Quanta and localized wave packets

[KipIngram]

Hi guys.  I've been away for a while - busy with work and family and so on, and I also got a little weary of the political and social narrow-mindedness that comes up here sometimes.  But the quest continues - I've still been prowling the internet for good papers on quantum theory and so on.  A few days ago I ran across Schrodinger's original 1926 paper, where he lays out quantum theory from ground zero, rather than via given axioms like most modern treatments use.  I've found the connection with optics to be a VERY helpful mental image.

So, if we start from the beginning and presume that matter is a wave phenomenon, and work through the development alluded to above, we wind up with Schrodinger's equation connected very clearly to classical mechanics.  I see how uncertainty arises - building a wave packet with a sharply defined position requires many frequency components, and each of those has a different momentum, so the  momentum becomes more and more fuzzy as we make the position more well-defined.

I'm left a little confused, though, trying to connect this to other stuff I've read.  Specifically, if we have a localized wave packet, with many frequency components, doesn't that automatically mean that we have many quanta of energy?  At least one for each frequency component?  Is it even possible for a single quantum to have a localized position?  It seems that a single quantum would have to reside in a single mode of the field, and that mode by definition is spread out over all of space.

Another angle on the same quandary.  Let's say we send a single quantum of an electron field through the double sit apparatus.  That quantum is spread out all over the place, and strikes the detector with an intensity given by the interference bands of the apparatus.  So far so good.  But when we get a flash in a specific spot on the screen, that seems to imply that there's a sharply localized event, which seems that it would require many quanta, associated with different modes so that they combine to give that very localized effect.  So how did we get from one quantum to many quanta?

I feel like there's something I'm missing here - but I know there are people here who can set me straight. Maybe the electron field passing through the apparatus has many quanta of energy, just all in that one mode?

I might should open another topic for this next question, but I'll put it here anyway and see how things go.  This "optical analogy" path that I read up on the last few days makes it very clear to me why electrons bound to atoms can only have specific energy levels - that comes right out of the solution to the equation.  So that degree of "quantization" is making a lot of sense to me.  But I think in the modern point of view quantization goes much further - it applies everywhere rather than just in isolated situations.  What sort of thought process do I use to step on to that more advanced point of view?

I hope all of you guys have been doing well - sorry to have been so absent.

Kip

 

[Rob McEachern]

You might find my unfinished slideshow "What went wrong with the “interpretation” of Quantum Theory?" ( http://vixra.org/pdf/1707.0162v1.pdf ) to be of interest, in answer to your questions. The central points are:

(1) De Broglie introduced superpositions into classical mechanics, to enable Fourier transform based descriptions of "tracking wave packets", attached to every particle.

(2) The Schrodinger equation is a purely mathematical, Fourier transform based description of motion - it is pure math, not physics; no laws of physics are involved in its derivation. It merely describes the motion of the tracking wave packet, attached to each particle.

(3) The double slits can be interpreted as particles scattering off a rippled field within the slits, rather than as any sort of wave interference phenomenon behind the slits.

(4) Unlike classical scattering, the limited information content of a quantum system (corresponding to a single bit of information) means that the particle and field cannot even detect the existence of each other (and thus cannot interact with each other) except at the few specific points in which it is possible for a particle to detect/recover at least one bit of information from the field. This is the explanation for the behavior described in David Bohm's 1951 book, concerning the discontinuous versus continuous deflection of a particle in a quantum field versus a classical field respectively.

[KipIngram]

Hi Rob.  Thanks for the reply.  Since reading Hobson's paper "There are no particles, there are only fields" I have, in fact, been operating with that as a working assumption.  So references to particles and fields aren't quite in my thought sphere right now.  Hobson describes arguments that assert the mere existence of an entity we could properly call a "particle" is incompatible with the combination of quantum theory and special relativity.  I find those arguments compelling, and am willing to operate for now on the presumption that everything is fields.  Schrodinger's paper pursued the same idea: that matter is wave based from the outset, and then the Schrodinger equation arose from analogy with the Hamilton-Jacobi equation.  Classical particle physics arises by direct analogy with the transition from true wave-based optics to (approximate) geometric optics.  All of that seems to have striking physical content to me.

I have done some reading on information approaches to quantum theory in the past, and there are some very interesting things going on there that I don't fully grasp yet.  I'm certainly open to more information on that front.

Anyway, our whole conception of "particles" arises from our day-to-day experience with macroscopic objects, and the work alluded to above shows how that behavior arises completely as a geometric optics like approximation of underlying wave essence.  We simply do not have day-to-day experience that has shaped our intuition that directly deals with atomic-scale systems, so it's entirely explainable why we find particles so satisfying.  But they don't appear to be "needed" as far as I can tell.

Cheers,

Kip

Edited November 19, 2017 by KipIngram

[swansont]

17 hours ago, KipIngram said:

Specifically, if we have a localized wave packet, with many frequency components, doesn't that automatically mean that we have many quanta of energy?

You can't have a well-defined frequency if you have one quantum. There will always be some frequency uncertainty. To have a pure frequency requires an infinitely long wave.

[KipIngram]

Hi swansont.  Yes, that makes sense, but some of the discussions here on QFT led me to believe that's exactly what QFT field modes were - fully space-filling, sinusoidal structures that changed everywhere instantaneously when that mode gains or loses a quantum.  I thought when a QFT field absorbed a quantum, that quantum was present everywhere in space per the shape of that particular mode.  I admit I'm very weak in all of this still, though, and your comment furthers my feeling that I'm missing some important piece of all of this.

Edited November 19, 2017 by KipIngram

[Mordred]

OK you are absolutely correct that the waveform you often see in textbooks isn't precisely telling the full story. The reality is that no waveform is ever singular. They never have just 1 frequency and wavelength. In fact there is always a small spread of frequencies. Most textbooks of a frequency is usually referring to the superposition state of these frequencies. So the localized wave packet is a superposition  of many wave functions in the same localized region. One must decompose a wave packet into the sine and cosine waves via Fourier analysis of which there is two broad categories 1) periodic 2) non periodic and non repeating the latter is the localized wave packet.

(little side note probably why the packet term is used)

The function A(k) is the Fourier transform of f(x) it gives the weighting of cosine functions with different values of k that make up the wave packet via [latex] f(x)=\int A(k)cos(kx)dk[/latex] an easy way to understand the above is A(k) gives the distribution of wave numbers k that make up the localized wave packet.

so Heisenburg [math]\Delta x\Delta p\ge\frac{\hbar}{2}[/math] this tells us we cannot precisely know the the position and momentum

De-Broglie tells us [math]p=\frac{h}{\lambda}=\hbar k[/math]

the uncertainty (range of values of the momentum) [math]\Delta p=\hbar\Delta k [/math]

where [math]\Delta k[/math] is the spread in k values in a wave packet that represent a particle. So in terms of the uncertainty the above becomes in terms of k as [math]\Delta x\Delta k\ge\frac{1}{2}[/math]

see for the above formulas etc the reference I used for the above

https://www.google.ca/url?sa=t&source=web&rct=j&url=https://ps.uci.edu/~cyu/p51A/LectureNotes/Chapter6/Chapter6_wf.pdf&ved=0ahUKEwiA67HWmszXAhUpxlQKHdasAPMQFgggMAE&usg=AOvVaw3EJ1PhBYQbEOvkZgKVgrga

The above being a quick brief from the reference. Its been a while since I last thought of this in terms of the HUP lol

Edited November 20, 2017 by Mordred

[KipIngram]

I'm not sure my question is coming clear here.  I already did understand the fact that in the real world there's never a perfect, infinite sinusoidal  mode, and that all real wave patterns are combinations of sinusoids such that they do not extend to infinity and so on.  The math nuances of Fourier theory I think I already understand fairly well.

Let me try again.  I've read many descriptions of experiments over the years (let's use the double slit experiment as a reference point), where mention is made of lowering the intensity of the beam until only single quanta move through the apparatus at a time.  The interference pattern still appears, and the customary explanation is that even though the interaction at the screen is localized (a flash), the business going on at the slits is that a spread out field wave passes through the slits, creates spherical wave fronts emerging from each slit, and that those waves interfere with each other at the screen.

So I want to take this literally - that one quantum is released through the slits.  My questions then are 1) how does that one quantum get spread across all of the components of a wave *packet*, so that it's spread out some for purposes of going through the slits but not spread out all the way to infinity the way a pure monochromatic beam would be, and 2) how do we get a localized effect at the screen, which would seem to require a very sharply localized wave packet and thus require many frequency components to be present.

I thought that a single quantum of energy could only be in one mode of the field - not spread out across a bunch of modes to get a packet.  Maybe this isn't strictly correct and is my problem.

Thanks,

Kip

[swansont]

1 hour ago, KipIngram said:

 I thought that a single quantum of energy could only be in one mode of the field - not spread out across a bunch of modes to get a packet.  Maybe this isn't strictly correct and is my problem.

Superposition of states. And I don't think that you are necessarily in more than one energy mode of the field. The superposition is in the possible paths.

 

[koti]

I’m confused... is superposition the only available answer to Kip’s question? I’d love to see some detail layed out, @Janus ?

[swansont]

According to Feynman path integrals, the photon takes all paths. Most of the results are zero because of interference. That gives you some possible results, which would be a superposition until you detect the photon. 

[KipIngram]

Maybe it will help if I try to identify where I think my misunderstanding might be.  After thinking about this, I believe I have assumed that it is necessary for a quantum of energy to reside in only one mode of a field - that if an entity is "many modes," as is necessary for it to be physically localized, it is of necessity many quanta.  But this morning I'm thinking maybe that's wrong - maybe the two things have nothing to do with one another.  Maybe a single quantity can still be a superposition of many modes, any one of which might be the one sampled by a momentum measurement, for instance.

After all, it is possible to measure the position of a single particle - say an electron - and an electron is a quantum of the electron field.  If we sharply measure its position, we have to call into service many different modes which we can superpose to produce that sharply localized position.  But it's still just one quantum.  Whereas if we sharply define its momentum, we may only need a small number of modes.

So maybe my mis-thinking from the outset was to get it in my mind that each quantum had to correspond to just one modal component of the field.

I think this is what swansont was getting at - even just one quantum can still be in a superposition of many modes.  Maybe - still kind of wandering in the dark, but I'm trying. 

Didn't see swansont's last reply when I posted my last one.  Yes, I see what you're getting at.  I think I'm falling into the beginner's trap of trying to view the quantum as something well-defined, perhaps well defined in a way that supports a precise position measurement, or a precise momentum measurement, but well-defined nonetheless.  And of course that's just wrong - it's all of those things at once, subject ot a probability distribution based on past events.  But it's still just one quantum.  When I asked if a quantum has to reside in a single mode, the answer is "no" - that would imply it actually *has* a precise momentum and a wholly undefined position.

So I guess the way we associate a modal distribution to a quantum depends on what we know about it - if we've done a precise position measurement we now use a tightly localized collection of modes to describe it.  It's position now has meaning because we've done that - but conversely its momentum is now very uncertain, because the frequencies we had to use in choosing those modes are diverse.

Didn't see swansont's last reply when I posted my last one.  Yes, I see what you're getting at.  I think I'm falling into the beginner's trap of trying to view the quantum as something well-defined, perhaps well defined in a way that supports a precise position measurement, or a precise momentum measurement, but well-defined nonetheless.  And of course that's just wrong - it's all of those things at once, subject ot a probability distribution based on past events.  But it's still just one quantum.  When I asked if a quantum has to reside in a single mode, the answer is "no" - that would imply it actually *has* a precise momentum and a wholly undefined position.

So I guess the way we associate a modal distribution to a quantum depends on what we know about it - if we've done a precise position measurement we now use a tightly localized collection of modes to describe it.  It's position now has meaning because we've done that - but conversely its momentum is now very uncertain, because the frequencies we had to use in choosing those modes are diverse.

I believe I thought myself into this trap as a result of reading some of the early papers (e.g., Schrodinger's 1926 paper).  Following the analogy between optics and geometric optics, it becomes tempting to try to attach some real, tangible nature to the wave function.  But bearing in mind that the wave function itself is unobservable seems to help lead out of the problem.  Maybe that's why modern treatments of the theory eschew that intuitive connection with classical mechanics and go straight for the clean axiomatic presentation.