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Aug
16
comment Is there a physical interpretation of Neumann boundary conditions for the free Schrodinger equation on a domain?
@udrv Also, yes, those more general conditions are called Robin boundary conditions.
Aug
16
comment Is there a physical interpretation of Neumann boundary conditions for the free Schrodinger equation on a domain?
So $n\cdot j = 0$ is equivalent to the condition that the boundary term vanishes when you integrate $\langle \Psi|\Psi^*\rangle$ by parts and that's exactly what you need to have a self-adjoint extension of the Hamiltonian.
Aug
15
comment Is there a physical interpretation of Neumann boundary conditions for the free Schrodinger equation on a domain?
@arivero What do you mean by "teleportation"? Identification of different sides of the domain, a la gluing a square into a torus?
Aug
15
awarded  Commentator
Aug
15
comment Is there a physical interpretation of Neumann boundary conditions for the free Schrodinger equation on a domain?
@udrv So is this the idea? For a domain $D$, there are essentially two ways of fencing in a particle: either impose an infinite potential outside $D$, or impose the condition that the probability current never cross the boundary. Either way, the particle stays confined to the domain.
Aug
13
comment Is there a physical interpretation of Neumann boundary conditions for the free Schrodinger equation on a domain?
@arivero I'm not sure if that intuition extends to higher dimensions. For instance, in a multiply connected domain in the plane, like a thickened figure-8, collapsing the boundary to a point produces a space that's not even a manifold. In 1d, any function on an interval can be extended to a periodic function, which then descends to a function on the appropriate circle, so the analogue in higher dimensions would be functions on the appropriate torus. That's my mathematical intuition, at least; I can't really speak to the physical intuition.
Aug
13
revised Is there a physical interpretation of Neumann boundary conditions for the free Schrodinger equation on a domain?
added 263 characters in body
Aug
13
comment Is there a physical interpretation of Neumann boundary conditions for the free Schrodinger equation on a domain?
Sorry about the sign issues. I added a note to the OP about this. I'm digesting your answer and will be back shortly with more questions, I'm sure.
Aug
13
comment Is there a physical interpretation of Neumann boundary conditions for the free Schrodinger equation on a domain?
@jac What about domains where the boundary is not totally geodesic? For instance, a disk in $\mathbb{R}^2$ where half the boundary is given Dirichlet and the other half is given Neumann?
Aug
13
revised Is there a physical interpretation of Neumann boundary conditions for the free Schrodinger equation on a domain?
added 263 characters in body
Aug
13
comment Is there a physical interpretation of Neumann boundary conditions for the free Schrodinger equation on a domain?
@arivero There's not room in the post to add a third column, so I added a postscript instead :)
Aug
11
awarded  Nice Question
May
11
revised Why do all the atoms of a radioactive substance not decay at the same time?
edited body
May
11
revised Why do all the atoms of a radioactive substance not decay at the same time?
deleted 251 characters in body
May
11
comment Why do all the atoms of a radioactive substance not decay at the same time?
I don't understand why you think me saying the derivative of an exponentially decrease is again an exponential decrease is controversial, but I have deleted the contentious section until I have time to rewrite it.
May
11
awarded  Nice Answer
May
11
comment Why do all the atoms of a radioactive substance not decay at the same time?
@user121330 Ah, thank you. My probability-fu was weak: the Poisson distribution measures the probability of decay in a given length of time.
May
11
revised Why do all the atoms of a radioactive substance not decay at the same time?
fix
May
11
awarded  Yearling
May
11
revised Why do all the atoms of a radioactive substance not decay at the same time?
(typo)