Understanding Matter Waves and the de Broglie Hypothesis in Quantum Physics

[TrickyDicky]

According to the de Broglie hypothesis particles like electrons can have a wavelength and be treated as waves for instance when confined in atoms.
What I fail to understand is what exactly waves, I mean, in what medium does the electron wave? is it empty space, like photons? Perhaps this is too basic, but I'm new at this quantum stuff.
Thanks

 

[arkajad]

For the system of n electrons (without spin) this is a complex-valued wave in a 3n-dimensional space. With spin the wave will have values in an appropriate multidimensional complex vector space.

From the original idea of "waves" to the present-day description it is a long way.

 

[TrickyDicky]

For the system of n electrons (without spin) this is a complex-valued wave in a 3n-dimensional space. With spin the wave will have values in an appropriate multidimensional complex vector space.

From the original idea of "waves" to the present-day description it is a long way.

I'm sure it is. You seem to be talking about the mathematical space, I was naive enough to think in terms of actual physical electrons in a physical space. I guess I'm too biased by real mechanical waves.
So in your opinion (and maybe that's the general understanding), the wave treatment of matter particles is just an abstract mathematical representation that gives good results and has nothing to do with an actual mechanical wave, so there is not a medium perturbed that "waves". Is that it?

 

[arkajad]

Who knows? Perhaps there are waves in some space that is less mathematical, but I do not think these will be "mechanical waves". But that's just speculation. For now we have what we have.

 

[G01]

VERY Qualitatively:

In quantum field theory, every particle is associated with some field. The particle is considered a disturbance in that field.

The photon is a quantized disturbance of the electromagnetic field. In the same way, an electron is considered to be the quantized disturbance of an "electron field."

So, it would be this "electron field" that is waving.

Also, on a side note, remember that waves can be real phenomena and yet not be mechanical waves. Consider EM waves, for example.

 

[arkajad]

So, it would be this "electron field" that is waving.

But, perhaps, itmay be worth to add that this "wave", that is the excited state of the quantized field, is a "wave" living in an infinite dimensional space. The field has lot of excited modes and some of these modes (for instance the infrared modes for the electromagnetic field) are not pleasant at all - they do not want to stay within denumerable dimensions.

 

[TrickyDicky]

VERY Qualitatively:

In quantum field theory, every particle is associated with some field. The particle is considered a disturbance in that field.

The photon is a quantized disturbance of the electromagnetic field. In the same way, an electron is considered to be the quantized disturbance of an "electron field."

So, it would be this "electron field" that is waving.

Also, on a side note, remember that waves can be real phenomena and yet not be mechanical waves. Consider EM waves, for example.

Thanks G01 for your qualitative comments, as it's often the case with this hard to put in words physical concepts they raise many questions.
The parallelism with the photon is fine but I was under the assuming that the photon is a field particle (a force carrier) and therefore its natural field is the EM field that has clear manifestations we all are used to. But I believe the electron or any other fermion are not force carriers, so I wonder what its "natural" field is and what are their manifestations. Could their natural field be the gravitational field since they have mass? But then there is active search for the graviton so I doubt that I'm on the right track here. Any comments welcome.

 

[G01]

Thanks G01 for your qualitative comments, as it's often the case with this hard to put in words physical concepts they raise many questions.
The parallelism with the photon is fine but I was under the assuming that the photon is a field particle (a force carrier) and therefore its natural field is the EM field that has clear manifestations we all are used to. But I believe the electron or any other fermion are not force carriers, so I wonder what its "natural" field is and what are their manifestations. Could their natural field be the gravitational field since they have mass? But then there is active search for the graviton so I doubt that I'm on the right track here. Any comments welcome.

No. In field theory, every particle, whether force carrier or not, fermion or boson, is defined as the quanta of a field. Here, this "electron field" is a separate thing from the gravitational field. The "electron field" is defined as the field for which the electrons are the quanta. That is all there is to it.

Through the Higgs mechanism, the "electron field" is given mass (thus the electrons have mass) and they are able to interact with other massive objects through the gravitational field.

However, I'm am basically at the level of an student in their first QFT course. I defer any more in depth description of the Higgs mechanism to someone more qualified.

 

[TrickyDicky]

No. In field theory, every particle, whether force carrier or not, fermion or boson, is defined as the quanta of a field. Here, this "electron field" is a separate thing from the gravitational field. The "electron field" is defined as the field for which the electrons are the quanta. That is all there is to it.

Yes, after reading some more in the wikipedia I think you are referring to the Fermionic or Dirac field that obeys canonical anticommutation relations and for which electrons (and the rest of fermions) are the quanta, unless there exists also a specific "electron field"?


Also, this "electron field" field is limited in extension to the wave packet of the electron, or it manifests somehow outside this space?

 

[arkajad]

The electron field (operator-valued) extends all over space and time. But it admits states ("wave functions", that is
excitations from the vacuum state) that, at a given time, are pretty well localized in space. The field variables are operators, like a simple $\hat{x}$-operator for the harmonic oscillator, "wave functions" are excited states (Hilbert space vectors) on which the field operators can act etc.

So, for instance if [itex]\hat{\Psi}(x)[/tex] is the field, and if $\xi$ is its state, then $\langle\xi|\hat{\Psi}^\dagger(x)\hat{\Psi}(x)|\xi\rangle$ is, roughly, "the number of particles at the spacetime point x when the field is in the state $\xi$".

 

[TrickyDicky]

The electron field (opearator-valued) extends all over space and time. But it admits states ("wave functions", that is
excitations from the vacuum state) that, at a given time, are pretty well localized in space.

Ah, ok, thanks. I should have figured that being a field it has to extend all over. I guess I was asking for the observable part, that I think is what you refer to in the second sentence as pretty well localized in space.
But thinking of it in these terms it is hard for me not to identify the electron field outside this localized excitation as the vacuum. So is the vacuum the meeting place of fields such as the EM field or the fermionic fields?

 

[arkajad]

You can make the vacuum "the meeting place" - if you wish. The photon creation operator will create from the vacuum a photon state, the electron creation operator will create from the vacuum an electron state. The vacuum can be made common if photons and electrons do not interact and it must be made common if you have them as fields interacting one with the other.

 

[TrickyDicky]

You can make the vacuum "the meeting place" - if you wish. The photon creation operator will create from the vacuum a photon state, the electron creation operator will create from the vacuum an electron state. The vacuum can be made common if photons and electrons do not interact and it must be made common if you have them as fields interacting one with the other.

Thanks, Arkajad. Your answers are helpful.