1. L10 – QM in 3 dimensions

2. V(r): separation of variables

4. The angular equation

5. The angular equation - f

6. The angular equation - q
Pl are the Legendre polynomials, defined by the Rodrigues formula:

9. Spherical harmonics
e = (-1)m for m>=0 and e=1 for m<0. The Y are orthogonal, so

10. The radial equation

11. L11 – The H atom
u(r) = rR(r)

12. Asymptotic behavior
u(r) = rR(r)

13. The radial solution u(r) = rR(r) n > l
is the q-th Laguerre polynomial.

14. The radial solution: energy

15. Ground state

16. n > 1

17. n > 1 (continued)
Since they are eigenvectors for different eigenvalues

18. Example

19. Graphs

20. Angular momentum
Poiche’
Poiche’ , l’equazione in f si puo’ scrivere