# Matter Waves and Electromagnetic Waves

kith
These are nice illustrations but I am looking for a rigorous analysis of the problem. Marcella tried to do this but he wasn't very successful. Or maybe he was successful if we allow his simple state preparation / measurement model. In any case, the critique of Rothman and Boughn shows how classical optics gives a more sophisticated picture.

I just found a paper from 2011 by Beau (http://arxiv.org/abs/1110.2346) which uses the path integral approach and seems promising.

atyy
These are nice illustrations but I am looking for a rigorous analysis of the problem. Marcella tried to do this but he wasn't very successful. Or maybe he was successful if we allow his simple state preparation / measurement model. In any case, the critique of Rothman and Boughn shows how classical optics gives a more sophisticated picture.

I just found a paper from 2011 by Beau (http://arxiv.org/abs/1110.2346) which uses the path integral approach and seems promising.

Actually I think Marcella is ok, since the delta function initial state is fixable (I mean, do we really need rigour here?) I think the main steps that Marcella skipped were that what's actually happening is a position measurement, whereas Marcella calculates a momentum measurement, which is only correct for the very large distance limit (Fraunhofer).

There's also some interesting stuff, necessarily increasingly unrigourous because they include more and more complicated real phenomena like http://arxiv.org/abs/quant-ph/0407245 and http://arxiv.org/abs/1405.4649.

kith
Actually I think Marcella is ok, since the delta function initial state is fixable (I mean, do we really need rigour here?)

I think the main steps that Marcella skipped were that what's actually happening is a position measurement, whereas Marcella calculates a momentum measurement, which is only correct for the very large distance limit.
In classical wave optics, the theoretical description of diffraction at the double slit is well-understood. In order to derive results similar to Marcella's, one solves the wave equation while taking the geometry of the slits into account and by using a number of approximations. I'd like to see a quantum description with the same level of detail. The large distance limit is only a part of this.

Also, there are questions specific to quantum mechanics. For example in the nice visualization you linked to above, a part of the wave is reflected backwards. Doesn't this tell us something about the nature of Marcella's state preparation? And what about dynamics? What can we say about state evolution and what about the time of the measurement? These questions are directly motivated by the postulates of QM, so discussing them is important if the double slit is taken as an example case of quantum behavior.

So no, I'm not satisfied with Marcella's paper. He gives a rough idea of how the formalism could be applied but he doesn't discuss all aspects. And his description is a significant step backwards from the sophisticated classical description.

bhobba
Mentor
The worst kind of people of all are those pretending you could understand physics (particularly quantum theory) without mathematics. Mathematics is the only language precise enough to describe nature adequately. There's no way without math! Without math you must believe, with math you can understand. Thus it's religion without math.

Very true.

In relation to the 6 yo comment I am taken back to a comment Feynman made about books that explain physics to lay persons. It doesn't matter how good the author is they will never be able to do that correctly without math - they will always run in problems and end up saying things that are not quite true. Even he fell into the inevitable trap in his excellent book - QED - The Strange Theory Of Light And Matter. He explained light passing through a medium by absorption and remission - which is wrong:

That's why I love Susskind's Theoretical Minimum books. They require a smattering a calculus but are correct.

Thanks
Bill

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vanhees71
atyy
And what about dynamics? What can we say about state evolution and what about the time of the measurement? These questions are directly motivated by the postulates of QM, so discussing them is important if the double slit is taken as an example case of quantum behavior.

For the dynamics, I was thinking that once the initial state, potential and boundary conditions are specified, it's just unitary evolution. Do you want an analytical solution? I was thinking a numerical solution would be good enough.

As for the observable, perhaps it should be something like ##x \otimes y \otimes 1## (at least conceptually, maybe not technically), where the screen is in the ##x##-##y## plane at a fixed ##z## distance from the slit? There should be a different distribution for each time of observation. The time of observation is not controlled by the experimenter, but it is possible to assign a time stamp to each appearance of a dot on the screen. So one can figure out when god or whoever decided to make something happen, and sort the data according to time stamps, and get different diffraction patterns corresponding to different times of observation, something like what is done in http://www.atomwave.org/rmparticle/ao%20refs/aifm%20refs%20sorted%20by%20topic/ungrouped%20papers/wigner%20function/KPM97.pdf [Broken] (Fig. 3).

But what is hard to get by that method is the density of dots on the plate after a long time. jostpuur and vanhees71 gave the solution for that in posts #57 and #71 of https://www.physicsforums.com/threads/position-eigenstates.764912/.

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kith
For the dynamics, I was thinking that once the initial state, potential and boundary conditions are specified, it's just unitary evolution. Do you want an analytical solution? I was thinking a numerical solution would be good enough.
My astonishment is mainly about pedagogics: the popularity of the double slit makes it seem that it is a standard case of quantum behavior. Yet almost no sources bother to connect it with the full machinery of QM, nor with the well-understood situation in classical optics. I don't think that giving the answers is especially hard although it can be if we make the problem more complex like in the two papers you cited.

vanhees71
Gold Member
It's true! There's a lack of a correct derivation. It's analogous to the optics derivation for classical em. waves, as e.g. given in Sommerfeld, Lectures on theoretical physics, vol. 4.

atyy
My astonishment is mainly about pedagogics: the popularity of the double slit makes it seem that it is a standard case of quantum behavior. Yet almost no sources bother to connect it with the full machinery of QM, nor with the well-understood situation in classical optics. I don't think that giving the answers is especially hard although it can be if we make the problem more complex like in the two papers you cited.

I think the pattern in the long time limit is not elementary. If you look at jostpuur's and vanhees71's solution the boundary condition is pretty slick.

An elementary solution is not obvious because it isn't obvious how observations happening at different times should be weighted. I wonder whether an elementary (ie. grungy brute force and not sophisticated) solution needs an ancilla, so that after a finite duration measurement interaction, the measurements on the ancilla at any sufficiently late time all yield nearly identical results. I think jostpuur's and vanhees71's boundary condition is a very slick way of modelling a strong interaction with the screen.

vanhees71
No - what its doing when not observed is anyone's guess - the theory is silent about it.

Quantum theory is a theory about this thing called a quantum state that encodes the probability of the outcomes of observations:
http://www.scottaaronson.com/democritus/lec9.html

When not observed - the theory says nothing.

There is no such thing as matter waves. It was an interim idea proposed by De-Broglie that was overthrown when Dirac came up with the transformation theory end of 1926. Schroedinger and Heisenbergs ideas were all subsumed in this more general theory.

Thanks
Bill

You stated above the theory is silent what is the electron in the atom doing when not observed. No problem with that.. but can we categorically say that the electron is not moving when not observed that is why it is not emitting electromagnetic wave (as we know moving charge radiate em wave)?

kith
An elementary solution is not obvious because it isn't obvious how observations happening at different times should be weighted. I wonder whether an elementary (ie. grungy brute force and not sophisticated) solution needs an ancilla, so that after a finite duration measurement interaction, the measurements on the ancilla at any sufficiently late time all yield nearly identical results. I think jostpuur's and vanhees71's boundary condition is a very slick way of modelling a strong interaction with the screen.
I wouldn't consider modeling the interaction with the screen to be part of the basic problem. In classical optics, we are interested in the intensity distribution at the location of the screen. The screen itself serves only as a tool to measure this quantity and isn't part of the analysis. Similarily in QM, the screen is the measurement apparatus and therefore not part of the quantum description. The difference between the cases is that we need to run the QM experiment many times in order to compare it with theory, but this doesn't change much.

Of course, we could try to understand what happens in the measurement process and model the interaction with the screen. But this goes way beyond the basic problem.

kith
You stated above the theory is silent what is the electron in the atom doing when not observed. No problem with that.. but can we categorically say that the electron is not moving when not observed that is why it is not emitting electromagnetic wave (as we know moving charge radiate em wave)?
Not categorically. What you say is true in the de Broglie-Bohm interpretation. There, the electron inside the atom is at rest and has an exact position. In the more standard Copenhagen interpretation, it has neither an exact position nor an exact momentum/velocity. You cannot say that it is at rest because this would require an exact velocity of zero.

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Not categorically. What you say is true in the de Broglie-Bohm interpretation. There, the electron inside the atom is at rest and has an exact position. In the more standard Copenhagen interpretation, it has neither an exact position nor an exact momentum/velocity. You cannot say that it is at rest because this would require an exact velocity of zero.

If the electron is at rest in the de Broglie-Bohm interpretation, why doesn't it fall down to the nucleus?
In Copenhagen, are you saying it is moving yet with no exact position and no exact momentum/velocity, but won't this cause it to sporadically radiate em wave?

kith
If the electron is at rest in the de Broglie-Bohm interpretation, why doesn't it fall down to the nucleus? In Copenhagen, are you saying it is moving yet with no exact position and no exact momentum/velocity, but won't this cause it to sporadically radiate em wave?
Your ideas that the electron should fall down to the nucleus or radiate are based on classical electromagnetism. Maybe there are answers which conform better to this intuition but in the end, it boils down to the fact that classical electromagnetism is wrong and that you have to use a quantum description for both the electron and the electromagnetic field.

One thing is that a electron in the ground state can't lose energy by radiating. If it gets into a state where its position is constrained to a volume smaller than that of the ground state, the uncertainty in momentum increases such that the energy of the electron is bigger than in the ground state. The existence of the ground state can thus be viewed as a direct consequence of the Heisenberg uncertainty principle.

My advice is to learn at least the basic maths of QM in order to see such things for yourself.

bhobba
Mentor
You stated above the theory is silent what is the electron in the atom doing when not observed. No problem with that.. but can we categorically say that the electron is not moving when not observed that is why it is not emitting electromagnetic wave (as we know moving charge radiate em wave)?

Of course not - that's what silent means.

In fact Bohmian Mechanics says it has a well defined position and momentum when not observed.

Thanks
Bill

bhobba
Mentor
If the electron is at rest in the de Broglie-Bohm interpretation, why doesn't it fall down to the nucleus?

Because its guided by a potential that prevents that.

In Copenhagen, are you saying it is moving yet with no exact position and no exact momentum/velocity, but won't this cause it to sporadically radiate em wave?

Its silent. That means it says nothing, it could be doing all sorts of things, dancing a jig, taking a trip to Mars and back, it doesn't matter, the interpretation doesn't worry about it..

Thanks
Bill

Because its guided by a potential that prevents that.

Its silent. That means it says nothing, it could be doing all sorts of things, dancing a jig, taking a trip to Mars and back, it doesn't matter, the interpretation doesn't worry about it..

Thanks
Bill

Perhaps it is better or easier to just think or imagine the electron doesn't exist as particle in between measurement.. and you just have the matter wave (or wave function) existing in the orbital... is it not incorrect to think this way?

bhobba
Mentor
Perhaps it is better or easier to just think or imagine the electron doesn't exist as particle in between measurement.. and you just have the matter wave (or wave function) existing in the orbital... is it not incorrect to think this way?

I tend to think that way - its perfectly OK.

Note though the wave-function is not necessarily real.

Thanks
Bill

I tend to think that way - its perfectly OK.

Note though the wave-function is not necessarily real.

Thanks
Bill

But if you have to think of the electron as not existing when not measured (or interacted), then the wave function has to be real or else the atoms would just fall apart. I can't imagine the electron not existing and yet the wave function not existing in the atom either... can you? think of this deeply...

bhobba
Mentor
But if you have to think of the electron as not existing when not measured (or interacted), then the wave function has to be real or else the atoms would just fall apart..

That doesn't follow. You a making all sorts of tacit assumptions that may or may not be true. In fact in atoms the electrons are entangled with the nucleus - we simply model them as separate to get a mathematical handle on the situation.

Thanks
Bill

vanhees71
Gold Member
If you have an atom, e.g., prepared to be at rest in your (inertial) reference frame, this means that its center of mass is not moving but that the nucleus and the electron are moving around each other (taken the average positions of these "particles" as their position). You don't need esoterics for this but just quantum theory in the minimal interpretation.

Now to see, what's about radiation emitted from the atom you have to work in full QED, i.e., you have to consider the system of the nucleus, the electrons, and the quantized radiation field. Provided the atom is isolated from its environment (FAPP), then it does not radiate if and only if its in the ground state. All other bound states of the perturbative treatment, where the interaction with the em. radiation field is neglected are in fact instable when the coupling to the radiation field is taken into account. The atom will rearrange itself in the ground state emitting one or more photons in this process. The photons have a small but finite width, which is inverse to the mean lifetime of the excited states.

Of course, it's not to be confused with classical bremsstrahlung from the charged particles within the atom. This phenomenon of the stability of atoms (in the strict sense in the ground state) cannot be understood in classical terms and this was one of the facts that lead to the discovery of quantum theory in 1925/26.

Bohrs model of 1911/12 and Sommerfeld's extension was an important step towards this discovery, but it's in almost all aspects wrong, even qualitatively. There are no "Bohr orbits", and consequently there's no necessity ad-hoc assumption about "orbits, where the electron doesn't rotate". The only way to understand the atom is quantum theory. You can go quite far with non-relativistic quantum theory in the semiclassical limit (i.e., treating the em. field as a classical Coulomb potential rather than the full QED treatment, and this can be proven from QED; see, e.g., the excellent QED treatment of the hydrogen atom in Weinberg's Quantum Theory of Fields, vol. 1).

My astonishment is mainly about pedagogics: the popularity of the double slit makes it seem that it is a standard case of quantum behavior. Yet almost no sources bother to connect it with the full machinery of QM, nor with the well-understood situation in classical optics.
I shared the astonishment, and then I realize that most depictions of the double slit experiment, not only in pop-sci but in many QM textbooks, use a highly distorted description and interpretation of the experiment. Maybe the intention is pedagogical simplification but it seems to be at the cost of seriously deviating from QM. And this trend is followed by many papers concentrating on "wich path" variants , "delayed choices" and "quantum erasers" experiments, they all seem to follow the naive introduction picture and extend a basic misunderstanding about QM.

They all rely on a very old and wrong particle-wave duality conception in wich wavefunctions are either behaving as classical waves or classical particles but never at the same time(complementarity),when one doesn't know which way the "classical particle"(since only classical particles have trajectories) travelled they'd be behaving as classical waves and you get classical superposition interference pattern, but if you manage to assign which slit it went thru it obviosly is behaving as a classical particle and you get a pattern compatible with what you'd obtain if you were shooting ping-pog balls thru two holes. This conception first fails to acknowledge the difference between classical and quantum superposition and second fails to abandon the concept of electrons(or any other fields) as classical particles from the moment it considers it can have a classical trajectory and it can be determined wich one it is.
All this made sense in the first years of QM but not now.
IOW the simplified versión treats the wave function in the double-slit as a pure state with two possibilities, wave or particle, when in fact there is degeneracy and one can set up the experiment in different ways wrt the relative phase so that interference patterns are more or less evident in the screen. I guess not many people tries the rigorous treatment of the experiment because most are comfortable with the old-style Copenhagen picture of collapse and complementarity.

vanhees71