A On tempered distributions and wavefunctions

[dumpling]

Very often in standard QM books, certain states, like exponentially growing ones are rejected on the basis that they are not in L^2 space.
On the other hand, scattered states are also not in L^2 spaces. This dichotomy can be repelled by using Rigged Hilbert spaces, and allowing tempered distributions.
On the other hand, in the case of central potentials, one cannot just throw away singular solutions, as they too are tempered solutions, as far as I know.

How then, do we have to decide which states are physical, even as a basis?

 

[A. Neumaier]

States at fixed time must be square integrable. Wave functions needed to analyze these need not.

 

[dumpling]

I know that, that is the whole idea behind allowing basis to be in the rigged Hilbert-space, is it not?

 

[A. Neumaier]

I know that, that is the whole idea behind allowing basis to be in the rigged Hilbert-space, is it not?

I don't understand what you are asking. The Dirac kets $|x\rangle$ or $|p\rangle$ don't form a basis but belong to the rigged Hilbert space and are used to describe states. They are physical in the sense that they are used by physicists for the analysis of quantum situations but not in the sense that they can be states of some physical system.

 

[dumpling]

What is the exact reasoning, that for example in the case of the radial wavefunction of hydrogen atom, we do not use singular solutions as basis, when some of those would be elements of the rigged hilbert-space?

 

[A. Neumaier]

In general one can use any basis but experience shows that well chosen ones lead to more tractable formulas. So one chooses according to what one knows from similar cases.

Typically, continuous ':bases' ' are most useful for scattering problems while discrete bases are more useful for bound state problems.