1. Hydrogen Atom PHY 361 2008-03-19

2. Outline  review of L z operator, eigenfunction, eigenvalues rotational kinetic energy traveling and standing waves  spherical coordinates definition Laplacian operator Schrödinger’s equation in spherical coordinates  separation of angular variables: L 2 and L z differential equations -spherical harmonics and eigenvalues -vector model of quantum angular momentum radial wavefunctions -effective radial potential – centrifugal `force’ -radial wave functions -hydrogenic orbitals

3. Spherical Coordinates

4. Cylindrical vs. Spherical Coordinates Schr ö dinger Equation: Laplacian: L z 2 / 2I L 2 / 2I

5. Spherical Harmonics L 2 Y lm =l(l+1)Y lm L z Y lm = m Y lm 1 x, y z x 2 +y 2, xy xz, yz 3z 2 -1 x, y xz, yz x 2 +y 2, xy s p d f …

6. Vector model of quantized angular momentum l = 0, 1, 2, … m = -1, -l+1, … l-1, l

7. Radial equation – effective potential

8. Radial hydrogenic wavefunctions

9. Putting radial and angular parts together 2p wave

10. Hydrogenic orbitals http://www.orbitals.com/orb/