Effective Potential and Angular Momentum

[dimensionless]

I have the following equation for potential energy. Actually it's for the effective potential energy.
<br /> &#039;V&#039;(r) = - \int F(r) dr - \int \frac{L^{2}}{m r^{3}}dr
<br /> &#039;V&#039;(r) = V(r) + \frac{L^{2}}{2 m r^{2}}<br />
Where does the second term on the right come from? What does it have to do with potential energy?

 

[Meir Achuz]

The L^2/2mr^2 comes ot of the reduction of the 3D Schrodinger equation to an effective !D equation. The state is assumed to be an eigenvalue of the angular momentum operator with eigenvalue L^2\Psi=l(l+1)\Psi, so the L^2 in your equation has the numerical value l(l+1).
The L^2/r^2 is the angular part of the Del^2 operator.
What is left is the radial equation.