Hydrogen Revisited

Now we understand quite a bit about the Sun’s spectrum – it is almost a black-body spectrum (a –la Planck) except there are absorption lines – finally we can also understand the splitting of the lines in Solar spots (Zeeman effect) which Bohr couldn’t explain. Zeeman Splitting

Quantum Numbers 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…

Quantization of Angular momentum 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!

Classical Thoughts A revolving charge gives rise to a magnetic field.

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

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

n=1

n=2

n=3

Constructing a wavefunction Spherical Harmonics

Probability of finding an electron in a "shell" between r and dr Normalization condition Average distance from nucleus