# About 'de Broglie hypothesis' and the double slit experiment

EDIT: As i mentioned in my original question, i do not have the background to fully understand @Timaeus answer (which was very detailed indeed). I would appreciate if someone could give a more 'classical physics' answer ,even not so detailed ,in order to clarify some things my self a little bit more. In addition i would like to know the difference between the term 'wave' and 'wavefunction' and how two uncharged particles would interfere in the experiment demonstrated by a single's particle's emission source.

Without enough theoretical background in physics , I post this question which actually has two related parts.

# Wavefunction and 'de Broglie hypothesis':

As far as I can understand the wavefunction of massless particles, is described by the magnitude's change of a certain particle's property, i.e. the wavefunction of a photon is described as the changing of the intensity of it's EM field over time. This wave moves through space with velocity $C$ and carries energy equal to $hf$.

On the other hand 'de Broglie hypothesis' suggests that 'all matter has wave properties' and the wavelength of this function is equal to $\lambda=h/p=h/mv\cdot \gamma^{-1}$. Now here comes my first question: In which particle's property is this wavefunction related to? Or does this wavefunction actually describes the particle's motion (~) with it's simultaneous transport through space with velocity $v$,carrying $\rm{KE}\;?$

b) In the double slit experiment that is demonstrated by a single's electrons emission source, what physical property is the interference pattern related to ? Is this pattern the result of the interference of accelerating electron's EM wave or something else? If the source emits 'n' single electrons, how many arrivals of matter do we detect on the screen?

• The waves you are associating with wavefunctions don't , in any way, carry energy. They are probability amplitude functions. However, they have expectation value of energy. – user36790 Mar 6 '16 at 13:07
• @aJ1974: You're part (a) is answered by the de Broglie relation: the wavelength depends upon the momentum of the quantum particle, and holds for both massive and massless particles. Part (b), concerning the double slit interference pattern, it is the probability amplitudes given by the wave function that are interfering. The de Broglie hypothesis inspired Shroedinger to create the Shroedinger equation; it becomes a fundamental part of the quantum wave picture. – Peter Diehr Mar 6 '16 at 13:14
• @aK1974, I just wanted to post one point, which you may already be aware of. de broglie associates a wave with every particle, but the wave speed and particle speeds are generally different. The wave speed is only c for light/photons, not for massive particles. The de broglie hypothesis leads to this equation $w=\dfrac{c^2}{v}$, where v is the particle velocity and w is the wave velocity. In the case of light/photons, $v=c$ hence solving for $w$ we get $w=c$ also. So for light the two speeds are the same, but for massive particles v is always less than c, and therefore w is greater than c. – Ameet Sharma Apr 1 '16 at 22:12

## 1 Answer

There are two approaches to quantum mechanics: nonrelativistic quantum mechanics, and quantum field theory.

In quantum field theory, there is one wave in physical space for each type of particle/antiparticle. A photon field, an electron/positron field, a muon/antimuon field, and so on. But the fields are operator valued, and quite complex. And there isn't a clean thing that corresponds to a single particle. The one field collectively represents all the photons and such in the whole universe.

In nonrelativistic quantum mechanics you have to pick a number of particles, such as $n$ particles and then each has a spin space $\mathbb C,$ or $\mathbb C^2,$ or $\mathbb C^3$ or $\mathbb C^k$ and then the wavefunction is a function from $\mathbb R^{3n}$ (note this is configuration space, which is much larger than physical space) and it goes into $\mathbb C^{k_1}\otimes\mathbb C^{k_2}\otimes\dots\otimes \mathbb C^{k_n}$.

This wave moves through space with velocity $C$ and carries energy equal to $hf$.

That never happens.

On the other hand 'de Broglie hypothesis' suggests that 'all matter has wave properties' and the wavelength of this function is equal to $\lambda=h/p=h/mv\cdot \gamma^{-1}$.

The $p$ is canonical momentum, not mechanical momentum. And even if it were mechanical momentum, the correct formulas are things like $\vec p=E\vec v/c^2$ (which holds for all particles) not $\vec p=\gamma m\vec v$ (which only holds for massive particles).

Just because some formulas are more popular and they hold for some special cases doesn't make them the correct formulas.

Now here comes my first question: In which particle's property is this wavefunction related to?

The wavefunction describes all properties of all particles.

Or does this wavefunction actually describes the particle's motion (~) with it's simultaneous transport through space with velocity $v$,carrying $\rm{KE}\;?$

It's a wave in configuration space. So the probability current describes flows in configuration space. A point in configuration space is an assignment of locations to all particles. So it's a flow from a configuration of all particles to another configuration of all particles.

b) In the double slit experiment that is demonstrated by a single's electrons emission source, what physical property is the interference pattern related to ?

The interference pattern is in the residuals of the locations of the particles.

Is this pattern the result of the interference of accelerating electron's EM wave or something else?

Something else. The interference of the spin in configuration space. With a residual over the locations of the screen. As measured by the frequency of a statistical ensemble which is related to the overall amplitude of a single instance of the ensemble.

If the source emits 'n' single electrons, how many arrivals of matter do we detect on the screen?

Less than $n$ if some hit the barrier on the way to the screen. The ensemble is all $n$ and in nonrelativistic quantum mechanics you get the prediction of the frequency of different locations from the wavefunction for just one electron.

• Thank you very much for your answer. To be honest I do not fully understand all the aspects of your answer but this obviously overcomes my current knowledge of the issue.I am keeping this as a start for further research! – user98038 Mar 6 '16 at 21:39
• @Timaeus, Can you explain the difference between mechanical and canonical momentum. In the De broglie's hypothesis with the equation $\lambda = \dfrac{h}{p}$, the $p$ is $\gamma m_{0}v$ for a massive particle as per Rindler's "Introduction to Special Relativity" p.83. where $v$ is the particle velocity. Is this wrong? – Ameet Sharma Apr 1 '16 at 21:51
• @AmeetSharma It can hold in some special cases. Such as for a chargeless particle or a zero vector potential. But it is a completely wrong characterization of the momentum in quantum mechanics. The wavelength in quantum mechanics is related to canonical momentum. Sometimes canonical momentum equals mechanical momentum. It does in the very simplest cases, which are often the first ones introduced. – Timaeus Apr 1 '16 at 22:32
• @Timaeus, thanks for the explanation. But historically, when de broglie came up with his hypothesis, was he thinking in terms of canonical momentum or mechanical momentum? – Ameet Sharma Apr 1 '16 at 22:42
• @AmeetSharma That's a question for the history of science stack exchange. But if the example he considered had a zero vector potential, then they aren't different for that special case. – Timaeus Apr 2 '16 at 1:15