Physically valid wave funtions

[kini.Amith]

Is the function sinx/(x^2) a physically valid wave function? since Born's conditions state that the function should be continuous and i think this function is discontinuous at x=0.

 

[vanhees71]

It's not a representant of a pure quantum state, because its square is not integrable, i.e., the integral
$$\int_{\mathbb{R}} \mathrm{d} x |\psi(x)|^2$$
doesn't exist due to the singularity at $x=0$.

That it is not continuous or even analytic is not that important, because you deal with functions in the sense of the Hilbert space of square integrable functions, and it doesn't matter, if they are singular on a set of Lebesgue measure 0.

 

[akhmeteli]

Is the function sinx/(x^2) a physically valid wave function? since Born's conditions state that the function should be continuous and i think this function is discontinuous at x=0.

I don't think this is "a physically valid wave function", not because it's discontinuous, but because it's not square-integrable.

 

[kini.Amith]

SO is continuity of the wave function not a necessary condition for it to represent a physical system?

 

[bhobba]

There are a couple of ways of looking at wave functions.

You have the Lebesgue square integrable view - valid wave functions are those that are square integrable in the Lebesque sense - they also form a Hilbert space. This is Von Neumann's approach you will find in his classic Mathematical Foundations of QM text. The wave function you gave is not valid in that approach.

Then there is the Rigged Hilbert space approach. The physically realisable states are considered to be a subset of square integrable functions so as to have nice mathematical properties such as being continuously differentiable, or even simply finite dimensional. Of course your function is not a physically realisable state in that approach either. However for mathematical convenience that space is extended to be the linear functionals defined on the physically realizable states. Note there is slightly more to it in that it must be such that you can define a norm on the resultant space - but these are fine points you can consult tomes on it eg:
http://physics.lamar.edu/rafa/webdis.pdf

In that sense it is a valid wavefunction.

Thanks
Bill