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.


1 Answer 1


You have to realize that wave equations and interference phenomena had been studied and understood by the nineteenth century.

Plane waves plane wave 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.

two plane waves

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

electrons 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.

  • $\begingroup$ Can an aperture be considered a physical fourier transform device? $\endgroup$
    – Fraïssé
    Nov 27, 2014 at 6:08
  • $\begingroup$ 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. $\endgroup$
    – Peter
    Nov 27, 2014 at 6:14
  • $\begingroup$ 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. $\endgroup$
    – anna v
    Nov 27, 2014 at 7:57
  • $\begingroup$ @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.en.wikipedia.org/wiki/Fourier_transform_spectroscopy $\endgroup$
    – anna v
    Nov 27, 2014 at 7:59
  • $\begingroup$ See the answers to this question physics.stackexchange.com/questions/437/… . For how the photons build up the classical wave see here motls.blogspot.com/2011/11/… $\endgroup$
    – anna v
    Nov 27, 2014 at 8:55

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