April 2

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.

energy in rotating coords

$$E = \frac{1}{2} m ({v_x}^2 + {v_y}^2) - C/r \\ \, = \frac{1}{2} m ({v_r}^2 + {v_\theta}^2) - C/r \\ $$

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.