglement a “yes, but only once” affair for each entangled objects pair/group?
and, if so indeed,
– Are the states-of-systems of entangled objects decided upon separation of the objects, not upon their measurment?
Dov Henis
(Comments From The 22nd Century)
Evolution, Natural Selection, Derive From Cosmic Expansion
http://darwiniana.com/2010/09/05/the-question-reductionists-fear/
Cosmic Evolution Simplified
http://www.the-scientist.com/community/posts/list/240/122.page#4427
Terms And Concepts:
Re Quantum Mechanics And Entanglement
A.
Essence Of Quantum Mechanics
The universe, and life within it, are not just conglomerations of mechanisms. The universe, and life within it, have come into being by the nature of energy-mass dualism, and their fate, their final outcome, is governed by this dualism. The genesis and, most probable cyclic, existence of the universe are governed by the energy-mass relationship.
Energy-mass relationship governs also the routes, the mechanisms, of cosmic and life evolutions.
Mechanisms do not set/determine the classical physics fate states. Mechanisms are routes of evolution between classical physics fate states. Quantum mechanics are mechanisms, probable, possible and actual mechanisms of getting from one to other classical physics states WITHIN the expanse from cosmic singularity to the maximum expanded universe and back to singularity states.
The universe is the archetype of quantum within classical physics. This is the fractal oneness of the universe. Astronomically there are two physics. A classical Newtonian physics behaviour of and between galactic clusters, and a quantum physics behaviour WITHIN the galactic clusters.
B.
Entanglement loophole closed
A long-distance experiment rejects a challenge to quantum physics.
http://www.sciencenews.org/view/generic/id/64935/title/Entanglement_loophole_closed
An old USSR joke:
Question: Is it true that the USSR-made car “Volga” makes a 90-degree turn at 100 km/hr?
Answer: Yes…. but only once.
– Is entanglement a “yes, but only once” affair for each entangled objects pair/group?
and, if so indeed,
– Are the states-of-systems of entangled objects decided upon separation of the objects, not upon their measurment?
Dov Henis
(Comments From The 22nd Century)
Evolution, Natural Selection, Derive From Cosmic Expansion
http://darwiniana.com/2010/09/05/the-question-reductionists-fear/
Cosmic Evolution Simplified
http://www.the-scientist.com/community/posts/list/240/122.page#4427
“A quantum particle can and will occupy multiple states right up until the instant that it is measured; after the measurement it is in one and only one state. ”
So, it this merely because the moment we measure the particle, it is in one state, then when we cease measurements it continues to alter states? Then this could just mean the frequency with which we measure the particle would have to change for us to measure it in another state…?
Or is it that the particle “decides” (though I don’t mean consciously) a state to exist in from the moment it is measured onward? If this second part is so, then it would defy some mode of logic- unless the very measurement process “pulls” the particle into a certain state of existence…
Or maybe the universe splits into two separate parts at the moment of the measurement; why not ? Seems just as logical as any part of quantum mechanics to me.
1. Measurements Do Not A Reality Make – go figure;
2.How Do We Propose to Debate the Details, yet deny the results?;
3. Who Benefits from Knowledge – the secret lab guy cloaked in mysterious symbols and equipment – or – the simple, yet helpful, sympathetic, courageous, mother raising children without social imperfections, programs and interventions of supposed higher understanding? (thrown in for kicks)
Regarding point 2, discussed in comments 1, 3, 6, 16, it is NOT true that in quantum physics EVERYTHING comes comes in discrete amounts nor even that MOST of the measurements give discrete outcomes as far as we do not define what we mean by ‘most’; it is true that what happens depends on the type of interaction and constraints: specifically the solutions of quantum system (eigenstate of an eigenvalue equation) have spectra (eigenvalues) that become discrete in general when the boundary conditions are of compact support. So of a field equation after you get the solutions you go to see what are the boundary conditions and if they are compact then your measurements are discrete: the energy and momentum of a particle that is free are continuous while if the particle is in a box of potential they are discrete. Discrete comes because the solution is bound and unbound solutions are continuous.