Go over chapter 6 homework.
Start chapter 7.
The text does not do a good job of explaining where the solution to the 3D Schrodinger comes from. I've added a section to the references with links to more complete explanations - please check those out.
$$ E = \frac{1}{2} m ({v_x}^2 + {v_y}^2) - C/r \\ \, = \frac{1}{2} m ({v_r}^2 + {v_\theta}^2) - C/r \\ $$
where \( v_\theta \) is the velocity around the origin and C/r is some central potential.
Since the angular momentum is \( L = r (m v_\theta) \) and is unchanging, we can also write this as
$$ E = \frac{1}{2} m ({v_r}^2 + \frac{L^2}{m^2 r^2}) - C/r \\ \, = \frac{1}{2} m {v_r}^2 + \frac{L^2}{2 m r^2} - C/r $$
Those last two terms can be thought of as an "effective potential" for radial motion in a rotating coordinate system.
Compare this with the radial part of the 3d schrodinger equation.
