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Mathematically, what happens at the screen of the 2-slit experiment? How would one represent the detecting screen at the receiving end? What boundary condition should be used there to represent the detecting screen?

When one solves the Schrodinger Equation, one needs to specify the boundary conditions. For the 2-slit experiment, there should be a potential with two small gaps representing the screen with the two slits, and there should be a potential representing the detector screen at the receiving end. In what way should one write these potentials, especially the one for the detector screen. That is, what would the boundary condition at the screen be?

There is a related question and answer at What is the wavefunction of the Young Double Slit experiment? but it does not address mathematically what happens at the detecting screen.

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  • $\begingroup$ This seems like a duplicate of the question you linked. $\endgroup$
    – auden
    Commented Aug 4, 2016 at 14:16
  • $\begingroup$ Sorry, but I would have to disagree with you about the math treatment of the other question, the answer given and the link in the comments at the end, seem pretty well worked to me. $\endgroup$
    – user108787
    Commented Aug 4, 2016 at 15:00
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    $\begingroup$ Possible duplicate of What is the wavefunction of the Young Double Slit experiment? $\endgroup$
    – auden
    Commented Aug 5, 2016 at 0:25
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    $\begingroup$ Why do you even need a screen to describe the light that emanates from the twin slits? Why not just describe the intensity of the light rays as a function of angle? The screen is just a means of visualizing that intensity map. $\endgroup$ Commented Aug 5, 2016 at 13:24
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    $\begingroup$ @PhysicsFootnotes While I agree that this post is not a duplicate, note that the existence or absence of answers for another question is not the criterion for marking as a duplicate. If another duplicate question exists, but you think there are no answers which address an important part of that question, simply post your new answer there. $\endgroup$
    – DanielSank
    Commented Aug 6, 2016 at 6:10

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In modeling a quantum experiment, Schrodinger's equation is not the end of the story. We use Schrodinger's equation (with an appropriate choice of potential and boundary conditions) to represent the evolving state of the system, however the measurement process (in this case a detection screen) is completely absent from this part of the model.

The detection screen is represented instead by an appropriate choice of self-adjoint operator, which in this case would be a multiplication operator corresponding to the axis along which the screen is aligned.

In other words, we model the double-slit screen by an appropriate choice of potential and boundary conditions on the wavefunctions, and we model the detection screen by a separate self-adjoint operator. The two components of the model unite at the point where we calculate probabilities using the Born rule.

As an interesting aside, if you're wondering how to model a detection screen itself as a quantum system (rather than as an 'external' observable), join the club! This is an unsolved problem in quantum mechanics which goes under the name of the Measurement Problem. Its successful resolution would be worthy of a Nobel Prize, so you are unlikely to get that question answered here I'm afraid ;-)

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  • $\begingroup$ What do you mean by model a detection screen as a quantum system? $\endgroup$ Commented Aug 6, 2016 at 4:05
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    $\begingroup$ I think this answer makes it all sound a bit more mysterious than it is. We understand that if a quantum system couples to a large number of degrees of freedom (i.e. a classical system) via operator $X$, then the quantum system's density matrix diagonalizes in the $X$ basis. $\endgroup$
    – DanielSank
    Commented Aug 6, 2016 at 6:12
  • $\begingroup$ wiley.com/college/halliday/0470469080/simulations/sim48/… $\endgroup$
    – Damon
    Commented Aug 18, 2016 at 17:04
  • $\begingroup$ @ Physics Footnotes. Thank you for your useful answer. I have a related question. In the one-slit experiment, there is also a diffraction pattern. There is an illustration here: wiley.com/college/halliday/0470469080/simulations/sim48/…. How is this diffraction pattern generated in the one-slit experiment? Again, what is the boundary condition for this in quantum mechanics? $\endgroup$
    – Damon
    Commented Aug 18, 2016 at 17:13
  • $\begingroup$ @Damon My explanation holds regardless of the pattern of slits, which will only influence the choice of wave function used to model the initial state of the system. The detection screen will still be modeled by the same old operator I described in my answer. $\endgroup$ Commented Aug 19, 2016 at 19:49
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The double slit result is a simplified version of the much more complicated set ups of particle physics experiments.In these experiments, the primary interaction region is modeled using quantum field theory, of point particles interacting in a summed series of feynman diagrams.

The outgoing particles leaving the interaction region towards the detectors are treated essentially as wavepackets, within an uncertainty region consistent with the Heisenberg uncertainty relations,HUP, i.e. modeled as a classical particle. The hits in the detectors are also treated macroscopically, again because the interaction region of the outgoing particle with the detector is way larger than the quantum mechanical constraints of the HUP.

The equivalent interaction region for the two slit experiments, where a quantum mechanical solution is necessary is the "electron scattering off two slits", with the specific geometrical bounds. The screen is the detector.

The interactions of the electrons on the screen , the measured points, are way over the bounds of the HUP and the electron wavepacket is well approximated by the point in (x,y) space.

If you are worrying that one should in principle write out one mathematical expression for the whole experiment, i.e.that there is an entanglement of the initial electron wavepacket impinging and going through the slit plane with the same wavepacket hitting the screen, the density matrix formalism helps clearing up this: when dimensions become macroscopic with respect to the HUP the off diagonal elements of the entanglement of the screen particles with the incoming electron wavepacket are zero, the phases are lost, exactly because the involved dimensions are macroscopic, as gauged by using the HUP.

So viewed as one experimental setup, the quantum mechanical interaction is at the level of the two slits where the distances have been chosen so as to be commensurate with the de broglie wavelength of the electron. The screen is just a detector.

Of course there is a quantum mechanical interaction where the electron hits the screen: atoms become ionized and the energy is distributed in a many body way. The density matrix for this part has nondiagonal elements, i.e. quantum mechanical phases, only with the wavefunctions of the local scattering. The two slit wavefunction has no off diagonal element in the density matrix with the wavefunctions of the local screen scatters.

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    $\begingroup$ This does not address the question, which is about the mathematical modeling involved in the double slit experiment. $\endgroup$
    – Danu
    Commented Aug 9, 2016 at 19:22
  • $\begingroup$ @Danu, it attempts to answer the question. Under the current low quality answer policy, it is an "acceptable" answer, though I certainly agree that it is not the best answer. $\endgroup$
    – auden
    Commented Aug 15, 2016 at 14:52
  • $\begingroup$ @anna v. Thank you for your useful answer. I have a related question. In the one-slit experiment, there is also a diffraction pattern. There is an illustration here: wiley.com/college/halliday/0470469080/simulations/sim48/…. How is this diffraction pattern generated in the one-slit experiment? Again, what is the boundary condition for this in quantum mechanics? $\endgroup$
    – Damon
    Commented Aug 18, 2016 at 17:16
  • $\begingroup$ For the classical wave, it is the scatter at the edge which introduces a phase with respect to the direct beam. For the single photon case it is as you say a boundary condition "single photon scattering on one slit " the boundary of the two edges of the slit introduces the corresponding phase difference in the wavefunction which gives the interference pattern in the probability distribution . $\endgroup$
    – anna v
    Commented Aug 18, 2016 at 18:59
  • $\begingroup$ @anna v. Thank you again for your answer to my questions raised in the my last comment. Could you please recommend some good text books or papers which cover the scattering at the edge? Your help will be much appreciated. $\endgroup$
    – Damon
    Commented Aug 19, 2016 at 11:39
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Mathematically, what happens at the screen of the 2-slit experiment?

Well, an electron has a wave nature, that's why we can diffract electrons. It isn't a point particle. But when we detect it on the screen, we see a dot. So, mathematically what happens at the screen is comething akin to a Fourier transform. See Steven Lehar's web page on that. Pay special attention to the optical Fourier Transform where the incident wave is converted into something pointlike:

enter image description here

How would one represent the detecting screen at the receiving end? What boundary condition should be used there to represent the detecting screen?

Sorry, I don't know.

When one solves the Schrodinger Equation, one needs to specify the boundary conditions. For the 2-slit experiment, there should be a potential with two small gaps representing the screen with the two slits...

Take a look at Ehrenberg and Siday's 1949 paper The Refractive Index in Electron Optics and the Principles of Dynamics*. An electron is depicted as plane waves going past a solenoid in figure 2. (Edit : not unlike the Gaussian beam Neurofuzzy referred to on the other question.) So I imagine one could emulate this and represent one electron as a plane waves going through two slits, much like water waves.

enter image description here

and there should be a potential representing the detector screen at the receiving end. In what way should one write these potentials, especially the one for the detector screen. That is, what would the boundary condition at the screen be?

Sorry, I don't know. But note that at the screen we have an interaction between two electromagnetic extended entities, and the combination looks pointlike.

There is a related question and answer at What is the wavefunction of the Young Double Slit experiment? but it does not address mathematically what happens at the detecting screen.

I took a look. Note that you send one electron through the apparatus, and you get one dot on the screen. See the Wikipedia article. So you might think the electron is pointlike. However you know the light going through a lens isn't pointlike. And when you see the pattern build up, you know the electron isn't pointlike either. After all, it's the Schrödinger wave equation, not the Schrödinger point-particle equation.

enter image description here

  • I've got a non-paywall link to the full paper somewhere. I'll dig it out tonight.
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    $\begingroup$ This post does not try to address the question, which is asking about the mathematical modeling of a double slit experiment. In particular, you explicitly say "Sorry, I don't know" to the main questions. That is a good indication that you should not be posting this as an "answer". $\endgroup$
    – Danu
    Commented Aug 9, 2016 at 19:21
  • $\begingroup$ I gave the answer. The answer to Mathematically, what happens at the screen of the 2-slit experiment? is a Fourier transform. $\endgroup$ Commented Aug 9, 2016 at 21:26

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