Chapter 10: Quantum physics 49 10.12 Atomic physics 10.12.1 Solutions The solutions of the Schr¨odinger equation in spherical coordinates if the potential energy is a function of r alone can be written as: ψ ( r, θ , ϕ )= R nl ( r ) Y l,m l ( θ , ϕ ) χ m s , with Y lm = C lm √ 2 π P m l (cos θ )e im ϕ For an atom or ion with one electron holds: R lm ( ρ )= C lm e-ρ / 2 ρ l L 2 l +1 n-l-1 ( ρ ) with ρ =2 rZ/na0 with a0 = ε0 h 2 / π m e e 2 . The L j i are the associated Laguere functions and the P m l are the associated Legendre polynomials: P | m | l ( x )=(1-x 2 ) m/ 2 d | m | dx | m | ± ( x 2-1) l ² ,L m n ( x )= (-1) m n ! ( n-m )! e-x x-m d n-m dx n-m (e-x x n ) The parity of these solutions is (-1) l . The functions are 2 n-1 ∑ l =0 (2 l +1)=2 n 2-folded degenerated. 10.12.2 Eigenvalue equations The eigenvalue equations for an atom or ion with with one electron are: Equation Eigenvalue Range H op ψ = E ψ E n = μe 4 Z 2 / 8 ε 20 h 2 n 2 n ≥ 1
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This note was uploaded on 01/30/2011 for the course PHYSICS 208 taught by Professor Ye during the Spring '10 term at Blinn College.