Normalization and singularities

[diegzumillo]

Hi There!

Being direct to the point: Does normalization removes singularities? Such as infinite.

I came up with this question because, while I was working with a not normalized function, I reached a very strange result. There are two points where the probability tends to infininte.

Maybe that's because the function is not normalized. (intuitively, I thought the normalization would only re-scale the 'curve', when plotted) Or did I mess up the calculation?

If anyone wants to see more than that first question, I can show the equations I'm working. So we can have a more specific discussion. :)

 

[Avodyne]

Hi There! Being direct to the point: Does normalization removes singularities?

No.

I came up with this question because, while I was working with a not normalized function, I reached a very strange result. There are two points where the probability tends to infininte.

That's possible if the probability distriubtion is integrable at these points. For example, consider a classical harmonic oscillator with [itex]x=a\cos(\omega t)[/itex]. If you look at it at a random time, the probability that you will find it between [itex]x[/itex] and [itex]x+dx[/itex] is [itex]P(x)dx[/itex], with

[tex]P(x)={1\over\pi}\,{1\over\sqrt{a^2-x^2}[/tex]

for [itex]-a\le x\le a[/itex], and [itex]P(x)=0[/itex] otherwise. [itex]P(x)[/itex] becomes infinite at [itex]x=\pm a[/itex], but still [itex]\int_{-a}^{+a}P(x)dx = 1[/itex].

 

[diegzumillo]

Thanks! :D

I didn't remember that example, it really clarifies things out.

Ok, singularities are not evil! They just have a 'not so trivial' interpretation.