2. Basic QM of the H-Atom (1)
Convenient Hamiltonian for H-atom:
All angular dependence in (, ) is contained in
(orbital angular momentum operator).
Schrödinger equation:
Look for solutions of the form (r,,) = R(r)·f(,)
LHS only depends on r, RHS only depends on ,.

3. Basic QM of the H-Atom (2)
Since the variables on both sides can be independently varied, both must be equal to the same constant .
RHS:
Eigenvalue equation of the angular momentum operator , so we know the solutions:
Y(,) are the Spherical Harmonics Yℓm(,);  = ħ2ℓ(ℓ+1)

4. Basic QM of the H-Atom (3)
Spherical Harmonics:
where the Pℓ|m|(cos ) are the “associated Legendre polynomials” defined by:
(ℓ = 0,1,2,3, ... ; |m| ≤ ℓ)
Useful recursion relation:
Useful integral:

5. Basic QM of the H-Atom (4)
With the knowledge of  = ħ2ℓ(ℓ+1), the LHS of the H-atom Schrödinger eqn. takes the form
New, effective potential

6. Basic QM of the H-Atom (5)
Solutions of the Radial Schrödinger equation:
= “associated Laguerre polynomials”
Radial wave functions Rnℓ(r):
At small r:  rℓ, Rnℓ(r)  0 for r0
At large r:  e-r, Rnℓ(r)  0 for r
Oscillating in between for n>2

7. Basic QM of the H-Atom (6)
from: I. N. Levine “Quantum Chemistry”

8. Basic QM of the H-Atom (7)
Quantum numbers: n, ℓ, m
n = 1,2,3,... principal quantum #
ℓ = 0,1,2,...,n-1 orbital angular momentum quantum #
m = -ℓ, -ℓ+1, ...., ℓ-1, ℓ projection quantum #
Without external fields, only n determines the energy:
independent of ℓ,m (same as in Bohr’s model)

9. Level diagram of the H-Atom
Basic QM of the H-Atom (8)
Level diagram of the H-Atom
degeneracy
from: T. Mayer-Kuckuk “Atomphysik”