Hydrogen Atom PHY

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

Spherical Coordinates

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

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 …

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

Radial equation – effective potential

Radial hydrogenic wavefunctions

Putting radial and angular parts together 2p wave

Hydrogenic orbitals