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 $x=a\cos(\omega t)$. If you look at it at a random time, the probability that you will find it between $x$ and $x+dx$ is $P(x)dx$, with

$$P(x)={1\over\pi}\,{1\over\sqrt{a^2-x^2}$$

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

 

[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.