(Disclaimer: I’m only a layman and I could be wrong.)
edgepflow said:
For the single particle case, I was reading that the "Many Worlds" interpretation of quantum mechanics suggests that an electron from "our universe" goes through one slit and an electron from a "parallel universe" goes through the other.
In the
Copenhagen interpretation we have a process called http://en.wikipedia.org/wiki/Wave_function_collapse" , i.e. in the case of the Double-slit experiment the wave function of the photon (
or electron) collapse, after interaction with an observer (
measurement), to one determinate outcome, to 'explain' the observation.
In
MWI there is no wave function collapse. The observer and the observed (particle) have become http://en.wikipedia.org/wiki/Quantum_entanglement" ; i.e. the state of the observer and the observed are correlated
after the observation is made.
In MWI this means that if the observer tries to measure which slit the photon passes through, then the observer is split into
observer1 that measures
left slit, and
observer2 that measures
right slit. These two observers are now in separate worlds, and can never interact again (
after the observation).
I don’t have the full understanding of MWI to explain what happens when the interfered wave hits the detector screen... Logically the observer ought to be split into every possible position that the photon can have on the screen. But that would mean no single MWI observer can ever see the full interference pattern!? (
In case where photons arrive one by one)
If a pro could explain this it would be great...
edgepflow said:
I wanted to try to figure out the probability density of the wave function that it goes through both slits at the same time. I think this might be called degeneracy but I am not sure. I studied basic quantum mechanics in my undergraduate engineering studies but this was awhile back. I have been reading "quantum Mechanics DeMystified" to try to get back into it (a good book for its purpose in my opinion). Maybe some others more versed in this area could give me some hints.
Well, as I understand this, the photon
must go through both slits at the same time – otherwise there would not be any interference pattern...??
http://en.wikipedia.org/wiki/Photon_dynamics_in_the_double-slit_experiment#Probability_for_a_single_photon"
"Some time before the discovery of quantum mechanics people realized that the connexion between light waves and photons must be of a statistical character. What they did not clearly realize, however, was that the wave function gives information about the probability of one photon being in a particular place and not the probable number of photons in that place. The importance of the distinction can be made clear in the following way. Suppose we have a beam of light consisting of a large number of photons split up into two components of equal intensity. On the assumption that the beam is connected with the probable number of photons in it, we should have half the total number going into each component. If the two components are now made to interfere, we should require a photon in one component to be able to interfere with one in the other. Sometimes these two photons would have to annihilate one another and other times they would have to produce four photons. This would contradict the conservation of energy. The new theory, which connects the wave function with probabilities for one photon gets over the difficulty by making each photon go partly into each of the two components. Each photon then interferes only with itself. Interference between two different photons never occurs."
http://en.wikipedia.org/wiki/Photon_dynamics_in_the_double-slit_experiment#Probability_amplitudes"
The probability for a photon to be in a particular polarization state depends on the fields as calculated by the classical Maxwell's equations. The photon probability density of a single-photon Fock state is related to the expectation value of the energy density of the equivalent E and B fields.
Simplified version of Maxwell's equations, written in terms of either the electric field
E or the magnetic field
B:
[tex]\nabla^2 \mathbf{E} \ - \ { 1 \over c^2 } {\partial^2 \mathbf{E} \over \partial t^2} \ \ = \ \ 0[/tex]
[tex]\nabla^2 \mathbf{B} \ - \ { 1 \over c^2 } {\partial^2 \mathbf{B} \over \partial t^2} \ \ = \ \ 0[/tex]
where c is the speed of light in the medium