4. Solve Numerically :
First normalize
Then evaluate

5. Average Values of Powers of the Coordinate

6. Kepler Coulomb : V(r ) = - k/r
Isotropic Harmonic Oscillator : V(r ) = Kr2/2

7. Numerical Solution: Hydrogen atom (3p)
E = - Ry / 9 ; L = 3ħ/2 ; V(r ) = - Ry a0 /r
Radial momentum ( Units : [energy] = Ry ; [length] = a0
Limits: A- = ; A+ =
Normalization : C = 84.82
Moments: r =
r -1 =
r -2 =
r -3 =

8. Numerical Solution: Isotropic SHO (nr=0 ; l=3)
E = 9ħ/2 ; L = 7ħ/2 ; V(r ) = m2 r2 / 2
Radial momentum ( Units : [energy] = ħ ; [length] = (ħ/m)1/2
Limits: A- = ; A+ =
Normalization : C = 2.215
Moments: r 2 =
r 4 =
r -2 =
r -4 =

9. Einstein-Brillouin-Keller Action Quantization
(1917) (1926) (1958)
Bohr-Sommerfeld-Wilson quantization used fuzzy math, neglecting caustics at turning points in librations. The correct semiclassical action quantization condition is:
where
i = (rotations)
Topological
Maslov Index
= (librations)
It yields astonishingly accurate results !!!