PH 401 Dr. Cecilia Vogel

Review Outline  Prove the radial H-atom solution  Spin  evidence  spin angular momentum  Spherically Symmetric Hamiltonian  H-atom for example  Eigenstates of H, L z, L 2  Degeneracy

Spherically Symmetric Problem  For H-atom  V depends only on r,  not  or .  and the Hamiltonian looks like.  =

Radial Eqn  Radial part of Scroed eqn   Solution is a polynomial*exponential  Claim:  E n =-13.6eV/n 2  lowest order p = ℓ

Terms of Radial Eqn   1 st term:  2 nd term:  3 rd term:

Radial Eqn  Thus, radial part of Scroed eqn   becomes (after canceling the exponential from each term)

Radial Eqn … p-2 power  Each power of r must balance in this equation separately  Lowest power is r p-2. Only two terms have this power  This implies (p+1)(p)=ℓ(ℓ+1)  or p= ℓ …proved!

Radial Eqn … n-1 power  Each power of r must balance in this equation separately  Highest power is r n-1. Only two terms have this power  This implies  when you plug in values of constants  E n =-13.6eV/n 2 …proved!

Degeneracy of Eigenstates  Consider n=5, 4 th excited state of H-atom  What are possible values of ?  For each, what are possible values of m ?  for each n &, how many different states are there? “subshell”  for each n, how many different states are there? “shell”  what is the degeneracy of 4 th excited state? 25?  In general if you count all the values of and m for a given n,  you would expect a degeneracy of n 2

Spin Quantum Number  Actually there turns out to be twice as many H-atom states as we just described. 2n 2.  Introduce another quantum number that can have two values  spin can be up or down (+½ or -½)  Spin also affects the energy in the presence of a magnetic field.

Spin Angular Momentum  Spin is like other forms of angular momentum, in the sense that  it acts like a magnet, affected by B-fields  it contributes to the angular momentum, when determining conservation thereof.  The eigenvalues of the magnitude of the vector are  for electron, s=1/2, so  An t he eigenvalues of the z-component are m s  where m s ranges from –s to s in integer steps  for electron, s=1/2, so m s =+½