Propagation of de Broglie waves

[Akash Pardasani]

I had a doubt regarding the propagation of de Broglie waves.
How do we know the speed of propagation of these waves and also do there characteristics( speed and amplitude and frequency) depend on the source particle ( or they depend only on the motion(momentum) of the source and no whatsoever relation with the "type" of particle)?

 

[dextercioby]

The concept of de Broglie waves is outdated, has been that way since the wave mechanics of Schrödinger has been reinterpreted correctly in the realms of serious QM (Born + Dirac + von Neumann). So your question has really no relevance. If you mean the deBroglie-Bohm <guiding waves> (the so-called <Bohmian mechanics>), that's a different (>1952) enchilada and here we have some very knowledgeable forum colleagues.

 

[bhobba]

Dextercioby is correct.

The problem is the semi historical way QM is often taught. They don't go back and point out things like de-Broglie waves they use to motivate the QM formalism are no longer correct in that formalism. This leads to all sorts of confusion in beginning students, so much so I don't think its the correct way to go about teaching it - but then again other methods have issues as well so its basically loose loose doesn't matter what you do.

But just as an example of another approach check out:
http://www.scottaaronson.com/democritus/lec9.html

Also the following gives the correct historical account of how QM came about:
http://www.lajpe.org/may08/09_Carlos_Madrid.pdf

De-Broglies ideas lead to Schroedinger's equation when someone challenged Schroedinger that if you have waves you need a wave equation. But Heisenberg came up with a totally different approach based on matrices. It was thought they were equivalent and Schroedinger made an attempt to show such but it wasn't entirely successful. At the end of 1926 Dirac made a better fist of it and it is what generally goes by the name QM these days - but mathematically it has issues with the so called Dirac Delta function. Von-Neumann fixed those up in his famous Mathematical Foundations Of Quantum Mechanics published in 1932 but it wasn't as elegant as Dirac's approach so physicists stuck with Dirac despite its mathematical problem. Mathematicians then went hard to work and fixed up the issues with Dirac and these days either method is mathematically just as valid.

The point though is Dirac's version (he called it the transformation theory) meant the issues that birthed it like de-Broglies were consigned to the scrap heap, and except as part of the history of QM is best forgotten about.

Thanks
Bill

 

[afcsimoes]

Dextercioby is correct.
De-Broglies ideas lead to Schroedinger's equation when someone challenged Schroedinger that if you have waves you need a wave equation. But Heisenberg came up with a totally different approach based on matrices. It was thought they were equivalent and Schroedinger made an attempt to show such but it wasn't entirely successful. At the end of 1926 Dirac made a better fist of it and it is what generally goes by the name QM these days - but mathematically it has issues with the so called Dirac Delta function. Von-Neumann fixed those up in his famous Mathematical Foundations Of Quantum Mechanics published in 1932 but it wasn't as elegant as Dirac's approach so physicists stuck with Dirac despite its mathematical problem. Mathematicians then went hard to work and fixed up the issues with Dirac and these days either method is mathematically just as valid

I have read that the today's mainstream QM is based on the infinite wave because it is the unique that has a defined frequency, with the sequente use of Fourier analisys. This lead to the lost of the locality.
I also has read that the wavelets math concept can bring the losted locality back again.

 

[vanhees71]

Whereever you read that, through the source away ;-). In modern non-relativistic QM (there's only one QM not a mainstream or whateverelse) a particle can be described by a wave function, obeying the Schrödinger equation. It is well defined and not infinite. It's a complex valued field $\psi(t,\vec{x})$. Its modulus squared, $|\psi(t,\vec{x})|^2$ gives the probability density to find a particle at the position $\vec{x}$ at time $t$.

I also don't know, what you mean by "lost locality". The Schrödinger equation is a partial differential equation and as such a local concept.

 

[Quotidian]

Hi there - I have been involved in a discussion of interpretations of physics on a philosophy forum. A question has come up, related to this thread, so I thought I might ask it here.

I am not a physics graduate. I have read documentry histories of the subject, like Manjit Kumar's 'Quantum' and David Lindley's 'Uncertainty', and have a reasonable grasp of some of the philosophical issues underlying the Bohr-Einstein debates, but I can't read physics equations.

Bearing that in mind, the question is this. The 'interference patterns' associated with the famous 'double-slit' experiment, are really patterns left on screens. The question is, what is causing those patterns? The answer is, sub-atomic particles. But if the particles are fired one-at-a-time, then how can they form an 'interference pattern'? How is a single particle interfering with itself? That seems to be the issue.

Could you say that when the particles are fired sequentially, they're still behaving as though they're part of a beam of particles.? It's as if time has been taken out of the equation, so that the sequence of particles behaves as though they're being fired together instead of being separated by time.

So is there any sense in which the probability distribution is causing the individual particle to behave as though it is part of a beam of particles? It seems as if the probability distribution is itself like the so-called 'pilot wave' - in other words, it determines all the possibilities, but only in the sense of constraining the possible paths that any individual particle takes, whether individually or as part of beam. But because it is simply probability, or possibility, it is not something that actually exists; it is on the borderline of potentiality and actuality.

Does this kind of thinking make any sense?

 

[bhobba]

But if the particles are fired one-at-a-time, then how can they form an 'interference pattern'? How is a single particle interfering with itself? That seems to be the issue.

The answer is the uncertainty principle and the principle of superposition. It is explained here:
https://arxiv.org/ftp/quant-ph/papers/0703/0703126.pdf

Physics unfortunately goes through various levels depending on how advanced the audience is. At the start and in popularization's you have the wave particle duality as the explanation of the double slit. Unfortunately it is one of the many myths of QM:
https://arxiv.org/abs/quant-ph/0609163

Then you have an explanation like the above aimed at a slightly more advanced reader. Unfortunately, while much much better (otherwise I would not use it) it to is wrong to an even more advanced reader:
https://arxiv.org/abs/1009.2408

To make matters worse to the very advanced reader, some of whom post here it, to is wrong. Physics unfortunately can be like that and is a big issue for guys like you. At your level take the paper I linked to is the explanation - forget the more advanced stuff.

Does this kind of thinking make any sense?

Not really. We know the why of the double slit very well and hopefully after reading my linked paper you will to.

If you REALLY want to learn QM get these three books in the following order:
https://www.amazon.com/dp/0471827223/?tag=pfamazon01-20
https://www.amazon.com/dp/0465075681/?tag=pfamazon01-20
https://www.amazon.com/dp/0465062903/?tag=pfamazon01-20

Then if you want to learn the real basis of modern physics, the truth that students when they learn about it usually in graduate school leaves them it in stunned silence, the truth that basically left Einstein speechless when he heard about one of its foundation discoveries, Noethers Theorem, get the following:
https://www.amazon.com/dp/3319192000/?tag=pfamazon01-20

But before that, to set the stage so to speak, get Landau Mechanics:
https://www.amazon.com/dp/0750628960/?tag=pfamazon01-20
If physicists could weep, they would weep over this book. The book is devastingly brief whilst deriving, in its few pages, all the great results of classical mechanics. Results that in other books take take up many more pages. I first came across Landau's mechanics many years ago as a brash undergrad. My prof at the time had given me this book but warned me that it's the kind of book that ages like wine. I've read this book several times since and I have found that indeed, each time is more rewarding than the last.'The reason for the brevity is that, as pointed out by previous reviewers, Landau derives mechanics from symmetry. Historically, it was long after the main bulk of mechanics was developed that Emmy Noether proved that symmetries underly every important quantity in physics. So instead of starting from concrete mechanical case-studies and generalising to the formal machinery of the Hamilton equations, Landau starts out from the most generic symmetry and dervies the mechanics. The 2nd laws of mechanics, for example, is derived as a consequence of the uniqueness of trajectories in the Lagragian. For some, this may seem too "mathematical" but in reality, it is a sign of sophisitication in physics if one can identify the underlying symmetries in a mechanical system. Thus this book represents the height of theoretical sophistication in that symmetries are used to derive so many physical results.'

I could say more but best you discover it for yourself from the references I gave.

It will change your view of the world and hopefully start a discussion with your philosophy friends on the REAL foundations and meaning of physics.

Thanks
Bill

 

[Quotidian]

Hey Bill - thanks for that - I guess the answer is, I am not going to learn physics at this stage in life. I really ought to think about other things. Thanks.

Altough, I would like an answer to this question, if it's possible:

'Could you say that when the particles are fired sequentially, they're still behaving as though they're part of a beam of particles?'

 

[bhobba]

'Could you say that when the particles are fired sequentially, they're still behaving as though they're part of a beam of particles?'

In QM what's going on when not observed we do not know - the theory is silent. So the answer is - we don't know.

Thanks
Bill

 

[Quotidian]

In QM what's going on when not observed we do not know - the theory is silent. So the answer is - we don't know.

Thanks
Bill

But, the result when you fire the beam of particles sequentially, and when you fire it all it once, is the same, is it not?

 

[hairyzussy]

How do you produce a negative probability using an electron probability wave?

 

[bhobba]

But, the result when you fire the beam of particles sequentially, and when you fire it all it once, is the same, is it not?

Obviously not - in a beam two or more can flash the screen at the same time.

If by behaving the same you mean it makes little difference to the analysis - yes (you do then have the possibility is collisions in a beam).

Thanks
Bill

 

[bhobba]

How do you produce a negative probability using an electron probability wave?

Have you heard of the Kolmogorov axioms? If not look them up and you will see its impossible - probability is positive.

During the early days of QM in relativistic problems you got negative probabilities which indicated you were in deep do do. The reason is it was the positive probability of an antiparticle. But it was also found the extension of QM to Quantum Field Theory (QFT) was a much better way to handle the issue.

This however is way way off topic - if you want to discuss it further stat a new thread.

Thanks
Bill

 

[afcsimoes]

If by behaving the same you mean it makes little difference to the analysis - yes (you do then have the possibility is collisions in a beam).

Thanks
Bill

So, if we made a first duble slit experiment using a beam and a second experiment firing the same amount of particles but in a one by one basis, then we wil finish with two conrete objective and real identical images of an interference patern?

 

[bhobba]

So, if we made a first duble slit experiment using a beam and a second experiment firing the same amount of particles but in a one by one basis, then we wil finish with two conrete objective and real identical images of an interference patern?

Yes.

Thanks
Bill

 

[Quotidian]

afcsimoes: So, if we made a first double slit experiment using a beam and a second experiment firing the same amount of particles but on a one by one basis, then we wil finish with two concrete objective and real identical images of an interference patern?

bhobba: Yes.

Thanks! So from this, can I presume that the interference pattern is not rate-dependent, i.e. the rate at which the photons are emitted doesn't affect the distribution?

 

[bhobba]

Thanks! So from this, can I presume that the interference pattern is not rate-dependent, i.e. the rate at which the photons are emitted doesn't affect the distribution?

Yes.

Thanks
Bill

 

[afcsimoes]

What is the explanation for that? At the second experiment, fired one by one, how can each electron interfere with all the previous and with all the futures? Or interfere with himself?
Alternatively, can it be justified with some uncertainty/probabilistic quantic event at/in the shooter? Or at the double slit? A shootted electron interfering with the electrons of the slit edge substance? (e.g. as if the electron's wave was a 4D helicoide)

 

[bhobba]

What is the explanation for that?

Its simply the same experiment done in parallel and repeated many times over.

I gave a link to the why of the double slit. The explanation works if there is one or many.

Thanks
Bill

 

[afcsimoes]

I will try to read and understand.
Thank you Bill
Armando

 

[fanieh]

The concept of de Broglie waves is outdated, has been that way since the wave mechanics of Schrödinger has been reinterpreted correctly in the realms of serious QM (Born + Dirac + von Neumann). So your question has really no relevance. If you mean the deBroglie-Bohm <guiding waves> (the so-called <Bohmian mechanics>), that's a different (>1952) enchilada and here we have some very knowledgeable forum colleagues.

Do particles in the deBroglie-Bohm have trajectories? It seems they are local particles yet it seems there is at the same time no trajectories. So do the particles just appear when measured? but is it not Bohmian mechanics is about trajectories?

 

[Quotidian]

So, to recap the question I had: it is well known that if you set up a double-slit experiment, then send the particles through the slits one-at-a-time, you still get interference patterns. This is a paradox, because there's no question of the particles interfering with one another, as they're sent separately. This paradox is consequence of superposition.

As a consequence, if you ran two instances of a double-slit experiment, the first being one-photon-at-a-time, the second being all the photons being fired at once - given that in all other respects the set-up was identical, could you detect the difference between the two experiments, on the basis of the interference patterns alone? Answer is 'no'.

What I find interesting about that, is that if the same interference pattern is created by either 'slow' or 'instantaneous' transmission of energy, then the factor of 'duration of the event' doesn't seem to have any consequences for the formation of the pattern. So the 'interference pattern' is not actually a consequence of 'interference' at all, because when the single photon test is run, there is nothing to interfere with the single photon - yet it produces the same pattern as if it is run en masse.

So what I am taking from that, is that the cause of the interference pattern, is not actually 'interference' in the sense that water waves 'interfere' with each other. The kinds of waves associated with water, are only an analogy for the 'wave function' that determines the 'interference pattern'.

So what determines the interference pattern really is the probabilities expressed by the wave function equation, and this is not a function of time (otherwise, the rate would effect it.) But more to the point, it's an illustration of how mere probabilities can be causally effective.

 

[physics pfan]

Hi bhobba,

Your wrote: "The point though is Dirac's version...meant the issues that birthed it like de-Broglies were consigned to the scrap heap"
Please explain the issues. A photon meets a slit, an electron meets a crystal lattice; both photon and electron diffract; hence small matter can assume a wave identity; this was de Broglie's main point.. What part of this goes to the scrap heap?

You also wrote: "...then again other methods have issues as well so its basically loose loose doesn't matter what you do." I think you meant 'lose' and not 'loose' and I don't think you can claim 'typo' here. You and the many are not clear on the difference...

physics pfan

 

[bhobba]

Please explain the issues.

All the phenomena which was explained by the so called wave particle duality was explained instead by Schrodinger's equation whose new explanation, while not known to Dirac (he derived from an assumption about Poisson brackets) is actually symmetry ie the probabilities of the Born Rule are frame independent which is so damn obvious nobody would really question it - but strictly speaking you are invoking Einsteins Principle Of Relativity - see Chapter 3 Ballentine. Note - its at the advanced undergraduate level - if you are not at that level you will just have to take my word for it. Schrodinger thought it was actually waves and his equation described those waves. As an aside he actually made two cancelling errors in his derivation, but that's another story - and his derivation was pretty much a crock anyway. But it soon became clear it wasn't about actual waves, and he became despondent because he thought that's what was going on - so despondent he wished he never had anything to do with it. To understand its not waves simply look at the hydrogen atom:
https://chemistry.stackexchange.com...-many-wave-functions-associated-with-hydrogen

None of those look like waves to me .

You also wrote: "...then again other methods have issues as well so its basically loose loose doesn't matter what you do." I think you meant 'lose' and not 'loose' and I don't think you can claim 'typo' here. You and the many are not clear on the difference...

Auto correction - its not perfect and doesn't pick up words that are correct spelling for something different - its lose, lose.

In don't know what you mean by the many do not know the difference - even someone in grade 1 here would know the difference.

My English is terrible, my worst subject at HS - I failed it. But its not so bad I don't know that one - it's just carelessness in trusting spell checkers - it happens to the best of us, even people with much better literacy skills than me.

Thanks
Bill

 

[Blue Scallop]

All the phenomena which was explained by the so called wave particle duality was explained instead by Schrodinger's equation whose new explanation, while not known to Dirac (he derived from an assumption about Poisson brackets) is actually symmetry ie the probabilities of the Born Rule are frame independent which is so damn obvious nobody would really question it - but strictly speaking you are invoking Einsteins Principle Of Relativity - see Chapter 3 Ballentine. Note - its at the advanced undergraduate level - if you are not at that level you will just have to take my word for it. Schrodinger thought it was actually waves and his equation described those waves. As an aside he actually made two cancelling errors in his derivation, but that's another story - and his derivation was pretty much a crock anyway. But it soon became clear it wasn't about actual waves, and he became despondent because he thought that's what was going on - so despondent he wished he never had anything to do with it. To understand its not waves simply look at the hydrogen atom:
https://chemistry.stackexchange.com...-many-wave-functions-associated-with-hydrogen

Can you please elaborate what you meant by Schrodinger actually "made two cancelling errors in his derivation"? so the two errors cancel out to become right? weird.. what is it

None of those look like waves to me .



Auto correction - its not perfect and doesn't pick up words that are correct spelling for something different - its lose, lose.

In don't know what you mean by the many do not know the difference - even someone in grade 1 here would know the difference.

My English is terrible, my worst subject at HS - I failed it. But its not so bad I don't know that one - it's just carelessness in trusting spell checkers - it happens to the best of us, even people with much better literacy skills than me.

Thanks
Bill

 

[bhobba]

Can you please elaborate what you meant by Schrodinger actually "made two cancelling errors in his derivation"? so the two errors cancel out to become right? weird.. what is it

See:
https://arxiv.org/pdf/1204.0653.pdf
'Schrodinger’s notes show that he was well aware that the solution of (8.12) gives correctly the bound state energies of the hydrogen atom before introducing in the anstaz concerning the hypothetical quantity J. This artifice compensates for the physically incorrect ansatz (8.3). The constant K should actually be the pure imaginary quantity −ih in which case (8.5) becomes the correct equation (7.27). Repeating Schrodinger’s stationary algorithm starting with the correct relation S = −ih ln ψ would then give the incorrect equation (8.5)! Indeed, since the H-J equation and the properties of the generating function S already follow from Hamilton’s equations which in turn are a consequence of Hamilton’s Principle — the condition that the action S should be stationary for arbitrary variations of space-time paths - it would seem that Schrodinger is attempting here to close a door that is already shut.'

But his whole derivation, as the paper explains, is basically a load of the proverbial. As one person said - his derivation is what it is - inspired guesswork.

Schrodinger of course was a genius - we generally can't go where they do. Something else than strictly logical reasoning is really in charge of the creative impulse - the reasons they give are just sort of a crutch - as one person calls it its like being a sleepwalker. Einstein, Newton, many of the greats were basically sleepwalkers.

Tthanks
Bill

 

[Physics Footnotes]

So, to recap the question I had: it is well known that if you set up a double-slit experiment, then send the particles through the slits one-at-a-time, you still get interference patterns. This is a paradox, because there's no question of the particles interfering with one another, as they're sent separately. This paradox is consequence of superposition.

As a consequence, if you ran two instances of a double-slit experiment, the first being one-photon-at-a-time, the second being all the photons being fired at once - given that in all other respects the set-up was identical, could you detect the difference between the two experiments, on the basis of the interference patterns alone? Answer is 'no'.

What I find interesting about that, is that if the same interference pattern is created by either 'slow' or 'instantaneous' transmission of energy, then the factor of 'duration of the event' doesn't seem to have any consequences for the formation of the pattern. So the 'interference pattern' is not actually a consequence of 'interference' at all, because when the single photon test is run, there is nothing to interfere with the single photon - yet it produces the same pattern as if it is run en masse.

So what I am taking from that, is that the cause of the interference pattern, is not actually 'interference' in the sense that water waves 'interfere' with each other. The kinds of waves associated with water, are only an analogy for the 'wave function' that determines the 'interference pattern'.

So what determines the interference pattern really is the probabilities expressed by the wave function equation, and this is not a function of time (otherwise, the rate would effect it.) But more to the point, it's an illustration of how mere probabilities can be causally effective.

I would put it like this: The outcomes of quantum mechanical experiments are individually unpredictable yet collectively synchronized. Whilst there is nothing inconsistent about this state of affairs at the level of probability theory (the quantum formalism getting the statistics right every time), any attempt to build a physical picture that joins the fence-posts of observations breaches any common-sense notion of causality, and this negates the whole point of having a physical picture.

 

[Blue Scallop]

See:
https://arxiv.org/pdf/1204.0653.pdf
'Schrodinger’s notes show that he was well aware that the solution of (8.12) gives correctly the bound state energies of the hydrogen atom before introducing in the anstaz concerning the hypothetical quantity J. This artifice compensates for the physically incorrect ansatz (8.3). The constant K should actually

In the same paragraph in the paper above, it was written "Schrodinger's notes [49].." the 49 refers to the Moore's book "Schroedinger Life and Thought".
https://www.amazon.com/dp/052135434X/?tag=pfamazon01-20
Have you read this before Bill? I just bought it (the kindle version is only $16) and it is more than a layman book so would take time to digest it. If you have read it.. what pages are the double error detailed. If you haven't read it.. try to get the book.. quoted a part as I was going over the book at random

"From 1943 to 1951, Schrodinger's research work was dedicated almost exclusively to the search for the a unified field theory that would encompass both gravitation and electromagnetism..."

I didn't know he was working on it same as Einstein and the book detailed it. Looks interesting.

be the pure imaginary quantity −ih in which case (8.5) becomes the correct equation (7.27). Repeating Schrodinger’s stationary algorithm starting with the correct relation S = −ih ln ψ would then give the incorrect equation (8.5)! Indeed, since the H-J equation and the properties of the generating function S already follow from Hamilton’s equations which in turn are a consequence of Hamilton’s Principle — the condition that the action S should be stationary for arbitrary variations of space-time paths - it would seem that Schrodinger is attempting here to close a door that is already shut.'

But his whole derivation, as the paper explains, is basically a load of the proverbial. As one person said - his derivation is what it is - inspired guesswork.

Schrodinger of course was a genius - we generally can't go where they do. Something else than strictly logical reasoning is really in charge of the creative impulse - the reasons they give are just sort of a crutch - as one person calls it its like being a sleepwalker. Einstein, Newton, many of the greats were basically sleepwalkers.

Tthanks
Bill

 

[bhobba]

" the 49 refers to the Moore's book "Schroedinger Life and Thought".

This is off topic and the mods may delete it, but quickly - no I haven't. However books on the life of great scientists are always worth a read. Reading another on Einstein right now (have read quite a few in my time) - very interesting:
https://www.amazon.com/dp/0393337685/?tag=pfamazon01-20

Thanks
Bill

 

[Blue Scallop]

This is off topic and the mods may delete it, but quickly - no I haven't. However books on the life of great scientists are always worth a read. Reading another on Einstein right now (have read quite a few in my time) - very interesting:
https://www.amazon.com/dp/0393337685/?tag=pfamazon01-20

Thanks
Bill

It's not so off topic because the book about Schrodinger also mentioned about propagation of de Broglie waves and Born and the full derivation of the Schrodinger Equation.

But to make it directly on topic. I'd like to ask about de Broglie waves whose modern interpretation is the bohm Debroglie guiding waves. You mentioned even if wave function is separate from the local particles.. wave function interacting with wave function still has the Hamiltonian of them interacting. But are there any interpretations where the Hamiltonians are not in the wave functions but in the particles? And to make it 100% on topic. What did de Broglie think about the Hamiltonian with regards to his initial de Broglie waves thoughts? Since Hamiltonian is very basic. It's possible de Broglie has thought of it.

 

[bhobba]

IBut are there any interpretations where the Hamiltonians are not in the wave functions but in the particles? And to make it 100% on topic. What did de Broglie think about the Hamiltonian with regards to his initial de Broglie waves thoughts? Since Hamiltonian is very basic. It's possible de Broglie has thought of it.

The Hamiltonian is in Schrodinger's equation which follows from symmetry principles.

This is now way off topic and I will go no further than point you to Chapter 3 of Ballentine.

De Broglies arguments are simply working out the idea if waves can act as particles maybe the reverse is true. He cobbled together relativity and other ideas to do that. It was his PhD thesis and the examiners said they couldn't even understand it but sent it to Einstein for comment. He recognized immediately its importance and gave it his full recommendation so he got his PhD and eventually a Nobel.

You can read the exact math he used in many places.

Thanks
Bill

 

[Blue Scallop]

The Hamiltonian is in Schrodinger's equation which follows from symmetry principles.

This is now way off topic and I will go no further than point you to Chapter 3 of Ballentine.

De Broglies arguments are simply working out the idea if waves can act as particles maybe the reverse is true. He cobbled together relativity and other ideas to do that. It was his PhD thesis and the examiners said they couldn't even understand it but sent it to Einstein for comment. He recognized immediately its importance and gave it his full recommendation so he got his PhD and eventually a Nobel.

You can read the exact math he used in many places.

Thanks
Bill

Ok let's be fully on topic. About the old de Broglie waves.. he has group velocity, particle velocity and phase velocity. What is the final modern form of the different velocities as they are still mentioned in QM now..

 

[Blue Scallop]

The Hamiltonian is in Schrodinger's equation which follows from symmetry principles.

This is now way off topic and I will go no further than point you to Chapter 3 of Ballentine.

De Broglies arguments are simply working out the idea if waves can act as particles maybe the reverse is true. He cobbled together relativity and other ideas to do that. It was his PhD thesis and the examiners said they couldn't even understand it but sent it to Einstein for comment. He recognized immediately its importance and gave it his full recommendation so he got his PhD and eventually a Nobel.

You can read the exact math he used in many places.

Thanks
Bill

I always wanted to ask about this so let me ask this now as I'm quite confused about it. See:

You can see this image anywhere. What is retained in the modern form?

 

[bhobba]

Ok let's be fully on topic. About the old de Broglie waves.. he has group velocity, particle velocity and phase velocity. What is the final modern form of the different velocities as they are still mentioned in QM now..

No.

The modern view has nothing to do with what birthed it.

The wave-function, even though it has wave, in its name has nothing to do with waves - is, technobabble now follows, the expansion of the state in terms of eigenfunctions of position. Nothing like what De-Broglie or Schrodinger thought - which is why Schrodinger became so despondent - it's very mathematical. The Von-Neumann's and Dirac's of the physics world were now in charge.

Even today we still 'argue' and struggle with what it means. Great progress has been made, especially in the area of decoherence - but questions still remain and endlessly discussed.

Thanks
Bill

 

[Blue Scallop]

No.

The modern view has nothing to do with what birthed it.

The wave-function, even though it has wave, in its name has nothing to do with waves - is, technobabble now follows, the expansion of the state in terms of eigenfunctions of position. Nothing like what De-Broglie or Schrodinger thought - which is why Schrodinger became so despondent - it's very mathematical. The Von-Neumann's and Dirac's of the physics world were now in charge.

Even today we still 'argue' and struggle with what it means. Great progress has been made, especially in the area of decoherence - but questions still remain and endlessly discussed.

Thanks
Bill

Ok so de-Broglie wave is outdated.. no problem.. but here's the confusing part.. how come Valentini and others are reviving the de Broglie Pilot wave thing. Is this the same concept as the outdated de-Broglie waves? If there is difference. What's their differences so one won't mix the two concepts and get confused. Thank you Bill.

 

[bhobba]

Ok so de-Broglie wave is outdated.. no problem.. but here's the confusing part.. how come Valentini and others are reviving the de Broglie Pilot wave thing. Is this the same concept as the outdated de-Broglie waves? If there is difference. What's their differences so one won't mix the two concepts and get confused. Thank you Bill.

Its never gone away - we have DBB.

But Dymytifyer is the person to ask about that.

BTW in DBB its not a wave - its a particle guided by a wave.

Thanks
Bill

 

[Blue Scallop]

Its never gone away - we have DBB.

But Dymytifyer is the person to ask about that.

BTW in DBB its not a wave - its a particle guided by a wave.

Thanks
Bill

Ah. So the difference with the original de Broglie wave is the original de Broglie wave considered particle as wave only and no particle.. yet DBB has both particle and wave.. but then de Broglie invented the pilot wave too.. did he never mean it's both particle and wave?

 

[vanhees71]

One has to be careful with the various "velocities" related to waves. One must be aware, what they mean and under which circumstances one can use these notions. Take e.g., "group velocity". A general solution for the time evolution of a wave equation often can be written as
$$\psi(t,\vec{x})=\int_{\mathbb{R}^3} \mathrm{d}^3 \vec{k} \frac{1}{(2 \pi)^{3/2}} \tilde{\psi}_0(\vec{k}) \exp[-\mathrm{i} \omega(\vec{k}) t + \mathrm{i} \vec{k} \cdot \vec{x}].$$
The reason is that often the Laplace operator occurs in the wave equation, and the plane waves are a complete set of generalized orthonormal functions.

E.g., for the free Schröinger equation (with $\hbar=1$),
$$\partial_t \psi(t,\vec{x})=-\frac{1}{2m} \Delta \psi(t,\vec{x}),$$
you get the dispersion relation
$$\omega(\vec{k})=\frac{1}{2m} \vec{k}^2.$$
To make sense of the group velocity you need the assumption that $\tilde{\psi}_0(\vec{k})$ is narrowly peaked around $\vec{k}_0$. Then one can expand the exponent in powers of $\vec{k}-\vec{k}_0$:
$$\mathrm{i} [\vec{k} \cdot \vec{x}-\omega(\vec{k})t] \simeq \mathrm{i} [\vec{k} \cdot \vec{x}-\omega_0 t - t (\vec{k}-\vec{k}_0) \cdot \vec{\nabla}_k \omega(\vec{k}_0)].$$
Then you get with $\vec{v}_g=\vec{\nabla}_k \omega(\vec{k}_0)$
$$\psi(t,\vec{x}) \simeq \exp[-\mathrm{i} (\omega_0 -\vec{k}_0 \cdot \vec{v}_g) t+\mathrm{i} \vec{k}_0 \cdot \vec{x}] \int_{\mathbb{R}^3} \mathrm{d}^3 \vec{k} \tilde{\psi}_0(\vec{k}) \exp[\mathrm{i} \vec{k} \cdot (\vec{x}-t \vec{v}_{\text{g}})],$$
i.e., finally
$$\psi(t,\vec{x}) \simeq \exp[-\mathrm{i} (\omega_0 -\vec{k}_0 \cdot \vec{v}_g) t+\mathrm{i} \vec{k}_0 \cdot \vec{x}] \psi_0(\vec{x}-\vec{v}_{\text{g}} t).$$
This means, in this approximation up to a phase the initial wave packet moves undeformed with group velocity $\vec{v}_{\text{g}}$. The position-probability density is
$$|\psi(t,\vec{x})| \simeq |\psi_0(\vec{x}-\vec{v}_{\text{g}} t)|.$$
It is important to note that this only holds for sufficiently narrow-peaked $\tilde{\psi}(\vec{k})$, i.e., for a particle with a quite well prepared momentum!

With the dispersion relation for the free-particle Schrödinger wave you get the expected result
$$\vec{v}_{\text{g}}=\frac{1}{m} \vec{k}_0,$$
i.e., the wave packet moves with this velocity, which is pretty intuitive from the classical particle picture.

One must, however be aware that this interpreation relies on the assumptions made, i.e., that you have a wave packet that is narrowly peaked in momentum space, such that dispersion can be neglected.

It's a good exercise to solve the initial-value problem for a Gaussian wave packet as the initial state exactly. Most conveniently one gives it in terms of the momentum representation,
$$\tilde{\psi}(\vec{k})=N \exp \left (-\frac{(\vec{k}-\vec{k}_0)^2}{4 \sigma_k^2} \right ),$$
where $N$ is an unimportant normalization factor and $\sigma_k$ the standard deviation of the momentum components. Here the time-evolution integral can be easily solved exactly since it's a Gaussian integral, and you can also easily compare it to the approximation involving the group velocity $\vec{v}_{\text{g}}$.

 

[Blue Scallop]

No.

The modern view has nothing to do with what birthed it.

The wave-function, even though it has wave, in its name has nothing to do with waves - is, technobabble now follows, the expansion of the state in terms of eigenfunctions of position. Nothing like what De-Broglie or Schrodinger thought - which is why Schrodinger became so despondent - it's very mathematical. The Von-Neumann's and Dirac's of the physics world were now in charge.

Even today we still 'argue' and struggle with what it means. Great progress has been made, especially in the area of decoherence - but questions still remain and endlessly discussed.

Thanks
Bill

Do you know what is the best book and biography about de Broglie with focus on his thinking process from the matter waves thesis days to proposing the pilot waves after Born suggested the probability interpretation in 1927? Something like... de Broglie, Life and Thoughts? anyone?

 

[OurBladesAreSharp]

The 'interference patterns' associated with the famous 'double-slit' experiment, are really patterns left on screens. The question is, what is causing those patterns? The answer is, sub-atomic particles. But if the particles are fired one-at-a-time, then how can they form an 'interference pattern'? How is a single particle interfering with itself? That seems to be the issue.

Think of the particle as a wave function which has non zero value at both slits and interferes with itself later to produce interference patterns

 

[vanhees71]

Well, but this is not the right picture either since it's not what is observed. If you send a single particle with some quite well-defined momentum (such that in the wave picture you get nice interference patterns) through a double slit, you won't see the interference pattern at the slit as suggested if you interprete the (position) space function as a classical field entity, i.e., rather than finding some smeared distribution you find a single spot on the screen, which suggests that the electron is rather something more particle like. This contradiction was dubbed "wave-particle duality", but it's an obsolete concept. Very quickly after de Broglie's PhD thesis, where he suggested the introduction of "matter waves" and which was so unusual at the time that the PhD committee sent it to Einstein for evaluation. Since Einstein enthusiastically liked it, de Broglie rightfully got his PhD and later the Nobel prize for his ideas.

Nevertheless, the classical wave picture is obviously wrong from the observation just described. On the other hand, due to the very fact that there are indeed interference patterns occurring when sending many electrons through the double slits (all carefully prepared in the same way with a pretty well-defined momentum) it's as obviously wrong to interpret electrons in terms of classical particles.

That's why Schrödingers wave function had to be intepreted differently to make sense, and this was the breakthrough for quantum theory as a consistent picture of matter and is due to Born, who interpreted the wave function as "probability amplitude", i.e., after solving the Schrödinger equation, given a Hamiltonian (including the interaction of the electron with the material forming the slits, usually simplified by infinitely high potential wells, which is reformulated simply to boundary conditions as in optical diffraction theory) you just know the probability distribution for a particle to be at any position at a given time $t$:
$$P(t,\vec{x}) \mathrm{d}^3 \vec{x}=|\psi(t,\vec{x})|^2 \mathrm{d}^3 \vec{x}$$
is the probability to find at time $t$ the electron in a little volume $\mathrm{d}^3 \vec{x}$ around position $\vec{x}$.

The surprising finding is that although there's only one electron running through the double slit at any time you find the interference pattern when running enough electrons through. This shows, it's not an interference phenomenon of two or more electrons but it's an interference effect of probability waves for a single electron. One has to get used to it. There is no simpler way to somehow explain this fact about electrons but it's, according to present knowledge, the only consistent description of this observed fact. In other words the, admittedly counterintuitive, basic rules of QT are fundamental laws of Nature that can not derived from "simpler" or "more intuitive" laws.