Proving Rnl(r) Has (n-l-1) Zeros - Interesting Proof

[Ed Quanta]

How can it be proved that Rnl(r) has (n-l-1) zeros (not counting those at r-0 and r= infinity?

I tried doing this inductively but found it hard to get anywhere since the equations for Rnl(r) aren't too pretty. For instance, Rnl(r)=Anlunl

where Anl= sqrt((n-l-1)!/(2n((n+l)!)^3)

this last equation seems to bear a similarity with what I am supposed to prove but I am having trouble with this. Help anyone?

 

[dextercioby]

I'm sorry for not sharing your enthusiasm,but i don't find that proof to be interesting...
Yu can find the general formula for R_{nl}(r) here

I think you can build the proof really easily.

Good luck!

Daniel.

 

[kanato]

How can it be proved that Rnl(r) has (n-l-1) zeros (not counting those at r-0 and r= infinity?

I tried doing this inductively but found it hard to get anywhere since the equations for Rnl(r) aren't too pretty. For instance, Rnl(r)=Anlunl

where Anl= sqrt((n-l-1)!/(2n((n+l)!)^3)

this last equation seems to bear a similarity with what I am supposed to prove but I am having trouble with this. Help anyone?

I don't think you need to worry about A_{nl} in your proof, since that is just a constant for a given n and l, and not dependent on r. I think what you want to look at is the Laguerre Polynomials, (specifically their order), and remember what the fundamental theorem of algebra says about how many zeros a polynomial will have, and somehow come up with an argument that those zeros will be real.

 

[mikeu]

The link Daniel provided states that
R_{n\ell}(r) = r^\ell \exp\left(\frac{zr}{na}\right) \sum_{j=0}^{n-\ell-1} b_jr^j.
The sum is simply the definition of a polynomial of degree n-\ell-1, which the fundamental theorem of algebra guarantees will have exactly n-\ell-1 complex roots (not necessarily distinct). The coefficients in front of the sum provide the zeros at r=0 and z\to-\infty, and the others are provided by the polynomial.

If you need to show that there are n-\ell-1 distinct zeros then there will obviously be something more to the proof (I would guess it involves looking at the definition of the b_j), but this might be a good starting point at least...

 

[Ed Quanta]

I'm sorry for not sharing your enthusiasm,but i don't find that proof to be interesting...

Ha ha, I just didn't know what to title it. Thanks for help.