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 ,.

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)

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:

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

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

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

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)

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”