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When choosing a self-adjoint extension of a Hamiltonian, in general one can obtain domains in which (i) the probabilities teleport* between points on the boundary and (ii) boundary conditions locally conserve probabilities.

The ones which locally conserve probability currents somehow seems nicer to me. But this is not at all an argument especially since tunneling is allowed in quantum mechanics.

Is there any fundamental physical reasoning one can use to discard teleporting boundary conditions?

Thanks in advance for any useful inputs.

*I have used terminology from discussion about a related question : Physical interpretation of different selfadjoint extensions

Community
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symanzik138
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1 Answers1

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Once you admit the tunnelling the "topology" of the problem changes to "particle in a circle with an infinite barrier". In fact, to avoid arguments with energy when crossing the barrier, it can be described as "particle in a circle with an infinite barrier in one point". So physically it can be argued to be a different problem.

arivero
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  • Note: I am not fully content with the description as an "one point barrier" because for some range of the parameter space it indeed looks as two half-lines separated across a gap, and the momentum operator looks as the distance operators in Connes non-commutative geometries. – arivero Aug 09 '15 at 11:05