The article leaves out a crucial item. When it says:
"The normal modes are plane waves traveling at the speed of light in a particular directions with a given frequency."
This is not quite correct. A correct statement would be that plane waves are one possible choice of "normal modes" for the quantum electromagnetic field. But not the only possible choice. (In more technical language, they are one possible choice of basis for the Hilbert space of states of the quantum electromagnetic field, but not the only possible choice.)
In fact, the plane wave states (whose technical name is "Fock states") are often not very useful, because they are very hard to produce experimentally and pretty much never occur naturally. The states that are closest to naturally occurring states and which are easy to produce experimentally (for example in a laser) are coherent states. But coherent states are nothing like plane waves.
Typically the normal modes of a system are whatever diagonalize its time evolution. Plane waves are a privileged basis for free particles because they diagonalize the free particle Hamiltonian. Unless time evolution commutes with annihilation (I have no idea how this could ever happen, but hey), coherent states are not normal modes.
>they are one possible choice of basis for the Hilbert space of states of the quantum electromagnetic field
This is true but it's looking at them from far too general of a perspective. Most of those bases don't diagonalize any interesting operator, much less the time evolution operator.
I am pretty sure you are confounding the basis of functions (harmonics) that one can use to describe an EM field, with the basis of a Hilbert space used to quantize them. You can have a mode (e.g. a plane wave) and the field of that particular mode can be quantized and you get to pick whether to use Fock states or coherent states or any other basis. In other words, "plane wave" refers to a particular mode of the field and Fock/coherent states refer to the quantum state of that particular mode. Two different plane waves are two completely different modes, e.g. two completely different Hilbert spaces, not basis vectors of a given Hilbert space.
> I am pretty sure you are confounding the basis of functions (harmonics) that one can use to describe an EM field, with the basis of a Hilbert space used to quantize them.
No, I'm not. The same mathematical functions can appear in both classical and quantum treatments, but I'm talking about the basis of the quantum Hilbert space.
> Two different plane waves are two completely different modes, e.g. two completely different Hilbert spaces
No. They are different Fock states in the same Hilbert space; in the Fock state basis for that Hilbert space, they are two different, orthogonal basis vectors. And that same Hilbert space can also have other choices of basis besides the Fock state basis.
Mode A has a Hilbert space spanned by the Fock states {|0⟩,|1⟩,|2⟩...} or equivalently by the coherent states {|α⟩, α∈C}. Mode A has its own creation and annihilation operator.
Mode B has its own Hilbert space for which we can use similar bases. Mode B has its own creation and annihilation operators that are different from the ones for A.
Let us just say both A and B are plane waves for the examples that follow.
If mode A and mode B are orthogonal (the way two different plane waves are, and this is a purely classical statement), then we have all their operators commute and their Hilbert spaces are completely different.
We can have a coherent state in mode A and it will still be a plane wave. Same with having a single photon state in mode B and it will still be a plane wave. We can not use kets related to mode B to describe anything related to mode A.
If we want the total Hilbert space, that would be the tensor product of Hilbert space A and B and the basis kets in it will be denoted like {|00⟩,|10⟩,|01⟩,|11⟩,|20⟩,|21⟩,|12⟩,|02⟩,|22⟩,...}.
I continue to claim you are confounding the basis of harmonic functions (modes) for the classical description of the EM field, with the basis of kets describing the quantization of one single mode. Said another way, a plane wave is a basis vector of the space of square-integrable functions, while Fock states and coherent states are basis vectors for a Hilbert space of a quantum oscillator.
Yes; but both of these Hilbert spaces are subspaces of the full Hilbert space of the quantum EM field. For each mode, there will be a subspace consisting of all the Hilbert space vectors you can get by applying the creation operator for that mode only to the vacuum state an arbitrary number of times. But it's the same vacuum state in each case; and the full Hilbert space includes vectors that you get by applying two or more different creation operators to that vacuum state.
> If we want the total Hilbert space, that would be the tensor product of Hilbert space A and B
No, it would be the symmetric Fock space constructed using the direct sum of Hilbert spaces A and B ("symmetric" because photons are bosons). See here:
> a plane wave is a basis vector of the space of square-integrable functions
No, it isn't. A plane wave, i.e., a function of the form exp(ikx), is not square integrable. When we use plane waves as a basis, we have to either expand our space of functions to allow them, i.e., beyond just the square integrable functions, or restrict how linear combinations of the basis functions can be formed in order to ensure that all linear combinations we actually make use of are square integrable.
> Fock states and coherent states are basis vectors for a Hilbert space of a quantum oscillator.
Two different choices of basis, yes (and coherent states are an "overdetermined" basis--I think that's the right term--since they are not all linearly independent of each other). But Fock states, or at least the ones belonging to subspaces like your A and B above, formed by applying creation operators for only one mode to the vacuum, are plane wave states, since each "mode" is a plane wave, so any Fock state containing nonzero amplitude for only one mode will also be a plane wave.
> but both of these Hilbert spaces are subspaces of the full Hilbert space of the quantum EM field
This is definitely wrong, and it is the source of our disagreement. They are not subspaces. If they were subspaces then we would use a direct sum to build the Hilbert space of the entire EM field, and this Hilbert space would have dimensions that are the sum of the dimensions of the subspaces.
Instead, we take the tensor product of the Hilbert space for each mode. This is a drastically different operation, leading to a very different basis, different dimensions, etc.
Notice, the link you provided yourself states that the total Hilbert space is a tensor product. Only after they state that, they present a direct sum over tensor-powers of the Hilbert space of one particle, which is very different from the statement you wrote, exactly because of this tensor-power operation. I concede that if I have to be rigorous I need to specify this whole symmetrizing step that I skipped, but that is just a detail, especially when we talk about distinguishable modes.
Concerning your (justified) nitpick on square-integrable functions: substitute "square-integrable over the finite size of the oscillator".
Edit: a number of additions and extra explanations.
I'm sorry, but I disagree. You are misreading the link I gave.
> the link you provided yourself states that the total Hilbert space is a tensor product
No, it doesn't. What it says is that "n-particle states" are vectors in a symmetrized (assuming the particles are bosons) tensor product of n single-particle Hilbert spaces. Then the full Hilbert space is the (Hilbert space completion of the) direct sum of all the n-particle state Hilbert spaces, for n from zero (the n = 0 Hilbert space is just the vacuum state) to infinity.
The subtlety here is actually the proper definition of the "single particle" Hilbert space. This Hilbert space is in fact the space of all possible "modes", or more precisely the single particle Hilbert space has a basis consisting of all possible "modes", where each "mode" is a plane wave with a different momentum 3-vector (and polarization, if we include polarization, but we can ignore that complication for now). So each "single particle state" is either a single "mode", or a linear combination of multiple "modes" (which is how we can form things like wave packets).
The "n particle states" are then states containing n particles, each of which can be in any one of the single-particle states, either a single "mode" or a linear combination of them (and each particle does not have to be in the same single-particle state as any other). The full Hilbert space is then, as above, the space of all possible n-particle states, for n from zero to infinity.
It should be evident from the above that the full Hilbert space has a subspace for each single "mode", consisting of all states that only contain particles in that "mode" alone. This subspace will, as I said before, consist of the vacuum state, plus all states that can be obtained by applying the creation operator for the chosen "mode" to the vacuum state an arbitrary number of times.
> substitute "square-integrable over the finite size of the oscillator"
Doesn't help; a plane wave can't exist in a region of finite size, so with your restriction plane wave states don't exist at all.
> symmetrized __tensor product__ of n single-particle Hilbert spaces H
And clarifying your statement by copying from the wiki page (emphasis mine):
> direct sum of the symmetric or antisymmetric tensors in the __tensor powers__ of a single-particle Hilbert space
(which is very different from the direct sum you suggest in your comment)
I work with all of this in my day job and the research work I produce (which does deal a lot with quantum description of multiples modes of light) is usually perceived as correct by my academic colleagues. I am honestly trying to see how to interpret what you are saying differently, in a way that does not clash with the established knowledge on the topic, but I can not, even though I have non-negligible experience in the field. At best, I assume we are talking past each other while describing the same thing, but I am certainly uncomfortable with your use of "subspace" and "direct sum".
Going to the very first comment of yours that started this: My claim is that a mode (something that has a creation operator) can very well contain one, or two photons, or a coherent state, or anything else from its Hilbert space. There is no correspondence between plane waves and Fock states. We can use Fock states with other modes instead of plane waves, and we can have plane waves with a quantum state that is a coherent state for instance.
> which is very different from the direct sum you suggest in your comment
I don't see how. I suspect that we are interpreting ordinary language words differently and talking past each other.
> a mode (something that has a creation operator)
And a corresponding annihilation operator. Yes, that's basically the definition I've been using.
> from its Hilbert space
But, as I said before, all of these states you describe are obtained by applying the creation operator for the mode to the vacuum state some number of times (and then, in some cases, forming linear combinations of states so obtained). But this process for all modes starts with the same vacuum state. There are not different vacuum states for different modes. So ultimately, all of these states belong to the same Hilbert space, since they're all obtained starting from the same vacuum state; and this Hilbert space will also contain states that have more than one "mode" in them--states obtained by applying creation operators for different modes to the vacuum state, or linear combinations of states so obtained.
Nah, GP is right, if he's talking about how you can decompose states into superpositions of coherent states of the whole field. These are different from coherent states of individual modes.
Specifically, a coherent state of a field is a state which is an eigenvector of the positive-frequency part of the field operator, when that operator is evaluated at any position.
See the sibling comment and my answer. A plane wave can very well be a plane wave "containing" one photon or containing multiple photons or containing a coherent state. There is a big difference between saying "I can use different sets of creation operators to form a basis for the field operator" (what I think you are saying) and the confounded statement mixing up basis of a Hilbert space with basis of integrable functions of the OP.
Unfortunately, like all such descriptions, this one misses the mark also.
I've never heard a good explanation for what a photon "is", or any particle for that matter. That's because physicists don't know either.
No, seriously. Ask a randomly selected set of physicists about any core concept in Quantum Mechanics and you'll discover that their answers diverge wildly. Ref: https://arxiv.org/abs/1301.1069
Basic things like: "Are the photons quantised or is it just the photon interaction with atomic orbitals that's quantised?" I've personally asked this question and had working physicists confidently state that either the former or latter is true, and the other is false.
(PS: Compare this to biologists, chemists, or engineers. They all 100% agree on all core concepts. So do astronomers, and physicists studying classical mechanics and relativity. It's only quantum mechanics that's so fragmented.)
Ask any working QM researcher to clearly show how "a single photon" interacts with an electron in an orbital and they will without a doubt draw you a cartoon diagram. Never in the history of theoretical physics has anyone, ever, anywhere made a numerical visualisation of this. This is because both the QM mental model and the mathematical theory applies only to ensembles in potential wells.
This is a very narrow scope of applicability that was explicitly called out in the QM papers back in the early 1900s, but is now glossed over. People assume that what applies to a special case is the hard rule that applies to all things. It doesn't. There are no photons in free space. There are no little hard particles of light. You can't have a point with a wavelength. You can't have a single pure frequency with a finite duration. This is all gibberish, it's just a simplification to make the maths tractable when done with pencil and paper for special cases! The output of this simplified model is a 0-dimensional scalar. It can't do output much else, such a simulation of a full 3D interaction of any type.
Trying to work backwards from such an oversimplified model in order to explain the deep nature of reality is futile. This is like pointing at the bandwidth measurement in Windows Task Manager and saying "we can understand how the Internet works by watching this number go up and down".
> I've never heard a good explanation for what a photon "is", or any particle for that matter. That's because physicists don't know either.
The aim of many/most physicists is to predict rather than to know. Biologists, Chemists, Engineers can agree all they want but their theories aren't tested to 5-sigma (I'm sure some are but that's not as good of a metaphor). We have extremely accurate and predictive models of how photons behave (Quantum Field Theory not solely Quantum Mechanics).
You're arguably right, we don't really know - however, we can only know what we can see with our eyes. There may be some all-encompassing structure hiding at any energy level we haven't yet reached but the explanation given is as good as it gets in terms of explainable what we have observed in nature.
As to Quantum Mechanics, the only place you'll find Physicists disagreeing about the "core concepts" is on the absolute fringes of what can be considered science. You will find Physicists arguing about the interpretation of those concepts but it's not like they don't work. As to that interpretation, there is disagreement but I think it is fair to say that very few physicists actually think about it - i.e. "Shut up and calculate" is a common outlook.
> PS: Compare this to biologists, chemists, or engineers. They all 100% agree on all core concepts.
Biologists certainly don't. What is a species? Is Linnean taxonomy useful or merely confusing and obsolete? What is a gene?
> I've never heard a good explanation for what a photon "is", or any particle for that matter.
What would such an explanation look like, and why do you think that there is such an explanation?
In many cases when people ask what something "is", they want a category defined in terms of categories they are already familiar with. We can build up a set of observations and build a formal model to summarize them whose terms we can map to the results of various observations. We name the various parts of our formal model for convenience. But there is no requirement that the model be able to satisfy queries that depend on things that don't exist in the domain of the model.
> Basic things like: "Are the photons quantised or is it just the photon interaction with atomic orbitals that's quantised?" I've personally asked this question and had working physicists confidently state that either the former or latter is true, and the other is false.
Sure, because you're not asking a question about anything observable, so it's a question of how people put names on the various internal parts of their model. It doesn't matter if people agree on that, because when anyone sits down to actually use the formal model to model something, vague naming of internal parts of it is irrelevant.
So I think the right answer is: stop looking at the photon! There's a charged thing wiggling somewhere, and the interaction between two charged things that wiggle is quantized. If I have an electron at one point making small oscillations and another electron at another point, I can calculate the transition probability of the first electron oscillating with one less quantum and the second one oscaillating with one more, or vice versa.
> Biologists certainly don't. What is a species? Is Linnean taxonomy useful or merely confusing and obsolete? What is a gene?
These are human definitions for classifications, and biologists agree on that. However, if you make a statement that "two groups of animals that can never reproduce successfully with each other are separate species" then nobody will argue that. Similarly, nobody argues about what the mechanism is for species reproduction success or failure. Everyone knows how chromosomes and DNA works. There's no debate.
Every chemist uses the same periodic table with the same elements. Etc...
> What would such an explanation look like
Equations that can be evaluated numerically that shows the behaviour of the EM field, in 4D spacetime, capable of demonstrating interactions with charged particles. That's all. That's not asking much.
> why do you think that there is such an explanation?
Because it's a thing that happens. There are atoms that have electrons that electromagnetically interact with light and change orbits.
Quantum Mechanics provides a highly simplified model of this in a very narrow use-case. That's not wrong, or mathematically invalid. But it's not the whole picture, and isn't a valid source from which to answer questions like "what is a photon".
That's not describing the nature of light in the real physical world, it describes the nature of an abstraction in a simplified model. The photon exists in the equations. That's why it's so hard to visualise and "pin down" with a simple description. It's a property of the model, not the world.
> We can build up a set of observations and build a formal model to summarize them whose terms we can map to the results of various observations.
The models in this case are tiny subsets of all physical behaviour and generally only cover certain categories of scalar observables. E.g.: "The detector went off or not". Hence the cartoon diagrams and focus on the observables covered by the model.
> But there is no requirement that the model be able to satisfy queries that depend on things that don't exist in the domain of the model.
I agree. But photons are a thing that exist in the model. What I'm saying is that the photons exist only in the model.
> Sure, because you're not asking a question about anything observable
See, you made the same mistake as everyone else! Atoms aren't the only things that exist! There are free particles in vacuum chambers, particle accelerators, outer space, etc... that aren't atoms. There's a device called a "free electron laser" that has a continuously adjustable output frequency!
> There's a charged thing wiggling somewhere, and the interaction between two charged things that wiggle is quantized.
Only if they're in potential wells. Even then it's not entirely clear if all interactions are quantised.
> I can calculate the transition probability
That output is a scalar. The electron is in an orbital though, right? With a defined angular momentum, position in space, and everything. Even if you abandon the concept of point particles, it has a bunch of properties. Now show me all of those properties changing as it absorbs the photon.
Classic QM can't, because it doesn't even try. It focuses on the narrow case of "near resonance" light and then handwaves over how the photons actually change their orbitals. Cartoon diagrams and statistics. That's fine if you're designing a laser or something, but that's not what is physically happening with a single electron around a single atom.
"Equations that can be evaluated numerically that shows the behavior of the EM field..."
it's called the Standard Model. You seem far too hung up on the deficiencies of QM, which have been well known for over half a century, so I'm not really sure the point you're trying to make. Every physicist knows that QM is incomplete so whether or not their forced conceptualization of that model agree or not is not really that interesting of a phenomenon.
Also, your comparison to chemistry and biology seem misguided. Chemistry, and the periodic table, are built on our understanding of physics and the constituent particles in each atom. Furthermore, comparing the difference in conceptual descriptions of QM to "but every chemist uses the periodic table" is ridiculous. Every physicist agrees on the accepted values for known physical constants, they all agree what the known fundamental particles are, they all agree what the known fundamental forces are... etc. It seems like you're bothered by the lack of conceptual guidance from physics but that isn't the point of the field. The point is to develop models that make highly accurate predictions about the physical world, from the unimaginably small to the unimaginably large (and they're the most accurate scientific models known to man!).
How you define a species is actually fascinating and there are different ways to do it, depending on what you know about an animal:
https://www.youtube.com/watch?v=9fOfFlMe6ek
> These are human definitions for classifications, and biologists agree on that.
I'm a biologist, and I don't agree.
> However, if you make a statement that "two groups of animals that can never reproduce successfully with each other are separate species" then nobody will argue that.
You're assuming animals with two sexes, and even there things like ring species throw a wrench into that.
> What I'm saying is that the photons exist only in the model.
and
> See, you made the same mistake as everyone else! Atoms aren't the only things that exist! There are free particles in vacuum chambers, particle accelerators, outer space, etc... that aren't atoms. There's a device called a "free electron laser" that has a continuously adjustable output frequency!
But those particles exist to us only as the results of measurements. We cannot impute a category to them that is definable in terms of familiar categories. They are no more or less real than the photon. I don't personally dislike the term "particle" for them, but "vacuum excitation" or similar just doesn't have the same cachet as "particle," and usually carry other undesirable baggage. Even "vacuum" carries problematic classical baggage.
> The electron is in an orbital though, right? With a defined angular momentum, position in space, and everything. Even if you abandon the concept of point particles, it has a bunch of properties. Now show me all of those properties changing as it absorbs the photon.
No, you can't impute properties like angular momentum to a state outside of measurement. We're used to thinking that we can because of classical systems where we take our sequence of measurements of a phenomenon over time and interpolate them infinitely finely and assume that as we do so the measurements converge to a continuous function. But since that limit doesn't exist for quantum mechanical phenomena, you can only speak in terms of measurement outcomes.
Plus, even if you could impute a property before measurement, you could only impute them to pure states. Almost all quantum states are not pure states.
So we can't show you those properties changing. All we can speak of is the probable outcomes of sets of measurements.
> That's fine if you're designing a laser or something, but that's not what is physically happening with a single electron around a single atom.
Is what you consider "what is physically happening" something theoretically observable?
> Even then it's not entirely clear if all interactions are quantised.
Of course. That's why continuous spectra are a thing.
I picked a bad example. Well, no, actually, a good one, but an example that does not lend well to online debate. And then I failed to make myself clear.
I already know about all of the corner cases, such as ring species, the fuzziness you get during speciation, sub-species, the constant reclassification of taxa, etc..
What I was trying to get at is that "species", like "photon" is a human word. A label for an abstraction that's useful.
No working biologist (sans a few creationist crazies) would disagree about that vast, vast majority of what makes a species a species. The chromosomes, the interbreeding capability, the genetics, DNA, etc... Where they disagree is on the finer points of classifications, such as whether two very similar specimens are a species, sub-species, or just a "trait" within a species. Those are just human labels, like the dewy decimal code. They're not the thing. They don't disagree that the specimens exist, that their DNA is different, etc...
Quantum theorists disagree with each other on things comparable to the existence of species. As in, some will say that there is such a thing as wave function collapse, others will say that this is nonsense and there's no such thing. They're not disagreeing on exactly when wave function occurs in some obscure corner case. No, they disagree on the existence of the concept itself.
This is a crucial distinction.
No two chemists will ever disagree that H2O is water or not. Ever. There's just no such argument, and hasn't been for well over a century.
In theoretical physics, specifically quantum mechanics and its cousins, there still is debate at this level, and shows no signs of resolution.
> But those particles exist to us only as the results of measurements...<snip>... you can only speak in terms of measurement outcomes.
That is a complex debate that's been had in the past: is it meaningful to speak of something existing if it cannot be measured, even in principle?
The issue with that is that sometimes people just state that certain categories of experiments are impossible, and then it turns out that oops.. no, it was possible, just really hard to do. This is my favourite example: https://tf.boulder.nist.gov/general/pdf/2723.pdf
Note the quote at the end: "Some observations can be made on the existence of ‘blinders,’both in theory and in experiment. Some theorists were accustomed to seeing problems in terms of the tools which had previously proven useful for describing experiments with large numbers of atoms, such as the optical Bloch equations. Their intuition seemed to fail for the case of single atoms, and they were unprepared to accept the phenomenon of macroscopic quantum jumps."
This is particularly common in the QM field, especially in relation to single particles or atoms. The tools are optimised for ensembles, so... people seem to forget that single particles exist. The experiments involving single interactions are crazy difficult, so some people will simply state that they're impossible, and it's difficult to argue with that when there are so few experimental results available to counter that.
Note that GR had some similarly "hard" experiments, such as the detection of gravitational waves. That doesn't mean that people seriously said that talking about gravitational waves was meaningless.
> Of course. That's why continuous spectra are a thing.
Okay, so you're on the photons-are-not-always-quantised side. I agree with you, but other people don't.
This is a fundamental disagreement much like Hinduism and Islam, to pick two random religions. Sure, there are silly people saying things like "all religions are equally valid", but.. no, no they are not. Some of them are just local church traditions, some are full on sects, and others aren't even vaguely compatible.
"Quantum theorists disagree with each other on things comparable to the existence of species. As in, some will say that there is such a thing as wave function collapse, others will say that this is nonsense and there's no such thing. They're not disagreeing on exactly when wave function occurs in some obscure corner case. No, they disagree on the existence of the concept itself."
This is an extremely uncharitable analogy. Physics is not a framework to guide your conceptual understanding of the world. It is human nature to try and conceptualize everything but there are limits to this and the interesting areas of physics operate beyond those limits. You can not conceptualize quantum interactions in a way that is meaningful because that process requires forcing our intuition (based on the macroscopic/classical world) into a space where it does not belong. This is why we create mathematical models to "describe" these types of systems and their interactions. To emphasize this, consider how illogical it is to try and "visualize" the fundamental building block that all visualizations are built with? There are visualizations of the corresponding mathematical models but they only provide insight into certain properties, they do not "show" you what a photon interaction looks like because the concept of physical visualization fundamentally doesn't make sense at the quantum level.
I mean, that's because the language of physics is mathematics. If you ask vague questions with inherently undefined terms, you shouldn't be surprised if you get different answers! I bet chefs wouldn't agree on whether a taco is a sandwich, either, but that doesn't mean they don't know what they're doing.
Furthermore, as someone with a lot of experience thinking about photons in many different limiting cases, your objections don't make sense to me, because in general, absolute statements like "There are no X" in physics don't make sense. For almost any value of X, X can be a useful way of thinking in some limit or other, and whether or not you want to invoke X depends on the calculation you're doing. When you say there is no X and everybody who thinks otherwise is wrong, you're just inviting a debate over semantics, which we physicists are naturally not interested in.
> I mean, that's because the language of physics is mathematics.
But it is not a study of it! Mathematics is a tool, not the thing being studied.
This is lost on some people.
There are properties of the real world that are studied, and there are properties of the mathematical models that are used.
They are not interchangeable.
Similarly, saying that the "this specific mathematics is all there is, just understand that" when a simple verbal statement can be made that is not covered by the mathematical model doesn't mean that the statement is somehow invalid, it means the model is incomplete or narrow in scope.
There is this downright perverse notion in QM that the models are what reality is. If the model can't answer something, then the question is meaningless, etc...
Meanwhile the model has assumptions and simplifications such as linearisation, no GR or even no SR, and so forth.
My point is simply this: The mathematics of QM is not equipped to answer fundamental questions like "what is a particle" or "what is a photon".
> whether or not you want to invoke X depends on the calculation you're doing.
I'm not debating that the simplified models aren't useful, widely applicable, numerically useful, make predictions, etc... I'm simply stating that physical reality behaves in single, definite way.
> Similarly, saying that the "this specific mathematics is all there is, just understand that" when a simple verbal statement can be made that is not covered by the mathematical model doesn't mean that the statement is somehow invalid, it means the model is incomplete or narrow in scope.
Not necessarily. It may mean that the verbal statement invokes a language game to define a category that has rules incompatible with the rules of the formal model.
> My point is simply this: The mathematics of QM is not equipped to answer fundamental questions like "what is a particle" or "what is a photon".
Sure it is. A particle in QM is something that obeys the formal structure laid out. Dijkstra had some very biting things to say about being unable to cope with true novelty because you're trapped in reasoning by analogy (EWD1036).
At a risk of reiterating the same argument: You keep agreeing with me and perhaps not realising it.
> incompatible with the rules of the formal model.
Yes. I agree! In many cases this is just means that the statement is "out-of-scope" for the model. That's fine. That however has no bearing on the validity of the statement, unless the model is all-encompassing. Quantum Mechanics is not all-encompassing, it is not a TOE, and this is well known. It's just that people forget this sometimes.
> A particle in QM
Again, you're agreeing with me. The QM model has a notion of particles that is both a mathematical abstraction and a simplification. It's not what the physical world truly does. It cannot "explain" particles.
> obeys the formal structure laid out
So if I were to write a computer program with things in it that obey the same rules, are these things suddenly real, physical particles just like electrons? How about phonons? Holes? Anti-particles?
Particles are abstractions. They're labels humans assign to certain mathematical structures in certain models. Some of them, like leptons, are real things that exist whether or not they're named. Some of them aren't quite as real. Photons are questionable.
Working physicists were surveyed and asked if they think photons are "just a useful abstraction" for orbital interactions or if they're truly real, in the sense that individual photons travel through space like little point particles. They didn't all agree.
Before you argue that I'm wrong somehow, first figure out why theoretical physicists don't agree about such a basic concept.
> Dijkstra had some very biting things
Dijsktra should have known better than to say that, because the converse of this is that it's easy to lose touch with reality by getting too deep in the weeds of abstract algebra.
"But it is not a study of it! Mathematics is a tool, not the thing being studied."
Who knows? Mathematics turns out to such an acute tool for understanding physics. One might be inclined to say that we live in a mathematical reality. The quoted statement is more a statement of philosophy than of fact.
One might be inclined to say this if one doesn't understand the difference between physical nature and our consciousness of physical nature.
Mathematics does not exist outside of human minds. It is a method for measuring and describing the world around us that happens to lend itself to the study of its own conceptual structure.
Nothing in nature follows mathematical laws. Protons don't consult logarithm tables before they move through space. Particles don't consult statistics textbooks before they "decide" if it's time to decay. They act according to their nature, and that's all.
> Ask any working QM researcher to clearly show how "a single photon" interacts with an electron in an orbital and they will without a doubt draw you a cartoon diagram. Never in the history of theoretical physics has anyone, ever, anywhere made a numerical visualisation of this.
I personally have made a simulation of this.
It would be called a semi-classical approximation, in that it's a simulation of a Schrodinger equation-described electron in a hydrogen (Coulomb) potential interacting with a classical EM wave. However I suspect that's how nature works anyway.
Yeah pretty much. (I say this not from a position of profound knowledge or anything like that, just that I'm not familiar with a good argument for why it is, not that I have looked into it in great detail. My knowledge of QFT is very basic)
Ah, fair enough. I was worried you might be professing some sort of expertise while claiming that.
I can assure you that whatever may be true of the electromagnetic field, it is not classical. We know that to about the same degree of certainty that we know that atoms aren't classical.
It's just that when considering atoms in an electromagnetic field, there are often situations where you can ignore the quantum mechanics of the field.
Examples abound, but I’d say my favourite is the electron g-factor which is an example of one of the most precise agreements between theory and experiment ever achieved by humans and it’s calculation relies heavily on the quantum mechanics of the electromagnetic field.
But there’s lot of other, more pedestrian examples. For example, you can (and people have) generated entangled pairs of photons and then performed measurements on this entangled pairs that violate the Bell inequalities, meaning that their correlation is not classical.
Looking back on this, I feel I should also mention that there's not any single piece of evidence that should be bullet-proof convincing evidence for the EM field's quantum nature, rather it's the totality of many many many different, independent results exploring various quantum mechanical properties of electromagnetism (and its high energy unification with the weak nuclear force), all of which converge on the basically undeniable fact that electromagnetism is a quantum mechanical phenomenon.
Of course, this isn't the full story. There is physics beyond the standard model. But whatever that physics is, it's description needs to reproduce quantum electrodynamics in the appropriate limit.
I should also mention that it's not even mathematically consistent to have a classical field like electromagnetism coupled to quantum matter. It's fine in certain approximation schemes, but the breakdown and limitations of such an approach is well documented.
Ok yeah I'll dig up the code and capture a video of it.
I'm currently working on a Dirac equation simulation which should give a more accurate picture of an electron.
Um ... I'd suggest rewording that, as this wording is incorrect. Cosmic background photons would likely want to have a word with you on this as well.
It is true that what a photon is in a physical sense is poorly understood (double slit experiment, etc.). The goal of physics is to try to narrow this range, so that we understand more, and have a smaller null space of "here be dragons and photons" that we don't quite grasp.
If you can accept analogies, it would be reasonable to assume that photons are the free space analogs of phonons. Phonons are excitons of an underlying system, with stress/strain tensors. Phonons interact with physical particles, scattering them. Phonon-like things are responsible for Cooper pair formation (effectively a Bose Einstein condensate of two electrons which are fermions, into a pair of fermions with net integer spin). Cooper pairs are responsible for superconductivity. I am not sure if phonons (which I believe are bosons) would be considered particles mediatiating interactions in solids, but I am guessing a good/strong case could be made, along those lines.
The analogy may be apt. But the question that immediately arises is, given that phonons are excitons of an underlying material, is there something analogous for electromagnetic waves?
Up to the very early 1900's, they called this thing the "aether".
Transverse waves (oscillating perpendicular to velocity vector of travel) of phonons are called optical modes. Longitudinal waves (think like sound waves, compression and rarefaction) are called acoustic. I don't know if we have a longitudinal equivalent in photons.
There are many questions remaining. We don't completely understand QM, up to, and including if QM is the right theory. These questions and tests are good. The more questions a theory answers correctly, the less likely it is to be incorrect. As Feynman noted, you only need one conclusive irrefutable result running at odds with a theory to render the theory defunct.
Even with that said, some "wrong" theories (like the "law" of Newtonian gravity) are immensely useful in practical matters, as long as you pay particularly careful attention to range of applicability. Feel happy you don't normally need to pull out full 10-D general relativity to go to the moon. Newtonian mechanics plus corrections works well enough (for now). This will change the farther we want to go.
This all said, there are many open questions in physics, including numerous things that you hear about every day. Anyone claiming, ever, in any field, that the "science is settled" is either not a scientist, a poor scientist, or someone with an agenda. Science is never settled. Good science is looking at data late at night in your office when you should be home with your family, and saying "hmmm, that's funny ..."
(recovering computational theoretical physicist here, who used to quantum mechanically simulate semiconductors on supercomputers ... quite inception-movie like)
Yes, it's well taught that wavelength is related to frequency. Higher frequency is higher energy energy and lower frequency is lower energy, but it's the frequency of what? Unlike ocean waves, photons do not have a spacial amplitude. A photon does however have a spacial wavelength. This distinction between photonic wavelength and amplitude I've never heard explicitly addressed.
My understanding is that a photon's amplitude is the likelihood of interaction at a specific point in space. High amplitude is likely interaction, low amplitude is low likelihood of interaction. This means at a photon's peaks/crests it has high interaction and at its troughs the photon has low interaction likelihood. These likelihoods are repeated every wavelength, a length measurable in physical space.
This is some of the reason why an antenna for a given frequency range needs to be a specific size. Too small and there's a good chance of missing a signal. A proper sized antenna, related to wavelength, guarantees high interaction somewhere along the length of the antenna.
When speaking of many photons, amplitude (like AM radio) is simply flux, how many photons at a particular frequency exist during a given unit of time. That's not related to the amplitude of a single photon and not what I'm talking about here.
> My understanding is that a photon's amplitude is the likelihood of interaction at a specific point in space.
Not really, because there isn't really a well-defined wave function for a photon in the position representation (which is the one that has the interpretation "the likelihood of detection at a specific point in space").
> When speaking of many photons, amplitude (like AM radio) is simply flux, how many photons at a particular frequency exist during a given unit of time.
There isn't a well-defined answer to this question, because a state like this (which is a coherent state) is not an eigenstate of the photon number operator. (Also, energy flux is really amplitude squared.)
A good, if somewhat lengthy, discussion of the "photon" concept can be found here:
> What is the amplitude of a single photon's wave?
That doesn't have a well-defined answer, unfortunately. Suppose the electromagnetic field contains a single photon. Its electric field then necessarily exists in a superposition of values, so it doesn't have a well-defined amplitude. It also doesn't have a definite "typical" amplitude either -- that depends on how spread out the photon is, and its frequency.
As a rough estimate, a photon of frequency omega has energy hbar omega, so if it's spread out over a volume V, then matching energy densities gives E^2 V ~ hbar omega, so you expect the rms value of E to be of order hbar omega / V.
We have to first specify what you mean by amplitude of single photon. Are you asking what would be the E-field observed from a single photon? That is well defined, however it depends on what photon you are talking about. Is it a photon confined to a cavity made of perfect mirrors? Is it a photon with well defined "position" and poorly defined "frequency"? In all these cases you can answer "what is the E field for a single photon", and you need mostly classical electromagnetism to answer this.
Are you asking about the probability amplitude of the state? This can kinda be answered in the confines of Quantum Mechanics by talking about Klein-Gordon's equation, but truly, it is a mess and not a particularly practical approach. To get useful results you probably need to switch to a more general formalism called Quantum Field Theory, and in that formalism the pop-sci handwavy talk about wave-functions breaks a bit, but it also has a well defined answer.
Ordinary sound waves don't have an amplitude defined by a spatial displacement either - they're pressure waves, where the amplitude is a measure of the change in a scalar that is defined for each position in the volume of interest (the pressure/density).
Similarly for an EM wave the amplitude is the change in a value that is defined for each position in the space of interest - the electromagnetic field strength, although this value is a vector not a scalar. (The EM field is always perpendicular to the direction of propagation of the wave, so for any propagation direction there's only one degree of freedom left which is the "polarisation angle").
Quanta are specifically the quantization of the EM field, so if you will, a single photon at least theoretically is the smallest amplitude of an EM wave you can have. You can see this mostly by noting the expectation of the number operator gives essentially the radiant intensity.
The author left off at precisely the point I was hoping they would get across: How the wave that extends everywhere in space, obviously doesn't. I mean, I know you can make a localised 'packet' or whatever with an infinite sum of sine waves, but then wouldn't you have to talk about infinite photons moving every time 'one photon' hits a detector?
It is a neat trick actually. If you superimpose a few of these waves with frequencies that are fairly close, then you will see how they cancel out in the distance, while they reinforce each other in the center. This is called a "wave packet". 3blue1brown was somewhat useful, somewhat tangential video on the topic https://www.youtube.com/watch?v=MBnnXbOM5S4
The more rigorous way to say "superimpose a few of these waves", is to say that there is some uncertainty in the frequency. This uncertainty in the frequency lets you have more certainty about the position (the wave packet is centered somewhere, instead of being completely distributed the way an infinite plane wave is)
It's an infinite sum of variations of the same photon, not an infinite amount of photons. When it hits the detector (or, really, when the researcher eventually gets entangled with the experiment) it is detected as a single interaction.
This is why the terminology is confusing. A "classical" photon just doesn't exist in the modern treatment but we still need it to talk about what we do.
The blog post was interesting, but I wouldn't say it explained "what is a photon", it explained more, why do photons have energies and propagation properties that they do. The first 2/3 of it I was wondering, why go through all this intro about springs...
Although, there is one thing that I have always searched for an explanation of -- why the speed of light is invariant with reference frame.
The idea that space is filled with springs might help me with that. If light/photons are the passage of a wave from spring to spring, then the speed of that propagation is always at the "spring speed" however you measure it, or however fast you are going. You can only ever receive the microscopic spring transmission speed of the photon at the place you receive it. (Although it brings up the complicated followup of, how are you traveling through the "springs", etc, I guess).
But I'm sure I'm wrong about that too -- if anything QM has taught me is that my attempts at physical intuitions are usually wrong about QM.
This is a nice article, but it feels like it leaves a half of the puzzle to the table. We know by everyday experience that photons _do_ have some time- and place-varying properties: my WiFi has a perfect strength here, but absolutely zero at my friend's house 5 kms away. The photons emitted by my LED lamp are not a static field: I can switch it off by pressing a button so it's time-varying.
The article acknowledges place only in this short sentence in the end:
> This means that adding a quantum to a system has an effect that extends across the entire system. That makes it problematic to talk about the location of a photon.
But then leaves the experience gap between the model and our everyday experience totally without discussion.
This is written by a theorist so they do leave the important bit out. Of course, you know just from fourier analysis the uncertainty principle. The thing is treating photons as plane waves is an approximation (a fine one for most things, particularly theoretical calculations). A better approximation is thinking of photons as wave packets that thus have some localization but not so defined as to make their bandwidth too wide (so they can be considered monochromatic).
The important thing to remember is that photons, like the very idea of a planewave is always an idealized approximation.
Btw. I have an impression that time- and place-local behaviour can be modelled as a collection of photons that destructively interfere "elsewhere" than where we tend to actually see light. Fourier transform seems to play a big role here.
Your statement is not very rigorous, but the intuition it provides is spot on. It is pretty much exactly this intuition I convey the first time I teach such topics.
One thing I remember is that as something approaches the speed of light, its mass increases. This is why we can never achieve the speed of light because as we approach the speed of light, our mass increases which requires more energy to accelerate.
If that’s the case, the how can photons have a mass? Wouldn’t something traveling at the speed of light need to have infinite mass, regardless of how light it started off with?
There is rest mass, which is the mass of an entity when it does not move. And there is relativistic mass, which is the rest mass plus the mass due to all of the kinetic energy of the entity. Photons have zero rest mass, but depending on the energy they carry, they have different relativistic masses.
By the way, only entities with zero rest mass can move at the speed of light. Actually, in relativity it is mathematically inconsistent for something with zero rest mass to exist unless it is moving at the speed of light.
It's "fake" mass. Regular waves have "mass" too, due to contraction (for 3D) or raising (for 2D) of conducting matter, so it's mass of contracted air/water/etc.
I'm surprised that nobody pointed out yet, that harmonic oscillators have discrete energy states as well.
I think that's why the explanation is somewhat makes things more confusing. The fundamentally different thing about quantum mechanics is that even waves in free-space have discrete energy states.
I think the video by 3blue1brown posted by someone else is a much better explanation.
For others who like to learn from video, here is lenghtier explanation of quantum fields, discrete energies, plane waves, etc. from Sean Carroll: https://www.youtube.com/watch?v=Dy1LNk_B6IE
If you have read that, you know the number of photons is just another quantum number, with aspects of uncertainty, superposition and entanglement, so this will not be a surprise:
But they key is right there in the beginning; photons are NOT small billiard balls moving along ballistic trajectories.
"Wave-particle duality" doesn't mean that you can think of flicking a light switch as alternatively an electromagnetic field propagating from the light bulb, or that the light bulb shoots out a stream of tiny billiard balls.
No, there's only the electromagnetic field. However, if you squint closely enough you see that the energy levels of the field are quantized (aka photons).
Now, back to the double slit experiment. So a wave propagates from the source, through the slits, interferes constructively or destructively, and then hits the detector screen. However, due to the quantum nature of, well, everything, the "hits the detector" interaction is quantized. The field "deposits" quantas of energy (photons) to the screen in a localized interaction (say, a photon causes an electron in the screen to jump to a higher energy level). If the detection apparatus is sensitive enough you'll see that single interaction as a dot on the screen, rather than a faint interference pattern (which you'll eventually start to see if you repeat the experiment long enough).
So: Lets forget about the "wave-particle duality" already. There's only fields. Fields which interact in a quantized fashion.
"The normal modes are plane waves traveling at the speed of light in a particular directions with a given frequency."
This is not quite correct. A correct statement would be that plane waves are one possible choice of "normal modes" for the quantum electromagnetic field. But not the only possible choice. (In more technical language, they are one possible choice of basis for the Hilbert space of states of the quantum electromagnetic field, but not the only possible choice.)
In fact, the plane wave states (whose technical name is "Fock states") are often not very useful, because they are very hard to produce experimentally and pretty much never occur naturally. The states that are closest to naturally occurring states and which are easy to produce experimentally (for example in a laser) are coherent states. But coherent states are nothing like plane waves.