# In the double-slit experiment of electrons (observed by photons), is it correct to say the collapse is caused by the momentum of the photons?

I'm working off the article, The Double Slit Experiment Demystified. Disproving the Quantum Consciousness connection. I think it's well-written, but I'm not convinced about this part:

So what is causing the change in the particle’s behaviour?

In the example given above, when electrons are fired at a fluorescent screen, when we bombard the electrons with photons the interaction changes the state of the system catastrophically as, despite lacking mass, photons do carry momentum.

Here the author strongly implicates the momentum of photons as being responsible for the collapse. But is that correct to say?

I have read that the quantum eraser experiments verified that obscuring which-way information restores wave-like behavior. So it feels to me it's more about some "outer" state (perhaps well-characterized as "information") that we don't yet understand that collapses the system, not per se the photon-electron interaction.

Or am I wrong, and in fact for example, the process of obscuring in the quantum eraser experiments somehow "undoes" the photon-electron interaction or restores the quantum state through another interaction?

• collapse is a shorthand way of saying "boundary conditions changed so different wavefunction solutions describe the system" . Wavefunctions change whenever there are interactions. not just the momentum of the photon, have to be conserved in the new wavefunction. – anna v Nov 26 '18 at 19:05
• I’m still stuck on why electrons hitting a fluorescent screen has anything at all to do with electrons hitting photons. It is just weird. – Jon Custer Nov 26 '18 at 21:37

Your question is all about the quantum measurement problem. The quantum measurement problem has had vast amount of ink spilled on it, to not much purpose. But I will spill some more. In a physical process intended to measure something, the interaction between measured system $$q$$ and measuring device $$m$$ has the generic form

$$\frac{1}{\sqrt{2}} (\psi_{q,1} + e^{i\theta} \psi_{q,2}) \phi_{m,0} \rightarrow \frac{1}{\sqrt{2}} (\psi_{q,1} \phi_{m,1} + e^{i\theta} \psi_{q,2} \phi_{m,2})$$

where $$0,1,2$$ label different states and subscripts $$q,m$$ refer to the different parts of the joint system. If we suppose we were able to now find out the state of $$m$$, then we would at the same time learn the state of $$q$$. So now we introduce a third system whose job is to measure $$m$$, and so on. The problem is that by such arguments we are led to an infinite chain of measuring apparatuses.

The usual way to 'cut' this chain (gordian knot?) is to notice that if we introduce a collapse postulate late enough in the chain, then it doesn't matter exactly where we say the collapse happens. 'Late enough' here means that we have some third system which undergoes a complex and thermodynamicaly irreversible process, like an avalanche. Standard photon detectors work via some sort of avalanche, for example. When I say it doesn't matter exactly where we introduce the notion of a collapse onto one state or another, I mean that it doesn't matter in the sense that there are no experimentally observable differences once the processes are practically irreversible. (This does not rule out the value of exploring what really goes on with large or complex quantum systems).

What is going on in a 'quantum eraser' is that someone is using the language of 'measurement' at an early stage, where all that has happened is that one simple quantum system has had a simple interaction with another, and the state is like the one I wrote above. We can now notice that among the possible states of $$m$$ there are the following pair:

$$\phi_{m,+} \equiv \frac{1}{\sqrt{2}} ( \phi_{m,1} + \phi_{m,2} )$$

$$\phi_{m,-} \equiv \frac{1}{\sqrt{2}} ( \phi_{m,1} - \phi_{m,2} )$$

After solving these equations for $$\phi_{m,1}$$ and $$\phi_{m,2}$$ we get that the state of $$q$$ and $$m$$ on the right hand side of my first equation is

$$\frac{1}{2}( \psi_{q,1} (\phi_{m,+} + \phi_{m,-}) + e^{i\theta} \psi_{q,2} (\phi_{m,+} - \phi_{m,-}) ) = \frac{1}{2}( (\psi_{q,1} + e^{i\theta} \psi_{q,2} )\phi_{m,+} + ( \psi_{q,1} - e^{i\theta} \psi_{q,2} )\phi_{m,-}) )$$

In the context of your question, keep in mind that $$q$$ here is your particle going through Young's slits, and $$m$$ is some sort of simple system, such as a single atom, whose state has become correlated with the path of the particle (i.e. which slit in Young's slits). So we might be inclined to say that such an atom 'measured' the particle, but I would prefer not to introduce the term 'measured' for such a simple, reversible, interaction. I would be willing to say that $$m$$ in this example contains 'which-path information'. Anyway, whatever words we use, one can now introduce a truly irreversible measurement of the 'which-path signaler' $$m$$ (e.g. the atom in my example) and then reveal that $$q$$ shows an interference pattern, in the following way. If $$m$$ is found in the + state, then $$q$$ has an interference pattern with phase $$\theta$$. If $$m$$ is found in the - state, then $$q$$ has an interference pattern with phase $$\theta + \pi$$. These two interference patterns exactly fill in each others gaps, so if we don't know the result of measuring $$m$$ and just add all the results, then we see no interference, just a flat distribution for $$q$$. On the other hand, if we retain only the $$+$$ outcomes for $$m$$, then we will find we have selected from our dataset in such a way that an interference pattern is revealed for $$q$$.

I think you should regard this as interesting but not completely mind-blowing. I say this because similar effects happen with classical waves when we have a property such as polarization which can be manipulated so as to change the way one wave adds to another when an interference is set up.

To connect the above to the photon momentum, argue as follows. If electrons (say) are passing through the slits, and the slits are also illuminated by a light field, then $$q$$ in the above is an electron, and $$m$$ could be a photon. The state $$\phi_{m,0}$$ could be a plane wave photon state travelling towards the slits; the states $$\phi_{m,1}$$ and $$\phi_{m,2}$$ could be outgoing approximately spherical-wave photon states, centred at slit 1 and 2 respectively. If the photon wavelength is large, then $$\phi_{m,1}$$ and $$\phi_{m,2}$$ cannot be very different (they would overlap even at the slits, owing to diffraction; in technical language we say they are not orthogonal). In that case an observation of the photons won't determine the electron state. If the photon wavelength is below about $$d$$, on the other hand, where $$d$$ is the slit separation, then an instrument such as a microscope could collect the photons and determine which photon came from which slit. However, in order to have sufficient optical resolution to achieve this, the lens that gathers the light has to be large. In consequence when the photon is detected in this way (so as to tell where it came from), the information about what direction the photon was travelling in is largely lost. This means the momentum component of the photon in the direction parallel to the slits is randomized by an amount $$\Delta p \simeq \hbar / d$$, and by conservation of momentum, this is also the amount by which the electron momentum parallel to the slits is randomized. This is just enough to wash out the interference pattern of the electrons.

If the photon field were measured another way, for example by using an instrument to determine accurately the direction in which each photon moves away from the slits, then one could extract from the dataset just those cases where the photon moved away in a given direction, and these cases would show interference fringes in the electron signal!

In my first argument I talked about entanglement between $$q$$ and $$m$$. In the argument in terms of momentum and its conservation one is getting a further perspective on what the entanglement between $$q$$ and $$m$$ is like: among other things, it conserves momentum. Also, each state of motion of any given particle satisfies Heisenberg's uncertainty principle, and in the case of photons this is exactly equivalent to the requirements of ordinary wave optics of classical physics. Those requirements tell us how instruments like microscopes work and what limits their imaging resolution.

Here the author strongly implicates the momentum of photons as being responsible for the collapse. But is that correct to say?

The fact that photons have momentum does not make them cause things to collapse. I think that the author was just emphasizing that photons can interact with electrons, which is necessary (but not sufficient) for collapse, and the fact that they can exchange momentum is evidence of this. However, having non-interacting particles with high momentum fly about won't do anything to the electrons.

The main ingredient in what is believed to be collapse today (or at least, a process experimentally indistinguishable from collapse with limited technology) is entanglement with a macroscopic system. In the example of the double slit experiment, the electron's position would become entangled with the state of the screen. Conceptually, that would be something like:

$$|\psi\rangle_{e^- and \, detector} = \lambda_1|x=1\rangle|D_1\rangle +\lambda_2|x=2\rangle|D_2\rangle +...$$

where $${|D_i\rangle}$$ are macroscopically distinguishable states of the detector (the screen), which are therefore orthonormal. Of course, this is just a rough sketch: really you'd have an integral since $$x$$ is a continuous degree of freedom, not to mention there isn't a macroscopically distinguishable state of the screen for every real number $$x$$.

The reality is that collapse is only partially understood, though, and there are many naive questions that aren't predicted by standard quantum mechanics or quantum field theory, like "when exactly does collapse occur?", or "how many particles make it macroscopic?". A current popular answer is that we can't tell when, because it could be at various stages in the interaction between the electron and the macroscopic system, as far forwards as the moment that the information reaches someone's brain. My personal opinion is that the idea of collapse is just an approximation to an underlying process which does not have to contain a discontinuous jump at a (seemingly) arbitrary time & with an arbitrary size of system.