Radial Eigenfunction; Differential Equation

[Paradox101]

Homework Statement

Show that the radial eigenfunction u_nr,l is a solution of the differential equation:
ħ²/2m_e×d²u_nr,l/dr²+[l(l+1)ħ²/2m_er² - e²/4πε₀r]u_nr,l=E_nr,lu_nr,l

Homework Equations

The radial function is R(r)=u(r)/r, so that the expression on the RHS is E×u.

The Attempt at a Solution

I know that this equation is the radial Schrödinger equation, and that the particle with angular momentum L behaves like a particle in one-dimensional effective potential.But, to prove it, i don't know where to start.
I tried comparing this potential with the analogous effective potential and set the first term, l(l+1)ħ²/2mr², but with no luck. Any ideas, please?

 

[BvU]

I had to look it up in Griffiths (https://archive.org/download/IntroductionToQuantumMechanics_718/6804026-Physics-Introduction-to-Quantum-Mechanics-Ph-1995.pdf) and the problem statement seems OK. I do expect, either in the problem statement, or in the relevant equations, an expression for the radial eigenfunction $u(n, r, l)$. Because the exercise wants you to show that it's a solution.

Your relevant equation R = u/r isn't of interrest, I'm afraid. They want you to differentiate u twice.

Embark on that task and post the work (in reasonable detail) if you still get stuck.

You may want to adopt the notation (4.55) in the link to avoid excessive amounts of $\hbar\over 2m$ etc. -- but then you need your u also in terms of $\rho$.

 

[Paradox101]

well, i had to look at the solⁿ on the back of the book. Never mind, i got it anyway.

 

[BvU]

Was it something like "differentiate u twice and verify that it satisfies the equation" or was it different ?

 

[Paradox101]

nope, it was different. The book u r referring to, is the one i have already read. This is a different book, and a quite old one too---- It doesn't have a proper cover.