I'm reading Manjit Kumar's Quantum, which presents the historical decelopment of a physical field I'm absolutely oblivious to, and it got me wondering about something regarding the locality of photons and their constituent waves.
If a photon is the sum of a wide range (infinite?) of simple sine waves, wouldn't those waves have to occur instantaneously over a long (infinitely?) distance? Or am I simply going too far, and our view of a spacely limited electromagnetic wave as being the sum of sine waves just that, our view? (Perhaps mine exclusively?)
Any help or a nudge in the right direction will be greatly appreciated.
Locality of a Photon's Constituent Waves
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Re: Locality of a Photon's Constituent Waves
Not exactly sure if this is what you're asking about, but any quanta (be it a photon or an electron or a Prince Charles) is a wavefunction extending over all space with a set expectation value. The wavefunction collapses (or, perhaps more accurately, resolves) to a single location once it is observed.
Re: Locality of a Photon's Constituent Waves
But is it, or is it more of a can be explained as? I'm struggling with the notion that a wavefunction should instantly occur "all over space".
Re: Locality of a Photon's Constituent Waves
Slothrop wrote:But is it, or is it more of a can be explained as?
Why do you think there's a difference? The behaviour of particles such as photons can be explained using quantum theory, in which the particle is represented by a wavefunction.
I'm struggling with the notion that a wavefunction should instantly occur "all over space".
It doesn't magically suddenly appear, it's just a thing defined everywhere and evolving according to the rules of the theory. Of course, wavefunction collapse does occur instantaneously (depending on the specific interpretation in play): if that freaks you out then, well yes, it should.
Bear in mind that in vanilla quantum mechanics, particles are not created or destroyed: for that you need to move up to quantum field theory, in which any individual photon is just an excitation of the field. Which again exists everywhere, forever, and evolves according to the rules.
Re: Locality of a Photon's Constituent Waves
elliptic wrote:Slothrop wrote:But is it, or is it more of a can be explained as?
Why do you think there's a difference? The behaviour of particles such as photons can be explained using quantum theory, in which the particle is represented by a wavefunction.
I get the feeling that it's a difference in that the wavefunctions are, in my mind, modeling tools for our understanding of reality, while it objectively is something completely different. What this "completele different" might be, I have no idea ... I just get the feeling that the wave particle duality is due to the fact that it's neither. (I realize these are hardly revolutionary thoughts.)
elliptic wrote:Bear in mind that in vanilla quantum mechanics, particles are not created or destroyed: for that you need to move up to quantum field theory, in which any individual photon is just an excitation of the field. Which again exists everywhere, forever, and evolves according to the rules.
Since I don't know these rules: do they imply a pebble causing instantaneous waves over the whole pond? Because that seems to be my main problem.
elliptic wrote:... if that freaks you out then, well yes, it should.
That is, honestly, comforting.
Re: Locality of a Photon's Constituent Waves
Slothrop wrote:I get the feeling that it's a difference in that the wavefunctions are, in my mind, modeling tools for our understanding of reality, while it objectively is something completely different. What this "completele different" might be, I have no idea ... I just get the feeling that the wave particle duality is due to the fact that it's neither. (I realize these are hardly revolutionary thoughts.)
To make things even worse for you the "wavefunction" formulation is just one of several equivalent ways of expressing the maths involved: the others are considerably more abstract. So yes, you kinda have to step back from asking how the world actually is, and just accept a good description of how it behaves.
You also have to be very wary of hoping for something more intuitive lying behind. We know that QM *is* a good description and also that you can't have a classical theory which reproduces all of its predictions (Bell's theorem).
Slothrop wrote:elliptic wrote:Bear in mind that in vanilla quantum mechanics, particles are not created or destroyed: for that you need to move up to quantum field theory, in which any individual photon is just an excitation of the field. Which again exists everywhere, forever, and evolves according to the rules.
Since I don't know these rules: do they imply a pebble causing instantaneous waves over the whole pond? Because that seems to be my main problem.
No, except for measurement/collapse the wavefunction evolves according to the appropriate wave equation (Schrodinger's equation in nonrelativistic QM) which gives nice smoothly evolving waves spreading out, interfering with each other and bouncing off the environment: very much like normal ripples in a pond.
Re: Locality of a Photon's Constituent Waves
elliptic wrote:...the wavefunction evolves according to the appropriate wave equation (Schrodinger's equation in nonrelativistic QM) which gives nice smoothly evolving waves spreading out, interfering with each other and bouncing off the environment: very much like normal ripples in a pond.
Out of curiosity: at what speed do the waves travel?
If I were to try to get a better overview of this, something a little deeper than the popular introductory book on physics, do you have any suggestions on where to begin?
Thanks for all the help, it's appreciated.
Re: Locality of a Photon's Constituent Waves
Well you can get a "speed" by matching the coefficients of the Schrödinger equation to those of the classical wave equation.
This gives you the speed as
sqrt(iħ/2m)
The sqrt(i) bit just tells you about the relative phase of the wave's time derivative so isn't particularly relevant if you want to think of it as a speed.
This wave is nondispersive which makes sense because we don't see particles getting wider with time. Passing through apertures will create waveguidelike effects and create dispersion ; the wave will also diffract giving us all these wonderful wavelike features as well.
This speed is only from the Schrödinger equation and so is nonrelativistic. I've also implicitly ignored any potential so this really only applies to nonrelativistic massive particles in a potentialless box.
This gives you the speed as
sqrt(iħ/2m)
The sqrt(i) bit just tells you about the relative phase of the wave's time derivative so isn't particularly relevant if you want to think of it as a speed.
This wave is nondispersive which makes sense because we don't see particles getting wider with time. Passing through apertures will create waveguidelike effects and create dispersion ; the wave will also diffract giving us all these wonderful wavelike features as well.
This speed is only from the Schrödinger equation and so is nonrelativistic. I've also implicitly ignored any potential so this really only applies to nonrelativistic massive particles in a potentialless box.
my pronouns are they
Magnanimous wrote:(fuck the macrons)
Re: Locality of a Photon's Constituent Waves
Slothrop wrote:elliptic wrote:...the wavefunction evolves according to the appropriate wave equation (Schrodinger's equation in nonrelativistic QM) which gives nice smoothly evolving waves spreading out, interfering with each other and bouncing off the environment: very much like normal ripples in a pond.
Out of curiosity: at what speed do the waves travel?.
For a free particle (eg. electron) with precisely defined momentum, you get a pure sine wave at a single wavelength traveling at exactly half the particle's classical velocity (yes, that does seem a bit odd). Because it's a pure sine wave its envelope is constant everywhere in space, which means maximum uncertainty in the electron's position.
If you want to localise the electron then you have to superpose more sine waves at different wavelengths to get a wave packet ie. a "bulge" in the combined envelope. Those component sine waves travel at varying speeds depending on their wavelengths (the technical term for that is "dispersion") and the spread of those speeds represents the uncertainty in momentum you now have. But the wave packet (bulge) propagates at a combined "group velocity" which turns out to match the classical velocity, as you'd expect.
If I were to try to get a better overview of this, something a little deeper than the popular introductory book on physics, do you have any suggestions on where to begin?
This seems quite a good resource for an undergradlevel treatment: http://www.eng.fsu.edu/~dommelen/quantum/style_a/II.html
Re: Locality of a Photon's Constituent Waves
Thanks for all the help, guys. I'll get back to you after I've sat down with the resources elliptic provided ... some questions and "what the?"s are bound to occur.
It's dawned on me that my understanding of how this whole thing works is less than stellar ... which is concerning given that I spent two years teaching introductory physics.
It's dawned on me that my understanding of how this whole thing works is less than stellar ... which is concerning given that I spent two years teaching introductory physics.
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