What is the mathematical basis for understanding the wave property of particles?

[Delta2]

The wave property of particles (like electrons) is due to :

1) The wave function
2) The underlying fermionic field
3) Just because of the existence of de broglie waves?

Or maybe someshow the above three cases are unified ?

 

[bhobba]

The wave particle duality, and the idea quantum objects are in a sense waves, was consigned to the dustbin of history when Dirac came up with his transformation theory in 1927.

Its basically a crock of the proverbial.

If you want to understand the true basis of QM the following is a much better place to start:
http://www.scottaaronson.com/democritus/lec9.html

Wavelike solutions that sometimes appear in equations are a result of Schroedinger's equation. Why Schroedinger's equation is a deep issue, but its true basis is actually the Principle Of Relativity, without going into the details - you will find those in Chapter 3 of Ballentine - Quantum Mechanics - A Modern Development:
https://www.amazon.com/dp/9810241054/?tag=pfamazon01-20

Thanks
Bill

 

[Simon Bridge]

"Wave particle duality" is still taught at senior secondary and freshman tertiary levels in many countries - as a historical stepping-stone. You don't get Dirac stuff until much later. You need to be aware, therefore, that it involves concepts which will be discarded later so don't take to so much to heart.

In that context, the historically puzzling wave-like behavior of objects like electrons is in the emerging statistics.
As such it is a label for an outcome, not a name for any underlying physics.

Of the three options in post #1, the idea of deBroglie waves should be discarded right away as misleading.

The fermionic field is not going to do it because not all particles are fermions - i.e. it's not a general enough example. Field theory does handle both particle-like and wave-like behavior in a unified way and can be thought of fuzzily as wave(-function) mechanics with relativity.

"The wavefunction" would be a solution to Schrodingers equation. The behavior, such as electron diffraction at slits, that gave rise to the "wave-particle duality" is an emergent effect from these.

The Scott lecture from bhobbas post shows a way of getting QM from a purely statistical approach. I'm not so sure this constitutes a "basis" for quantum mechanics, but it does summarize the main mathematical ideas in an intreguing way. You need freshman statistics and algebra for it.

To answer the question properly, we need the context. The numbered list appears like homework for example.

Aside: I think I'm going to have to get a copy of Ballentine.
I would have suggested the Feynman lecture series on QED pending context.

 

[Delta2]

Thanks for the answers. This is not homework, i was just reading a indroductory book in Quantum Mechanics (written in Greek by Greek physicist) and also read some things in wikipedia about QFT and the fermionic fields and was wondering how to explain the wave nature of particles. (I know photon is also a particle (though a boson) and some say its wave function is the EM-field, that's how i came up with the fermionic field option),

 

[Simon Bridge]

Try: http://vega.org.uk/video/subseries/8
But in terms of the text you are reading, the wave-behavior is from the wave-functions.
The wave functions tell you about the statistics - you can think of them as sort of like probability density functions if probabilities are allowed to take negative values.

Notes:
There is also such a thing as a bosonic field too.
In QM, an EM field is not something photons have, it's something they are.
You should get used to the basic framework before trying to join the dots.

 

[atyy]

The wave property of particles (like electrons) is due to :

1) The wave function
2) The underlying fermionic field
3) Just because of the existence of de broglie waves?

Or maybe someshow the above three cases are unified ?

For non-relativistic physics, 1 and 2 are different but equivalent ways of saying the same thing.

For relativistic physics, we have to use 2.

3 is informal and loose language, and can correspond to 1 for non-relativistic physics of a single particle, or it can correspond to 2 in the Heisenberg picture.

 

[bhobba]

The Scott lecture from bhobbas post shows a way of getting QM from a purely statistical approach. I'm not so sure this constitutes a "basis" for quantum mechanics, but it does summarize the main mathematical ideas in an intreguing way. You need freshman statistics and algebra for it.

Yes - basis is too strong a word.

A better description would a 'nice' place to start that avoids historical baggage that needs to be unlearned later.

Thanks
Bill

 

[bhobba]

Aside: I think I'm going to have to get a copy of Ballentine

A note to the OP. Ballentine is my favourite QM book and had a big effect on me.

It is however well and truly graduate level. I studied it after reading Dirac's Principles of Quantum Mechanics and Von-Neumann - Mathematical Foundations of QM which are also advanced and a number of intermediate books like Griffiths. Then their was a detour into so called Rigged Hilbert Space.

Undoubtedly Simon will have no troubles with it - but without a good previous exposure to QM and a good math background it will be hard going.

Thanks
Bill

 

[Simon Bridge]

Yes - basis is too strong a word.

A better description would a 'nice' place to start that avoids historical baggage that needs to be unlearned later.

Thanks
Bill

Very abstract maths though ... you have to really get into the mathy descriptions of physics. Some people will get it though - even at an introductory level. But you then have to go right to matrix mechanics wouldn't you, where most courses start with wave mechanics?

Have you tried it on a class?

 

[WannabeNewton]

Thanks for the answers. This is not homework, i was just reading a indroductory book in Quantum Mechanics (written in Greek by Greek physicist) and also read some things in wikipedia about QFT and the fermionic fields and was wondering how to explain the wave nature of particles.

Consider a scalar field $\varphi$. Given the vacuum state $|0 \rangle$ of the free theory one can show that $\varphi (x) |0 \rangle$ creates a particle $|k\rangle$, the one associated with this field, at the event $x$. But the particle of course won't be localized since $|k \rangle$ is a momentum eigenstate. Indeed if we look at the matrix element $\langle 0 | \varphi(x) |k \rangle$ then we will find it to be a plane wave, the exact same plane wave for a free particle propagating through free space in QM (with a different normalization since we want it to be Lorentz invariant in QFT). So QFT incorporates the wave-nature of particles in the same way as QM. This can be readily generalized to fermion fields $\psi_a$.

 

[bhobba]

Very abstract maths though ... you have to really get into the mathy descriptions of physics. Some people will get it though - even at an introductory level.

Good point. I am the type that thinks mathematically - others aren't necessarily like that.

But you then have to go right to matrix mechanics wouldn't you, where most courses start with wave mechanics?

Hmmm. Yes you probably would. You associate each outcome with some real number yi. Then form the matrix O with the yi as diagonal elements. Its then easy to see E(O) = <u|O|u>. It would also require introduction of the Dirac notation before the physics. I wouldn't attempt it without knowledge of linear algebra. Then you basically have the two axioms as found in Ballentine. From that you would follow something similar to Ballentine.

Thinking about it, it would be better suited to a quantum physics course for mathematicians with a background in Linear Algebra, or physics students double majoring in math and physics.

Have you tried it on a class?

I am not a lecturer - simply a retired guy with an interest in this stuff.

The reason it probably appeals to me is my background, and inclination, is applied math. Its the reason I had the detour into Rigged Hilbert Spaces. Von-Neumann was scathing in his book about the Dirac Delta function and something in me made me want to the bottom of it.

Thanks
Bill