Solve Laguerre Equation for -a/z Potential | Quantum Mechanics

[alevu]

Homework Statement

I am doing the quantum mechanics and meet the Schrödinger question :
When the potential is given as U = -a/z ,the Schroedinger equation looks like

[- hbar^2 /(2m)] d^2 / dz^2 Psi(z) - a/z Psi(z) = E Psi(z).
And the thing here is that I couldn't solve this equation . Help me please

Homework Equations

The Attempt at a Solution

Phi^k_n (x) = e^{-x} x^{(k+1)/2} L^k_n (x) satisfies

(d^2 / dx^2 )Phi^k_n (x) + [ -1/4 + (2n +k+1)/(2x) - (k^2 -1)/(4x^2 ) ] Phi^k_n (x) = 0.

Therefore, if we let k = 1, then we get an equation similar to the above Schroedinger equation.

 

[gomunkul51]

http://tutorial.math.lamar.edu/Classes/DE/SeriesSolutions.aspx

I think this is what you need, it was hard to read your equation.