# Two paths having the same phase in the path integral approach

In the path integral approach to Quantum Mechanics, can two distinctly different paths of the possible infinite paths have the same phase, i.e can there be a bimodal distribution of the phases associated with each path? And Why?

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Isn't this the point of the double-slit experiment? The amplitudes for paths passing through the two slits will constructively interfere if they have the same phase. –  Scott Carnahan Jul 20 '11 at 10:56

## 2 Answers

Phases are a number, paths are an infinite dimensional space. Solving the equation "phase of path = c" gives an infinite dimensional space of paths with the exact same phase. I suppose you meant the classical question "are their two nearby classical trajectories with the same action", which is completely different.

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sure they can;

you can simply put a half-reflecting mirror in front of a source of light (like a laser), so 50% of the wave amplitude will go straight while the rest is being reflected. Both paths are bimodal, and they can still interfere with each other, if the path difference at the moment of mixing both again is the right one.

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I guess you got the wrong amplitude. I was talking about the amplitude, which when squared, gives the probability for the particle to go from 'a' to 'b'. So, rephrasing the question, are there two paths (not close to each other; distinct) to go from 'a' to 'b' such that both their phases are the same? –  Sameer Rao Jul 20 '11 at 6:20
nothing i said implied a field intensity (which is proportional to the probability density - not the amplitude). When you split two rays with a mirror, you are splitting two paths with probability amplitude, not the squared one, otherwise you won't be able to get interference when mixing them up again (that is why i brought up the mixing thing, otherwise you don't get the interference, which is what validates that the paths are complex amplitudes in the first place) –  lurscher Jul 20 '11 at 14:40