Legendre equation and angular momentum

[Angsaar]

Hi all,

I've been doing a math problem about the Legendre differential equation, and finding there are two linearly independent solutions. When I was taught about quantum mechanics the polynomial solutions were introduced to me as the basis for spherical harmonics and consequently the eigenfunctions of the hydrogen atom, etc etc.

But if there's a second type of solution (involving ln((1+x)/(1-x)) ), what significance to QM do they have? I've not been able to find any mention of them in the context of physics, only in math textbooks.

Thanks for any help!

 

[dextercioby]

So what is your equation...?Post it,so i can have a clear picture of you're trying to say.

Daniel.

 

[Angsaar]

(1-x^2) (d^2P/dx^2) - 2x (dP/dx) + l(l+1) P = 0

l being the total angular momentum quantum number, and P being the solution to the theta part of the eigenfunction from the Schrödinger equation after using separation of variables, with x=cos theta

I found the first solution type, say y1, using a series method, and the second, say y2, by doing y2(x)=y1(x)v(x) and finding out what v(x) had to be.

 

[dextercioby]

If you're going to claim that the second independent solution to Laplace equation is also elligible to describe physically possible quantum states,then u should check it out:do these new wavefunctions obey the orthonormality condition that a complete system of eigenfunctions of the H atom's Hamiltonian must fulfil...?My guess is NO,this thing being similar to the "radial wavefunctions issue" and those Whittaker functions...

Think about it...

Daniel.

 

[Meir Achuz]

The Legendre functions of the second kind, usually designated as Q_n(x), are irregular at the poles. Because of this, they can't be included in a normalizable wave function.
The same argument holds for non-integral l for the P_l(x).
That's also why they don't usually come up in EM theory.
They are used sometimes in scattering theory (along with non-integral l)
when the scattering amplitude is continued into the complex plane.