Electrons and the Double Slit experiment questions

[TrickyDicky]

I'm going to try a non-QFT response, but please don't forget that there is some serious hand-waving in this answer - I'm just trying to convince you that cannot always trust first-quantization stuff when applied to photons.

Localizing something is not the same thing as saying that it is in an eigenstate of a position operator. When a particle passes through a pinhole, that narrows down the region of space in which it may be found, but the position is still uncertain because of the non-zero size of the pinhole itself. The only way to really nail the position down to that delta function that you're thinking of is if the pinhole is of size zero (that is, model the transmissability through the screen with a delta function). You can make this work with a point particle such as an electron - but you will get zero electromagnetic field passing through a size-zero hole, so no photon at all instead of a photon of precisely known position.

Another problem you'll find is that the entire formalism of position delta functions found by superimposing all possible momentum states and vice versa that you're suggesting we use comes from solving Schrodinger's $H\Psi=E\Psi$. However, that's not a relativistic equation, so there's no particular reason to expect it to work for photons - and if you do try, you'll find yourself looking at a $p^2/2m$ term, a fairly strong hint that you shouldn't be applying this when $m$ is equal to zero.

Not entering into the photon position issue, your two examples are a hand-wavey form of saying the position delta function formalism is only possible in a time-independent setting like the tunneling zero size pinhole first example or the TISE $H\Psi=E\Psi$ second example. But again, the double slit experiment cannot be simplified in that form even in hand-waving: the wave function is perturbed in a time-dependent way in its time evolution by the slits. This makes it collapse to the measured dot at the screen. Ignoring this is keeping with the misleading picture of the experiment that so many sources perpetuate.

 

[bhobba]

I'm scratching my head at this. Where do you get photons are less of a point particle(in the QFT, particle physics current sense of elementary particle) than electrons?

They do not have position like you expect of a point particle - in fact that pretty much the definition of a point particle.

Thanks
Bill

 

[TrickyDicky]

They do not have position like you expect of a point particle - in fact that pretty much the definition of a point particle.

Thanks
Bill

That might be a definition of a classical point-particle, but since you are referring to QFT and position is not an observable there, that cannot be a difference between electrons and photons in QFT.
All it's happenning here is that photons are not easily(if at all, but some people have tried) describable in terms of NRQM wave functions, something not particularly surprising.

 

[bhobba]

That might be a definition of a classical point-particle, but since you are referring to QFT and position is not an observable there, that cannot be a difference between electrons and photons in QFT.

Come again.

I gave a link that showed position is an observable for electrons but not for photons. Obviously that is the difference.

Nugatory gave a hand-wavey argument - but that's all it is.

Thanks
Bill

 

[TrickyDicky]

Come again.

I gave a link that showed position is an observable for electrons but not for photons. Obviously that is the difference.

You are not even making the distinction between NRQM and QFT. There are not photons in NRQM, EM interaction is treated semiclassically there.

 

[bhobba]

You are not even making the distinction between NRQM and QFT. There are not photons in NRQM, EM interaction is treated semiclassically there.

Exactly how does that change anything?

Thanks
Bill

 

[TrickyDicky]

Exactly how does that change anything?

You need to specify in which setting you frame your assertions, you seem to be switching from non-relativistic QM to QFT as it fits you. Now in nrqm there are no photons to begin with in a stric sense for the mathematical treatment, so the problem doesn't arise in the terms you refer to. It so happens that phenomenologically, in terms of interference in the double slit experiment doesn't make much difference to do the experiment with photons or electrons at a first approximation(Davisson-Germer and Young experiments equivalence) .

If your assertions where framed in QFT terms(as you hinted at several times by alluding to QFT texts) they don't make much sense, since position is not an observable regardless if one talks about electrons or photons. Now can yo decide if you are referring to the nrqm or the QFT setting when speaking about position?

 

[bhobba]

You need to specify in which setting you frame your assertions, you seem to be switching from non-relativistic QM to QFT as it fits you

The link I gave was crystal clear - QFT.

But since position is not an observable for photons an ordinary wave-function does not exist.

Thanks
Bill

 

[TrickyDicky]

The link I gave was crystal clear - QFT.

But since position is not an observable for photons an ordinary wave-function does not exist.

Thanks
Bill

Nevermind, I simply figured you were aware that position was not an observable for electrons or any other particle in QFT either.

 

[bhobba]

Nevermind, I simply figured you were aware that position was not an observable for electrons or any other particle in QFT.

Did you read the link I gave?

I know its quite advanced but the conclusion was clear - for massive particles:
'Since the physical irreducible representations of the Poincare group are uniquely determined by mass and spin, we see that in the massive case, a position operator must always exist'

For photons:
'Therefore, the concept of a photon position is necessarily subjective, since it depends on the POVM used, hence on the way the measurement is performed. It does not describe something objective.'

Its not a bog deal. Most don't worry about it in the double slit experiment. Its simply, like I said, we have some advanced posters on this forum so I like to be exact.

Thanks
Bill

 

[TrickyDicky]

Did you read the link I gave?

I just did, it is obvious it is not exactly about QFT but about relativistic QM of particles. In the RQM formalism you have position eigenstates, in QFT position is a label on operators.

Its not a bog deal. Most don't worry about it in the double slit experiment. Its simply, like I said, we have some advanced posters on this forum so I like to be exact.

That is commendable, and while I agree it is no big deal in the context of this thread, it is advisable to get it right when choosing a reference, I recommend pages 25-6 of Srednicki.

 

[Howard]

Not an answer, just some basics. An electron is a field permeating space. We perceive it as a particle or wave. It's best not to think of fundamental entities as particles or waves, but as fields. Vibration in a field is perceived as a wave or particle depending on the experimental conditions. Fields react to create other fields or properties, eg, the Higgs field reacts with the electron field to give the electron its mass.
Regards, Howard.