How Do You Find <r^2> for Uncertainty in Position?

[swain1]

When trying to work out the uncerainty in position of the expectation value I have read that you have to find <r^2> as well as <r>^2. I have worked out the value of 3a/2 for <r> but what do I have to do to find <r^2>. Do I just sqare the whole function before I integrate?
Also as I am integrating I found, in a book a table that had a general form of integrations between 0 and infinity. I used it without giving it much thought but how would you evaluate this? Thanks

 

[Tomsk]

Yes to find <r^2> you integrate r^2 p(r) dr instead of rp(r)dr. Or for any function of r, it's just int[f(r)p(r)dr].

I'm guessing the table was for the gaussian interal int[exp(-ax^2)] a>0? The integral can't be evaluated directly, you have to do some weird substitutions, if you want to know how to do it I can show you. It's easiest to remember the the soln is sqrt(pi/a), from 0 to infinity is half this, and for x^2n exp(-ax^2), differentiate both sides wrt a.

 

[swain1]

Ok thanks for that. I have done the inegral and worked out the uncertainty. The value I have got is sqrt(3)a/2. It seems like an awful lot to me. Anyway, do you know if this is right. I am doing this for the ground state of hydrogen. If I have got the correct value. Why is it so large?

 

[Tomsk]

Not entirely sure, what exactly was the integral? And what is a? I haven't done any quantum, not sure what this is about exactly.

 

[stunner5000pt]

$$\Delta r = \sqrt{<r^2> - <r>^2}$$