Interaction between wavefunction of two particles

[San K]

What I am trying to find out is: whether the superposition of a singe photon (across the all the paths --that the photon can possibly take) interfere/interact with the superposition of another single photon... assuming that the superimposed paths of both the photons are close enough to interfere.

or better still ...lets take the DBB interpretation...which posits that the wave-function travels both the slits.

http://upload.wikimedia.org/wikipedia/commons/0/02/Doppelspalt.svg

Would this wave-function interfere with the wave-function of another photon?Ques 1. Do wave-functions (or better still - matter waves) of two different photons interact?... assuming that their paths are close enough to interfere.

for example -- would the wave-function of Photon A interact with the wave-function of Photon B?

or in other words...

Would the superposition of photon A (across the paths) interact with the superposition of photon B (across the paths)

assuming that -- in the experimental setup -- the paths of A and B are within range of interference/interaction

As an example let's say

we pass photon A via a double slit (labelled as DSA)
at the same time
we pass photon B via a double slit (labelled as DSB)

Lets say the DSA and DSB are (constructed/placed) close enough for interference.

Would there be some sort of interaction/interference between wave-function of A and wave-function of B?Ques 2. Does the waves-function of Photon A interact with the probability waves of Electron B?

as an example, we can repeat the above experiment and photon B is not replaced with an electron B.

Ques 3. Is the wave-function modelling (what some interpretations would conceptualize as) matter waves?

Note: in the below post -- matter waves, wave-functions, probability waves etc...have been used interchangeably. There might be a better term for it and a better way to put the idea across.

 

[physiologistK]

I just entered this forum and I do not yet know how I might fit in here. Trying to understand QM on the level of language rather than by mathematics seems to be difficult.

Yet I would like to try to give an answer to your question.

Experimental fact is this: In a double slit experiment you can reduce the intensity of the incoming radiation to a level, so that you will almost never have two 'particles' running at the same time. Still the interference pattern will be produced. So one might conclude that a particle is able to interact with itself.

On the other hand I thought it might be easier to think, that the wave function is a property of the experimental set up or the space of this set up, rather than a property of the particle. An electron entering this space would then just behave according to the rules of the wave function conveying the properties of the space . The question of how A interacts with A or B would then become irrelevant.

[There are anyway no fixed identities of particles, if I got it right. The electrons form a collective. And if there is only one electron at a time, it is still a collective. Therefore it seems natural to me to see the properties of collective behaviour as a property of the space rather than the property of the electron.]

The conditions under which wave functions can be added are not clear to me. May be asking for that might be my first question on this forum. I always thought, that it might have something to do with coherence. But I am not sure what this really means.

 

[Simon Bridge]

You appear to be misunderstanding core concepts in quantum mechanics.

1. wavefunctions do not interact at all. Thus the answer to pretty much every question you have asked is "no".

2. interference and diffraction is an emergent property from many individual particles - it is not something that happens to individual particles.

3. there is no such thing as "matter waves".

It sounds like you have been watching Discover Channel.

 

[San K]

Welcome to the forum physiologistK

As you correctly point out, the question is:

Does the wavefunction of A interact with the wavefunction of B?

Simon says no.

I just entered this forum and I do not yet know how I might fit in here. Trying to understand QM on the level of language rather than by mathematics seems to be difficult.

Yet I would like to try to give an answer to your question.

Experimental fact is this: In a double slit experiment you can reduce the intensity of the incoming radiation to a level, so that you will almost never have two 'particles' running at the same time. Still the interference pattern will be produced. So one might conclude that a particle is able to interact with itself.

On the other hand I thought it might be easier to think, that the wave function is a property of the experimental set up or the space of this set up, rather than a property of the particle. An electron entering this space would then just behave according to the rules of the wave function conveying the properties of the space . The question of how A interacts with A or B would then become irrelevant.

[There are anyway no fixed identities of particles, if I got it right. The electrons form a collective. And if there is only one electron at a time, it is still a collective. Therefore it seems natural to me to see the properties of collective behaviour as a property of the space rather than the property of the electron.]

The conditions under which wave functions can be added are not clear to me. May be asking for that might be my first question on this forum. I always thought, that it might have something to do with coherence. But I am not sure what this really means.

 

[vanhees71]

Indeed, a two-particle system is not described by two single-particle wave functions but, e.g., by one wave function $\Psi(\vec{x}_1,\sigma_1;\vec{x}_2,\sigma_2)$, where $(\vec{x}_1,\sigma_1)$ are position and spin projection of the first and $(\vec{x}_2,\sigma_2)$ those of the second particle.

The physical meaning of such a two-particle wave function is that
$$\mathrm{d} P=|\Psi(\vec{x}_1,\sigma_1;\vec{x}_2,\sigma_2)|^2 \mathrm{d}^3 \vec{x}_1 \mathrm{d}^3 \vec{x}_2$$
to find the first particle in a volume element $\mathrm{d}^3 \vec{x}_1$ around $\vec{x}_1$ with spin projection $\sigma_1$ and the second particle in a volume element $\mathrm{d}^3 \vec{x}_2$ around $\vec{x}_2$ with spin projection $\sigma_2$.

 

[San K]

Thanks Vanhees and Simon.

Vanhees -- by two-particle: do you mean not necessarily entangled?

My post is referring to two non-entangled photons.

 

[physiologistK]

San K, thanks for the welcome.

I guess, what Simon critisizes in the first place, is that you seem to use the terms interaction and interference synonymously. You mean interference. The term interaction is preserved in physics for something else, something that would most likely destroy the interference pattern.

In textbooks one can find descriptions of electrons A and B arising at slit 1 and 2 as semispherical waves. If you let them interfere (Ψ = Ψ_A + Ψ_B), they give the correct outcome/interference pattern for the double slit experiment. Therefore this view may have some justification.

However I do not like this view. The notion of two distinguishable electrons at the two slits, having traveled on two distinguished paths to their slits, contradicts the inner meaning of the quantum state.

Therefore I prefer the view, that the wave function is somehow prior to the particle. It is even there without particle. In bound states this is normally accepted: An atom has its electron shells. Whether the orbitals in the shells are really occupied by electrons, is a secondary question. And in QFT, as far as I understand, it is similar: You have a quantum structure even without quantum particles. And only therefore you can have non-zero vacuum energy.

I am not sure whether Simon was directly opposing such thinking. He is of course right that one needs many particles in order to make the interference pattern visible at the end of the experiment. I hope I get critized if something of my views is wrong.

And then the topic about single versus multiple particle functions: My view was that an electron beam, as used in a double slit experiment, say with 10000 electrons corresponds to a single particle wave function, not a 10000 particle wave function. There is also nothing like a Pauli exclusion principle acting on such a beam. Not sure why this is so. May be because electrons of the beam are well defined in their impulse, but very purely defined in their position.

 

[San K]

However I do not like this view. The notion of two distinguishable electrons at the two slits, having traveled on two distinguished paths to their slits, contradicts the inner meaning of the quantum state.

Therefore I prefer the view, that the wave function is somehow prior to the particle. It is even there without particle. In bound states this is normally accepted: An atom has its electron shells. Whether the orbitals in the shells are really occupied by electrons, is a secondary question. And in QFT, as far as I understand, it is similar: You have a quantum structure even without quantum particles. And only therefore you can have non-zero vacuum energy.

I am not sure whether Simon was directly opposing such thinking. He is of course right that one needs many particles in order to make the interference pattern visible at the end of the experiment. I hope I get critized if something of my views is wrong.

And then the topic about single versus multiple particle functions: My view was that an electron beam, as used in a double slit experiment, say with 10000 electrons corresponds to a single particle wave function, not a 10000 particle wave function. There is also nothing like a Pauli exclusion principle acting on such a beam. Not sure why this is so. May be because electrons of the beam are well defined in their impulse, but very purely defined in their position.

The idea of indistinguishably and single wave-function sounds good. It seems to answer a lot of questions.

Does it integrate with DBB (De Broglie Bohm) interpretation as well?

In de Broglie–Bohm theory, the wavefunction travels through both slits, but each particle has a well-defined trajectory that passes through exactly one of the slits.

The wave function interferes (ok, not interacts) with itself and guides the particles in such a way that the particles avoid the regions in which the interference is destructive and are attracted to the regions in which the interference is constructive, resulting in the interference pattern on the detector screen.

I read somewhere how Pauli ties into this.

If we can call Heisenberg uncertainty principle HUP, then we can call Paulis exclusion principle as PEP...:)

Have to go however will get back on Pauli and the rest.

 

[physiologistK]

I cannot answer your question. I try to understand QM in the interpretation, that is most standard. And this is the Kopenhagen interpretation. I do not have the capacity to go beyond that. Trying to understand means for me, that I am always on the receiving lower end. There is some conventionalist element in this - I acknowledge that. But I do not see how I could get beyond that. I would say the Kopenhagen interpretation is a refusal to give any interpretation. It retracts itself behind epistiomiological boundaries. It seems this is both its virtue and its weakness. It is objectivism without ontology. But then almost 100 years later it appears as if we have not found anything better. Therefore I am forced to acknowledge this virtue.

If I would try to approach DBB. What would I find? Elementary particles have rediscovered their ability to run on trajectories like particles in classical physics. But does this mean that quantum world and macroscopic world have found common ground?

I imagine a hydrogene atom. The wave function suggests to us an image of an electrone cloud surrounding the nucleus in a symmetrical way. This sounds just fine. But then DBB requires us to imagine an additional view of a silly electron, that follows defined orbits around the nucleus. What is this good for? If I understand correctly, this electron is only allowed to make its orbits, because it has been stripped of the privileges, that a classical particle has, as carrying charge or mass. Therefore I wonder. Is this really a particle or is it just another mathematical construct.

What does it mean that there are different interpretations of QM. Does it mean that one has a good relationship to god and is therefore true. Or does it mean that all interpretations need to be thought of being true at the same time? If Kopenhagen and DBB are true at the same time, then it means that we are unable to decide, whether we live in a deterministic universe or not, because any occurrence of indeterminacy can be explained by either randomness occurring spontaneously everywhere (Kopenhagen), or it can be explained by lack of complete knowledge about the initial conditions (Bohm).

Such a conclusion might not be without meaning. But does it help me to understand electrons?. No.

 

[Simon Bridge]

The idea of indistinguishably and single wave-function sounds good. It seems to answer a lot of questions.

Does it integrate with DBB (De Broglie Bohm) interpretation as well?

What do you mean "integrate"?

In de Broglie–Bohm theory, the wavefunction travels through both slits, but each particle has a well-defined trajectory that passes through exactly one of the slits.

The wave function interferes (ok, not interacts) with itself and guides the particles in such a way that the particles avoid the regions in which the interference is destructive and are attracted to the regions in which the interference is constructive, resulting in the interference pattern on the detector screen.

Citation? Where did you get this from?
That's a pretty glib description - I wouldn't put a lot of weight on it.

The wavefunction exists in all space at the same time - it does not "travel" in the usual intuitive sense - though the time-evolution of the wavefunction may see the amplitude peaks, wavefronts etc move around. It should not be thought of like a water wave.

The wavefunction does not "interfere with itself" any more than water-waves through two holes in a barrier interfere with themselves. The main trouble with 2-slit interference is that the exact process whereby a plane-wave states on one side of the barrier turns into the superposition/interfering-waves state on the other side is not easy to describe.
It's like when you do conservation of momentum for collisions - the before and after states of the crash are easy to describe but the actual process where the objects go from one to the other can be very complicated. Often too complicated to model in detail.

The best way to understand interference at slits in QM is to see it as a specific example of what generally happens all the time in nature. Probably the best treatment is from Richard Feynman ... also see the ]Marcella treatment.

It is not a good idea to use classical intuition to interpret QM.

If we can call Heisenberg uncertainty principle HUP, then we can call Paulis exclusion principle as PEP...:)

That's what we do, yes.
http://www.allacronyms.com/PEP/Pauli_Exclusion_Principle/1068494

 

[San K]

thanks Simon, good post. Will take some time to go over it.

Citation? Where did you get this from?
That's a pretty glib description - I wouldn't put a lot of weight on it.

The DBB part above is from Wikipedia.

http://en.wikipedia.org/wiki/De_Broglie–Bohm_theory

 

[Simon Bridge]

OK - Wikipedia is relating an outdated and very simplistic model, badly.
It is still commonly used as a "stepping stone" idea to help students transition from classical to quantum mechanics and as part of teaching the "history of science". PF is full of threads from people who have been mislead by this.

Have you watched the (Feynman) lecture series I gave you?
It includes the most comprehensive QM treatment of 2-slit diffraction that is still accessible.

 

[San K]

Have you watched the (Feynman) lecture series I gave you?
It includes the most comprehensive QM treatment of 2-slit diffraction that is still accessible.

There are four parts, and I am busy, it will take some time.

Is any of the four parts more relevant to double-slit? (seems like the first one)

 

[Simon Bridge]

You'll need the setup unless you've seen QED descriptions before.
I think you should see the whole thing - so just take your time.
The talk about reflection is particularly useful for conceptualizing.

The short version is just that we know the "before" and the "after" and we have a bunch of rules for figuring out one from the other ... but we don't know the physical process that connects the two situations.

QM texts usually devote about a sentence or so to this when they say that a full Hamiltonian for the two slits is too hard to calculate, that the effect is the same as wave optics anyway, and proceed according to wave optics.

 

[San K]

I wonder if transcripts, of the videos, are available. Tried searching but could not find.

Bill Gates has put, some of, the Feynman lectures online.

 

[Simon Bridge]

The "Feynman Lectures" were a series of stage-1 college lectures delivered by Feynman.
They are famous and very good for people who have already completed an introductory course - but it should be realized that the lecture course was something of a failure in it's day.

The videos I linked are on QED - and some of their details are reproduced in the book "The Character of Physical Law". The material does not translate well to text - it's the way he bilds the picture that is useful for that particular format. I could have just directed you to a textbook on quantum mechanics.

 

[eloheim]

@physiologistK above:
Methinks you know at least a thing or two. ;-) Also your points to DBB are spot on from my vantage-point. And I know you said as much but it's easy to overlook that Copenhagen's greatest attribute must be its place in history.

To the original q: Sounds a lot like the '2 particles in 3d space or just 1 in 6d?' topics concerning the ontology of configuration space.