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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/na 0 with a 0 = ε 0 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 , 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 2 0 h 2 n 2 n 1 L z op Y lm = L z Y lm L z = m l ¯ h - l m l l L 2 op Y lm = L 2 Y lm L 2 = l ( l + 1)¯ h 2 l < n S z op χ = S z χ S z = m s ¯ h m s = ± 1 2 S 2 op χ = S 2 χ S 2 = s ( s + 1)¯ h 2 s = 1 2 10.12.3 Spin-orbit interaction The total momentum is given by J = L + M . The total magnetic dipole moment of an electron is then M = M L + M S = - ( e/ 2 m e )( L + g S S ) where g S = 2 . 0023 is the gyromagnetic ratio of the electron. Further holds:
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