Every other popular science book and intro level text on QM starts with the double slit experiment. It is always just stated as a fact that experiments have been done, actual data is never presented in the many books I have seen. I am yet to see a derivation of the Schrodinger equation and a solution for this problem. Can anyone recommend on line resources and papers that show the detail of the problem, the solution and experimental data. I don't have access to the professional literature but can order papers for a fee.
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Peter, you might want to look up the article in AJP (No 57, pp. 117–120) Demonstration of single‐electron buildup of an interference pattern describing the famous experiment by Akira Tonomura (1989), who showed that single electron is a particle and not a wave, and that the interference pattern is not created by a single particle, but rather the consequence of accumulation of thousands of them. annav's answer contains the pictures from this experiment, and here is the video: https://www.youtube.com/watch?v=ZJ-0PBRuthc. – bright magus Nov 27 '14 at 12:16
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You need to pay to see the paper, but you will find a little bit more here on SE in this answer to the question "Is the wave-particle duality a real duality?" and the in links provided inside. – bright magus Nov 27 '14 at 12:19
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You have to realize that wave equations and interference phenomena had been studied and understood by the nineteenth century.
Plane waves 
are the simplest mathematical solution of wave equations,
where k, is the wave’s wave number or more specifically the angular wave number and equals 2π/λ, where λ is the wavelength of the wave. k, has the units of radians per unit distance and is a measure of how rapidly the disturbance changes over a given distance at a particular point in time.
A list of explanations of the other terms exists in the link.
A plane wave hitting two slits will produce interference patterns in the context ( equations and boundary conditions) that they are a solution of. It is not necessary to derive the solutions over and over again since for most interference patterns the plane wave assumption is a good approximation.
The Schrodinger equation is a wave equation , and plane waves are solutions of the equation. Again in the double slit experiments the plane wave solutions are used to model the impinging particles on the two slits, quantum mechanically . The difference between classical plane waves and quantum mechanical plane waves lies in the postulates of the theoretical model used for studying the patterns.
Classical equations predict variations of energy density in space at a given time, it is the amplitude that changes by construction of the theoretical models. In quantum mechanics the wave pattern predicts a probability distribution in space at time t. Thus light waves and photons impinging on the two slits will produce the same interference pattern, because the probability distributions for photons are from a solution of a quantized Maxwell's equation , but as has been shown with single photons at a time, when viewed quantum mechanically the two slits give a probability distribution in space for photons , which eventually builds up the classical ( energy deposition ) interference by each of the zillion photons leaving the appropriate energy at the appropriate (x,y) of the screen (the red histograms in the video). Physics is continuous between quantum mechanics and classical.
The mathematics for the probability distributions for electrons impinging on a double slit
are again plane wave solutions of the Schrodinger equation, the only complicated concept being that they represent probabilities of finding the particle in space at time t, not energies or mass and this is evident in the slow build up of the interference pattern.
anna v
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thanks anna v, I understand what the solution looks like, my question is as I understand QM you define a Hamiltonian to describe the problem and maths away. I can not conceive what a Hamiltonian for a photon incident on a double slit would like in order to solve the equation to yield all the results above. – Peter Nov 27 '14 at 06:14
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It is maxwell's equations in the potential form turned into operators operating on the psi of the photon. This is the general one. One has to solve it for the specific experimental situation. The bounds is a plane wave impinging on two slits. The mathematics is exactly the same as shown in the links above , except for the interpretation as a probability amplitude. cosines and sines do not know whether they are used to model a quantum phenomenon or a classical one. Only the user knows. – anna v Nov 27 '14 at 07:57
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@IllegalImmigrant it is a boundary condition. The two slit experiments create conditions of a coherent plane wave, laser light for example, because the solutions are simple. For more complicated experiments yes, fourier transforms simplify the solutions.http://en.wikipedia.org/wiki/Fourier_transform_spectroscopy – anna v Nov 27 '14 at 07:59
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See the answers to this question http://physics.stackexchange.com/questions/437/what-equation-describes-the-wavefunction-of-a-single-photon . For how the photons build up the classical wave see here http://motls.blogspot.com/2011/11/how-classical-fields-particles-emerge.html – anna v Nov 27 '14 at 08:55
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@annav: "Again in the double slit experiments the solutions are plane waves impinging on the two slits, quantum mechanical." If you watch the video from the experiment by A. Tanamura, the photos of which you used in your answer, you will see clearly that it has proved that electron is not a "plane wave" but only a particle, as single electron never creates the interference pattern. See also this answer to "Is the wave-particle duality a real duality?", where you will find more on the subject. – bright magus Nov 27 '14 at 13:08
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@brightmagus but that is exactly what I am saying. The plane waves are PROBABILITY waves, each electron follows a plane wave probability solution, one entry. – anna v Nov 27 '14 at 13:14
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@brightmagus the solutions are plane waves, the interpretations is that they represent the probability distribution of the impinging particles . Maybe it is my syntax but I think that is what I ams saying. The solutions, not the electrons. – anna v Nov 27 '14 at 13:57
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@annav: There are no "plane waves impinging on the two slits, quantum mechanically". The solutions do not travel through the two slits. They are probabilities, and not physical objects to be found anywhere. If the problem is in the wording, then I suggest you fine-tune it to avoid confusion. If 25 years after Tonamura's experiment there is still a widespread conviction that particle is also a wave, the blame is in continued up-keeping of the appearance that such abstract concepts are physical reality. – bright magus Nov 27 '14 at 14:06