What is the v(p) polynomial in radial wave function

[lonewolf219]

I think the solution to the radial Schrödinger equation includes a form of the Laguerre polynomials, the polynomial v(ρ). Does anyone know what this v(ρ) polynomial is called? The only information my book gives is: "The polynomial v(ρ) is a function well known to applied mathematicians."

Also, I noticed that when solving for the coefficients of v(ρ), we used a quantity called j_{max}=n-l-1. Then the value of j_{max} would coincide with the degree of the polynomial. But why does the degree of the polynomial depend on the values of quantum numbers n and l?

 

[Bill_K]

I think the solution to the radial Schrödinger equation includes a form of the Laguerre polynomials, the polynomial v(ρ). Does anyone know what this v(ρ) polynomial is called? The only information my book gives is: "The polynomial v(ρ) is a function well known to applied mathematicians."

They're known as associated Laguerre polynomials.

Also, I noticed that when solving for the coefficients of v(ρ), we used a quantity called j_{max}=n-l-1. Then the value of j_{max} would coincide with the degree of the polynomial. But why does the degree of the polynomial depend on the values of quantum numbers n and l?

It's the other way around, isn't it? The relationship states that the quantum number n depends on ℓ and j_max.

 

[lonewolf219]

Thanks Bill_K for the reply... I appreciate it!