The separated radial and orbital parts of the Schrodinger equation: Note that the angular momentum equation does not depend on the form of the potential, but it does relate the “magnetic quantum number” to the angular momentum quantum number. The radial equation does depend on the form of the potential…it relates the total energy to the angular momentum. Note that the magnetic quantum number is independent of the energy. This leads to degenerate states.

The Schrodinger equation for a spherically symmetric potential: where: The final separated forms: Gives us three quantum numbers…analogous to Ex, Ey, and Ez in the cartesian case…

Lz must be less than L, and cannot equal L, otherwise Lx=Ly=0, and all three components of the momentum would be known simultaneously in violation of the uncertainty principle!

e v A revolving charge gives rise to a magnetic field.

the potential expression for kinetic energy kinetic plus potential energy gives the total energy positionxx momentump potential energyUU(x) kinetic energyK total energyE observable operator

Eigenvalues are a constant of the motion. Consider the time independent Schrodinger equation… When applied to a wavefunction, this expression yields E…energy eigenvalues For example these are the energy eigenvalues you find when applying this to a simple harmonic oscillator potential

n=1 E=-13.6 eV n=2 E= -3.4 eV n=3 E= -1.5 eV n=4 E= -0.8 eV allowed transitions forbidden transition l=3l=2l=1 angular momentum must be conserved …photons carry angular momentum.

Energy: Assuming that no more than one electron can occupy each state, there are a total of states. There are a total of orbitals within each subshell. There are a total of n subshells. Preview: Actually, we will find in Chapter 9 that two electrons can occupy the same orbital if they have different “spin”.