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The double-slit "thought experiment" described by Feynman in Lectures on Physics Volume 3 section I-6 Watching the electrons consists in firing electrons through a double-slit to observe the interference of electron waves, and watching them after passing the slits with a light source placed behind the double-slit, at equal distance of each slit. As electric charges scatter light, one can "detect" which slit the electron went through if the photon wavelength is small enough.

Question: has this "thought experiment" been simulated by solving numerically the underlying Schrödinger equation? I am aware about numerical experiments of the double-slit, but did not find any including the interaction between the electrons and the photons just after the double-slit.

The numerical simulation can address other types of particles (replacing for example the photon by a slow electron to avoid relativistic equations), the crucial point being the simulation of the observation (here the photons being scattered by the electrons) and its effect on the wavefunction. Its interest could be in particular to better understand in which precise way the observation progressively becomes inoperative when the photon wavelength increases.

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  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$ – David Z Sep 23 '16 at 14:03
  • $\begingroup$ Something similar has actually been done experimentally with fullerenes, see doi: 10.1038/44348 Scattered photons would end up entangled with your electrons. Measuring those photons could destroy your interference pattern, if the measurement can tell you which slit the electron went through. $\endgroup$ – Arturs C. Mar 22 '17 at 11:07
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I'll pull my remarks from the (now migrated) comment thread in, since this has yet to be properly addressed. To answer your main question,

has this "thought experiment" been simulated by solving numerically the underlying Schrödinger equation?

I would say:

  • the precise simulation you propose (or close variations of it) probably has been performed at some point (probably multiple points, by multiple people);
  • it probably hasn't been published;
  • if it did get published, it won't make for a particularly interesting paper; and
  • if it hasn't been published, it's a mountain of a task to show credibly that there is no such paper in the literature.

Now, the reason for the above is that the simulation you propose just isn't very interesting. You say that

Its interest could be in particular to better understand in which precise way the observation progressively becomes inoperative when the photon wavelength increases.

but that is not the case: anything that such a simulation could show us, we already understand.

It is well understood, since the days of von Neumann, that within formal, unitary quantum mechanics, the effect of measurements is to cause entanglement: if the system is in a superposition of $|A=1\rangle$ and $|A=2\rangle$, say, and you measure $\hat A$, with some detector that goes to $\left|\uparrow\right>$ on $A=1$ and to $\left|\downarrow\right>$ on $A=2$, what you're really generating is the superposition $$ |\Psi\rangle = \alpha \left|A=1\right>\left|\uparrow\right> + \beta \left|A=2\right>\left|\downarrow\right>, $$ i.e. an entangled state between the system and the detector. This state can no longer show interference, because if you take the inner product between the two components, i.e. if you do $$ \bigg< \left|A=1\right>\left|\uparrow\right> ,\, \left|A=2\right>\left|\downarrow\right> \bigg> = \left<A=1|A=2\right> \left<\uparrow\middle|\downarrow\right> = 0 $$ you get zero, because the detector states are orthogonal. This completely kills any chance of interference, but the wavefunction hasn't "collapsed" (yet); if you do a projective measurement on the detector (forcing its wavefunction to "collapse", whatever that means), that extends to the system as well, but that is external to the simulation.

Let me reiterate the point: anything you could simulate, while keeping to unitary quantum mechanics, would completely fit within the scheme above, and it would not add to our understanding of the thought experiment.

If you do want your simulation to include some form of wavefunction "collapse" (or decoherence or whatever you want to call it), i.e. if you want your simulation to actually say anything useful about the measurement problem, then you're going to need to decide how you handle the information encoded in your which-way detector, and this is where your scheme goes south: in order to simulate anything at all, you essentially need to pre-bake some resolution of the measurement problem into your simulation. Whatever results you get back will just be a restatement of the premises you feed in, and will be subject to all the flaws of the premises.

Given this, all that such a simulation would provide is a very expensive visualization aid for processes that can already be understood analytically, and for which the main difficulty is conceptual. The creation of visualization aids is not without merit, but this one would offer very little towards the resolution of the true conceptual problems of the measurement problem, which have much more to do with what projective measurements mean than with photons and double slits.

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  • $\begingroup$ +1 great answer, particularly the last paragraph I would sign my name under. However I disagree with the second last paragraph. You could (practical realisability aside ;) ) put your detector and all the necessary environments in your your simultation. I don't see how that requires a pre-baked resolution of the measurement problem. The dynamics of such a system should surely be all there is to it? Of course this all hinges on realisability and if we could simulate such a system we would probably know the solution of the measurement problem, but that is really the reverse of your paragraph. $\endgroup$ – Wolpertinger Mar 23 '17 at 10:10
  • $\begingroup$ @Mrphlng The thing is, you can try and get as fancy as you want with your simulated detector, but it still won't get you past the fact that (unless you're manually breaking unitarity) the simulation's outcome will be entanglement between the particle and the detectors, and the interpretation of that entangled state is precisely where the measurement problem lies. You end up essentially dancing around the Wigner's Friend paradox instead of tackling it head on on a conceptual level, like you ought to. $\endgroup$ – Emilio Pisanty Mar 23 '17 at 10:24
  • $\begingroup$ Surely if the system you have is a good detector the collective probability distribution for your detector observable after decoherence has kicked in will just be very peaked and very stable with time? (i.e. classical without using that word). I don't see the conceptual issue. Or am I getting something wrong here? $\endgroup$ – Wolpertinger Mar 23 '17 at 10:36
  • $\begingroup$ Example: if I shoot my photon on a photographic film they entangle for a bit, decoherence comes, photon runs away/gets absorbed somewhere, at the end the state of the photographic film has changed and there is a black spot somewhere. Also would be happy to continue this in chat if you're not too bored of the topic, since it seems to develop into a discussion that might be useful (at least for me ;) ) but not very relevant to the question. $\endgroup$ – Wolpertinger Mar 23 '17 at 10:37
  • $\begingroup$ @Mrphlng I have nothing to add beyond my comment above and what's in the answer. What you propose presupposes a resolution to the measurement problem, so the simulation would not add anything. $\endgroup$ – Emilio Pisanty Mar 23 '17 at 10:52
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The loss of interference is not due to collapse because collapse is a myth.

The loss of interference can be simulated by quantum mechanical models that include the interaction that transfers information about the measured system, which cause decoherence. These papers are not specifically about decoherence in electrons but they are about decoherence in general double slit type experiments

https://arxiv.org/abs/1606.09442

https://arxiv.org/abs/quant-ph/0310095

There are papers about dehoherence of electrons specifically in electronic Mach-Zehnder interferometers. These are a couple of samples of a very large literature on the topic:

https://arxiv.org/abs/0801.2338

https://arxiv.org/abs/1105.2587

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  • $\begingroup$ Thank you very much for the links. Additional links to the full papers would be very welcome! $\endgroup$ – user130529 Mar 23 '17 at 7:53
  • $\begingroup$ You can get the full papers by clicking on the PDF link in the links above. the PDF link is in a box on the right. $\endgroup$ – alanf Mar 23 '17 at 8:25
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The effect Feynman describes is not calculated using the Schrödinger equation. Instead, it is a result of "wave function collapse" which is a result of the measurement of the electron's position. The collapse is a separate postulate that cannot be derived from the Schrödinger equation. This is the standard Copenhagen interpretation anyway.

Everett, in his "Relative State" paper, showed that in fact it can be derived from the Schrödinger equation, although you have to include the observer in your wave equation. This leads to the Many Worlds interpretation of quantum mechanics.

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  • $\begingroup$ Your second assert is exactly what I had in mind: using the Schrödinger equation including the observer in the numerical simulation (the observer here being simply the incident photon). As a side remark, I am not sure that including the observer necessarily leads to the multiverse interpretation. The validity of applying the Schrödinger equation to the whole system is independent of the interpretation (the multiverse interpretation doesn't seem to rely on facts). $\endgroup$ – user130529 Jan 23 '17 at 11:23
  • $\begingroup$ @claudechuber right, it's just one way of seeing all that huge configuration space of the setup+observer: there are subspaces where the wavefunction of the setup collapsed in one way, other subspaces with another way of collapse etc.. $\endgroup$ – Ruslan Jan 23 '17 at 12:03
  • $\begingroup$ @claudechuber I didn't mean that including the observer leads to Many Worlds. I meant that including the observer was necessary in order to explain Feynman's effect without collapse. $\endgroup$ – Bruce Jan 25 '17 at 2:05
  • $\begingroup$ Thank you Bruce, I fully agree with you. It would be so exciting if one day someone performs such a numerical simulation! $\endgroup$ – user130529 Jan 25 '17 at 7:45
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What Feynman is talking about is the "wave function collapse" which is due to measurement.

Suppose, you let electrons pass through two slits, but didn't observe it. There is no way of knowing which slit the electron will pass through. The superposition principle tells you that the electrons actually pass through both the slits at once. The interference of the wave functions result to the interference pattern of electrons on the screen.

Consider, you could observe the electrons without actually disturbing the state of it (purely hypothetical). Even, in that situation there occurs the collapse of the wave function. Now, you could actually see which slit the electrons pass from. Thus, you will observe no interference pattern on the screen.

enter image description here

The picture shows what happens in the case. Please know this is a purely hypothetical situation. I just want to show the wave function collapses with any types of measurement.

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  • $\begingroup$ The wavefunction squared xx* is a probability distribution, not a prediction of a single measurement, but an accumulation of measurements. A single electron has a probability of going through one or the other. No one has seen an electron spread over space, as it is a point particle, even experimentally to a high accuracy. $\endgroup$ – anna v Mar 22 '17 at 11:45
  • $\begingroup$ QM isn't all about probability. Schrodinger said his cat is alive and dead both at the same time before observation (exists in a superposition of both alive and dead) and the superposition collapses with measurement). Similarly, an electron in a box exists in multiple positions at the same time (superposition once again). In the same way the electron also passes through both the slits at once. And it is the interference of the wave function that occurs in the other side of the slit. $\endgroup$ – Deechit Poudel Mar 22 '17 at 13:44
  • $\begingroup$ It was a gross analogy, the cat used as an elaborate detector instead of a geiger counter, to plot the probability of decay, which would need many boxes with cats. In any case quantum mechanics has been clarified since S. times, and he is not a pope. Superposition is for the wavefunction, not for the measured particle which needs the square. $\endgroup$ – anna v Mar 22 '17 at 16:30
  • $\begingroup$ S.position is actually for the states of a particle. 'A particle exists in a s.position of many states before measurement and its the measurement that collapses it.' This is the very foundation of QM. Each state can individually be described by a unique w.function and its the superposition of all these w.functions that gives the resultant. The superposition is more weighted for some states than the other. Thus, the particle on observation has more probability to turn up in that state. But, previously it WAS in the superposition of all the states it could possibly turn up after observation $\endgroup$ – Deechit Poudel Mar 22 '17 at 17:36
  • $\begingroup$ Schrodinger still is the pope for me. $\endgroup$ – Deechit Poudel Mar 22 '17 at 17:37
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The double-slit "thought experiment" ... consists in firing electrons through a double-slit ... and watching them after passing the slits with a light source placed behind the double-slit...

To generalize this it is said that any interaction with the electron changes its straight path of movement.

As electric charges scatter light, one can "detect" which slit the electron went through...

What else can deflect electrons from their path? Moving electrons could be influenced by EM radiation as well as by electric and magnetic fields. For magnetic fields this is described by the Lorentz force. The mentioned by you "Numerical Simulation of the Double Slit Interference with Ultracold Atoms" consists a lot of calculations Before the slits and Behind the slits. But what is about calculations of the electrons interaction With the slit or With the edges of the slit?

It is a strange thing that there has to be or better there is an interaction between the moving through the slit particles and the slits edges but this interaction is not an object of consideration.

Just my two cents, it has to be calculated the interaction between the moving particles and the magnetic and electric fields of the surface electrons of the (sharp) edges. Then it could be explained easily that an intensity distribution happens even behind single (sharp) edges and even over time with one by one moving particles.

Numerical simulation of the double-slit experiment including watching the electrons ...

... in their interaction with the slit seems to be the right way to bring the endless discussions of double slit experiments to a good end.

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