Hydrogen Atom PHY 361 2008-03-19

Outline review of Lz spherical coordinates 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: L2 and Lz differential equations spherical harmonics and eigenvalues vector model of quantum angular momentum radial wavefunctions effective radial potential – centrifugal `force’ radial wave functions hydrogenic orbitals I will start with a brief reminder of where we were last, and explain the different options available for polarizing the beam. Then I will go on and show the progress we have made towards a feasible design of the novel splitter polarizer geometry. We have also done new detailed simulations of the different designs and recalculated the costing of each for comparison. Based on this new information, we have arrived at a choice for the baseline design.

Spherical Coordinates

Cylindrical vs. Spherical Coordinates Laplacian: Schrödinger Equation: Lz2 / 2I L2 / 2I

Spherical Harmonics L2 Ylm=l(l+1)Ylm Lz Ylm= m Ylm s 1 x, y p z x, y x2+y2, xy xz, yz d 3z2-1 xz, yz x2+y2, xy f …

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

Potential energy function and bound states

Radial equation – effective potential

Radial hydrogenic wavefunctions

Putting radial and angular parts together 2p wave

Ground state wavefunction

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

Selection rules and transitions difference in energy states measured from atomic transitions – E = h f atomic spectroscopy only certain transitions are allowed