Particle or Wave? Duality Explained

[PeroK]

I don't know what Griffiths wants to attack in that footnote, but he doesn't seem to be questioning the validity of the fundamental equations like E = hf.

He's not attacking anything. He's simply teaching QM without the wave-particle duality.

In Sakurai's book it doesn't even get a mention.

The equation E = hf and the corresponding one for momentum are the foundation of quantum theory,

They are not the foundation of quantum mechanics.

 

[Ddddx]

They certainly are the foundation of quantum mechanics. You will probably recognize them more easily if I write them in the form

H |psi> = ih ∂_t |psi>

P = -ih ∂_x

Or other equivalent ways.

 

[vanhees71]

Here, you mix up several things in a very confused way. Neither of these formulas have anything to do with "wave-particle dualism", whatever this should mean. It's really good advise not to learn these old-fashioned concept when starting to learn quantum theory.

Conerning your two formulas: The first one describes the time evolution of an abstract state ket (by the way, one should indeed learn as soon as possible the abstract formalism a la Dirac, because then such confusion as demonstrated in this thread is avoided early on) in the Schrödinger picture.

The 2nd equation gives the momentum operator in the position representation. The correct formula is
$$\hat{\vec{p}}=-\mathrm{i} \hbar \vec{\nabla}.$$

In the position formulation your 1st formula reads
$$\mathrm{i} \hbar \partial_t \psi(t,\vec{x})=\hat{H} \psi(t,\vec{x}).$$
Now for free particles you have
$$\hat{H}=\frac{\hat{\vec{p}}^2}{2m}=-\hbar^2 \Delta,$$
and then you can find plain-wave solution for this free-particle Schrödinger equation by making the Ansatz
$$\psi(t,\vec{x})=A \exp(-\mathrm{i} \omega t+\mathrm{i} \vec{k} \cdot \vec{x}).$$
Inserting this into the equation leads to
$$\hbar \omega=\frac{\hbar^2}{2m} \vec{k}^2.$$
Now what's also clear is that this solution are (generalized) eigenfunctions of the Hamiltonian (energy operator) and the momentum operator, i.e., you can indeed (for this special case!) identify
$$E=\hbar \omega, \quad \vec{p}=\hbar \vec{k}$$
as energy and momentum of a non-relativistic particle, but it has still nothing to do with any kind of "wave-particle duality". Rather the wave function has the probabilistic meaning that $|\psi(t,\vec{x})|^2$ is the probability distribution for the position of the particle, but this holds of course only for square-integrable wave functions, not for the energy-momentum eigenmodes of free particles, but that's another subtlety.

 

[Aufbauwerk 2045]

I learned in QM that we only have a right to talk about what we can observe. The rest is metaphysics. Mathematically we use the Hilbert Space. Observables are operators in HS.

As for words like "particle and "wave" it's fine to use them as long as we realize their limitations. We invented these words to describe everyday experience. To say that this "thing" must be either a "particle" or a "wave" but not both is just an attempt to describe complicated nature with our primitive words and primitive logic.

 

[bhobba]

Some of the answers hae been very good. There is no wave-particle duality.

However some hisorical perspective may shed some light on why it still hangs around and what the early pioneers thought.

A good book to read is the folllowing:
https://www.amazon.com/dp/1491531045/?tag=pfamazon01-20

In 1924, in his PhD thesis, Louis de Broglie suggested that just as light exhibits wave and particle properties, all microscopic material particles such as electrons, protons, atoms, molecules etc, have also dual character. His examiners didn't know what to make of it but a copy made its way to Einstein (the above book gives the exact detail). He too did not believe it but recognized immediately it was an important breakthrough and highly recommended it. He knew it was wrong but believed, correctly, it was an important step, but not the final solution to the quantum puzzle. Einsteins intuition, as always was of the highest caliber - in fact his ability to penetrate to the heart of a problem was unmatched - not perfect - but better than anyone else's, even the greatest of scientists like Von-Neumann and Feynman who also were known for that ability.

Then this guy Schrodinger entered in the picture. At a lecture someone suggested if matter has wave properties then it should obey a wave equation. Using false reasoning he obtained the correct answer that now goes by the name of Schrodinger's equation:
https://arxiv.org/abs/1204.0653

We can derive it much simpler these days by writing down the most general relativistically invariant field equation in a single complex field, and believe it or not you get the Klein Gordon equation. Take the classical limit and you get Schrodinger's equation - you can find the detail here:
https://www.amazon.com/dp/3319192000/?tag=pfamazon01-20

More can be said about this approach but this is not the thread for it - suffice to say waves have nothing to do with it - it's that complex field and what in the damnation it means that's the issue.

Anyway from that point on things moved quickly:
http://www.lajpe.org/may08/09_Carlos_Madrid.pdf

It was then apparent, just as Einsteins intuition told him, wave-particle duality was wrong - or at the most of limited applicability.

But due to the semi historical approach taken in most beginner texts and popularization's they never point this out. It quite bad really and people are left with this insidious incorrect misconception. A misconception BTW even Einstein knew from the start was wrong.

Just as a personal aside I much prefer a non traditional presentation that avoids the whole thing such as in the book on symmetry above and the following:
http://www.scottaaronson.com/democritus/lec9.html

That avoids the confusion right from the start.

Thanks
Bill

 

[Stavros Kiri]

... as energy and momentum of a non-relativistic particle, but it has still nothing to do with any kind of "wave-particle duality".

Why not? It's a particle, we know that (or a quantum object, or quantum particle). And

... duality". Rather the wave function has the probabilistic meaning that |ψ(t,⃗x)|2|ψ(t,x→)|2|\psi(t,\vec{x})|^2 is the probability distribution for the position of the particle, but this holds of course only for square-integrable wave functions, not for the energy-momentum eigenmodes of free particles, but that's another subtlety.

Thus probability waves ...

Thus both, thus duality, q.e.d. (=quite easily demonstrated)

 

[bhobba]

The book is about 1960, and here, what need had John Bell to dedicate a book if, since 1925 these problems were already solved?

Because physicists often speak in loose language, and virtually all other physicists had been exposed to the semi-historical approach of beginner texts where they do not go to the trouble to explicitly correct it. You are supposed to sort of figure it out for yourself - and most do, but still speak loosely using it.

What do you think is more likely - the many professors who post here that teach this stuff to students are wrong, or you are misinterpreting papers like Bell? The latter is much much more reasonable, and in this case provably true - are waves complex valued and travel in a complex Hilbert space? That's the wave-like solutions you get to Schrodinger's equation, and most are not wave-like eg the solutions for the Hydrogen atom:
http://darksilverflame.deviantart.com/art/The-Shapes-Of-Hydrogen-Poster-327297786

They don't look very much like waves to me. As a matter of fact only the free particle solution looks wave-like. There is a deep reason for that explained in the following lectures:


Experts reading it know the intent of what he saying, they know its wrong, although not usually pointed out even Einstein knew it was wrong.

I am unaware of any QM textbook at the advanced level, and I have read quite a few, that talks about it. My, and many others who post here, reference is Ballentine. Not a single mention of it. It is very widely respected as one of the best, maybe even the best, textbook written on QM.

Thanks
Bill

 

[bhobba]

Why not? It's a particle, we know that (or a quantum object, or quantum particle).

You already contradicted yourself. Its a particle or quantum object or quantum particle.

Particles have definite position and momentum, quantum objects do not. Sometimes we can measure the position of a quantum object (you can't for photons - it has no position observable) and we say it is displaying particle like characteristics. But mostly it behaves nothing like a particle.

They are really excitation's of a quantum field - and that's what is meant by quantum particle. These excitation's do not obey wave-particle duality. In fact they do not obey any mental picture I am aware of at all, although physicists speak loosely of other falsehoods associated with quantum fields like virtual particles to have an intuitive picture, only in the math can it be described correctly. That's actually quite important - the development of intuition is vital - but its just that - an aid to intuition.

Thanks
Bill

 

[Stavros Kiri]

You already contradicted yourself. Its a particle or quantum object or quantum particle.

It's not a contradiction. There are classical (point-like) particles and quantum particles (or quantum objects). 'Particle' is a constituent of "matter" (that's why we say "Particle Physics" ...). [Then, matter versus waves, fields, energy etc., all can virtually be unified ... , but not quite ... ; there is unity and oppotition.]

Particles have definite position and momentum, quantum objects do not.

These are classical particles ...

 

[Stavros Kiri]

That's the wave-like solutions you get to Schrodinger's equation, and most are not wave-like eg the solutions for the Hydrogen atom:
http://darksilverflame.deviantart.com/art/The-Shapes-Of-Hydrogen-Poster-327297786

They don't look very much like waves to me. As a matter of fact only the free particle solution looks wave-like.

What is your definition of 'wave'? See #15 above: https://www.physicsforums.com/threads/particle-or-wave.907971/#post-5719052

 

[bhobba]

It's not a contradiction. There are classical (point-like) particles and quantum particles (or quantum objects). 'Particle' is a constituent of "matter" (that's why we say "Particle Physics" ...). [Then, matter versus waves, fields, energy etc., all can virtually be unified ... , but not quite ... ; there is unity and oppotition.]

There is nothing more silly that arguing semantics.

Now, explain to me how an excitation in a quantum field obeys the wave particle duality?

In it precisely define what you mean by a particle, a wave, and duality. I am willing to accept pretty much any reasonable definition - but melding them together - well let's see what you come up with and if it fits the intuitive idea of those concepts. That's what the issue boils down to - you can define your terms to say whatever you like - but is it what physicists usually mean?

Thanks
Bill

 

[bhobba]

What is your definition of 'wave'? See #15 above: https://www.physicsforums.com/threads/particle-or-wave.907971/#post-5719052

The undulation of some medium like water waves. Even in classical fields like the EM field. Schrodinger originally thought that it's what his equation described. When it was proven it didn't he lamented he ever became involved in this whole mess that Dirac and others had turned into something entirely different.

But I am willing to accept any reasonable definition you come up with for wave - fire away. Here is one - its waves in a quantum field - we can examine that - but if you do define it that way and understand QFT you will immediately see difficulties - but fire away.

Thanks
Bill

 

[Stavros Kiri]

Now, explain to me how an excitation in a quantum field obeys the wave particle duality?

It is a particle if considered as a part of matter, but it is actually a virtual particle. However see previous posts above (especially #31 https://www.physicsforums.com/threads/particle-or-wave.907971/page-2#post-5719577 ).

It is a wave because it has a wave function, and obeys a wave equation (thus it is a possible solution to it). Physical interpretation: probability wave.

I think that says it all!

Note: It can also just be a wave-packet, not always wave.

 

[bhobba]

but it is actually a virtual particle.

Really? You know that virtual particles are simply the pictorial representation of terms in a Dyson series used purely as an aid to calculation? Are you willing to call lines that appear when physicists do theoretical calculations particles? If that's your definition its far too weird for me and I think the vast majority of physicists.

Be aware that anything written about QFT outside a QFT textbook is likely wrong - virtual particles as actual particles is just one example. If you go down the QFT path I suggest first learning some of its basics from a proper textbook eg:
https://www.amazon.com/dp/0984513957/?tag=pfamazon01-20

Thanks
Bill

 

[Nugatory]

Closed pending moderation.

[updated]
We're going to leave this thread closed.

The original question was "How anything can be a particle or wave at the same time?" and that has been answered with as much accuracy is possible in a B-level thread: That's not what the modern formulation of quantum mechanics says.