Let's get it right about Electron Waves

[rohanprabhu]

'Electron Waves' is something I heard of in my 9th or 10th standard, and fooled around in 12th standard. But the thing is, I never really understood what a 'Matter wave' was.. because none of the textbooks ever say anything about it. There are two things I can make of it:

i] An electron follows a path like that of a wave.. as in, it's position about an axis oscillates, while it's distance from another axis perpendicular to it increases with a rate equivalent to the 'wave velocity'. Although this can be understood easily, it just doesn't explain electron diffraction for me [maybe.. somebody else could explain it to me].

ii] The properties of an 'electron' vary. As in, there is an electron, and at a distance, the electron's properties, as in, it's mass, it's charge all vary as a spatial function. However, this is also attractive enough, it doesn't quite satisfy the quantisation of charge. Nor does it explain to me as to how these properties vary, as in, what is the 'disturbance' that propagates throughout space.

iii] I am wrong in both these ideas and it's something else...

any help is appreciated.
regards,
rohan

 

[dst]

I thought the de Broglie wavelength corresponded to the 'wavelength' of the electron's wavefunction?

 

[rohanprabhu]

right.. do not deny it.. but then the 'electron wavefunction'.. what is it? What exactly varies or what exactly is the property that varies as a function of space coordinates.

 

[Manchot]

The wavefunction evaluated at a certain point in space gives the probability that a position measurement of the electron will determine that it is located at said point in space.

 

[G01]

The wavefunction evaluated at a certain point in space gives the probability that a position measurement of the electron will determine that it is located at said point in space.

This is not completely correct. The absolute square of the wave function gives the probability density of the particle, not the wave function itself:

$$P=\int\Psi *\Psi dx=\int |\Psi|^2 dx$$ The wave function is not a physically measurable quantity, while it's absolute square is.

So, while we can ask "What is waving?" and similar questions, we won't be able to get an answer, since it isn't physically measurable. The point is though, that electrons and can exhibit wavelike properties. The wave function is the mathematical description of those properties.

This view I'm presenting is based on the Copenhagen Interpretation. Other interpretations, like the Bohm Interpretation, have differing views.

 

[Manchot]

Whoops, I'm embarrassed.

 

[Domenicaccio]

This is not completely correct. The absolute square of the wave function gives the probability density of the particle, not the wave function itself:

$$P=\int\Psi *\Psi dx=\int |\Psi|^2 dx$$


The wave function is not a physically measurable quantity, while it's absolute square is.

So, while we can ask "What is waving?" and similar questions, we won't be able to get an answer, since it isn't physically measurable. The point is though, that electrons and can exhibit wavelike properties. The wave function is the mathematical description of those properties.

This view I'm presenting is based on the Copenhagen Interpretation. Other interpretations, like the Bohm Interpretation, have differing views.

Maybe we can say it is possible to measure it, or at least to VIEW it somehow (I mean the absolute square, not phy itself), because this is what we actually see when watching a diffraction figure.

We can imagine to have our particle beam so faint, that at first only 1 electron (or whatever, every particle gives diffraction) is shot by the source towards the slit: we will see this 1st electron hit the final screen somewhere, apparently in a random position.

Then we shoot a 2nd electron, it will pass through the slit and hit the screen at another position, apparently random again (and it could be ANYWHERE on the screen).

But after a while, when we have already shot a lot of electrons, even if EACH electron can hit the screen anywhere, we will start noticing that certain areas of the screen are hit more frequently than others (we'll start seeing the diffraction figure), and we can therefore say, in the classical sense of probability, that each electron had/has a higher probability of hitting the screen in certain areas rather than others.

Hence we can sort-of say that the diffraction figure itself is a "projection" of the phy squared. We can even calculate lambda (wavelength) quite easily from the diffraction lines separation and the slit-screen distance.

 

[G01]

Maybe we can say it is possible to measure it, or at least to VIEW it somehow (I mean the absolute square, not phy itself), because this is what we actually see when watching a diffraction figure.

We can imagine to have our particle beam so faint, that at first only 1 electron (or whatever, every particle gives diffraction) is shot by the source towards the slit: we will see this 1st electron hit the final screen somewhere, apparently in a random position.

Then we shoot a 2nd electron, it will pass through the slit and hit the screen at another position, apparently random again (and it could be ANYWHERE on the screen).

But after a while, when we have already shot a lot of electrons, even if EACH electron can hit the screen anywhere, we will start noticing that certain areas of the screen are hit more frequently than others (we'll start seeing the diffraction figure), and we can therefore say, in the classical sense of probability, that each electron had/has a higher probability of hitting the screen in certain areas rather than others.

Hence we can sort-of say that the diffraction figure itself is a "projection" of the phy squared. We can even calculate lambda (wavelength) quite easily from the diffraction lines separation and the slit-screen distance.

That is my point.

$$\Psi^*\Psi$$ is observable. We can measure that and see it in cases like the diffraction pattern you mentioned, but we can't observe the wave function itself.

The wave function can't be observable. In many cases it can even have non-real values.

 

[Domenicaccio]

That is my point.

$$\Psi^*\Psi$$ is observable. We can measure that and see it in cases like the diffraction pattern you mentioned, but we can't observe the wave function itself.

The wave function can't be observable. In many cases it can even have non-real values.

But it's not much of a problem, is it? What information does $$\Psi$$ has that $$\Psi^*\Psi$$ doesn't have? I think only the phase/arg, but what is the physical meaning of the phase of $$\Psi$$?

 

[G01]

But it's not much of a problem, is it? What information does $$\Psi$$ has that $$\Psi^*\Psi$$ doesn't have? I think only the phase/arg, but what is the physical meaning of the phase of $$\Psi$$?

It really isn't a problem, just a pedagogical point. I just wanted to point out that the probability density is given by the absolute square of the wave function, not the wave function itself.

 

[f95toli]

But it's not much of a problem, is it? What information does $$\Psi$$ has that $$\Psi^*\Psi$$ doesn't have? I think only the phase/arg, but what is the physical meaning of the phase of $$\Psi$$?

The fact that the wavefunction has a phase is what makes interference possible. Many (most?) of the "weird" phenomena in QM are due to this fact. Note that the phase can change even when the magnitude does not which give rise to some interesting effects; the Aharonov-Bohm effect is one striking example.