Chapter4_11 - PHY4604 R D Field The Hydrogen Atom –...

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Unformatted text preview: PHY4604 R. D. Field The Hydrogen Atom – Quantum Mechanics (2) Radial Wave Function: We see that α 2mc2 2αmc 1 h 2κ 2 2α 2 mc2 = =− =− En = − and κ = , ρ02 2m 2n 2 hρ0 r0n where I used ρ0 = 2n and where r0 is the Bohr radius r0 = D / α . Thus, r 1 1 l +1 − ρ j ρ = κr = and Rnl (r ) = U nl ( ρ ) = ρ e ∑ c j ρ j , r0n r r j =0 where jmax = l + 1 - n, n = 1, 2, 3, … and l = 0, 1, 2, …, n-1. max Complete Wave Function: The overall wave function is ψ nlm (r , θ , φ ) = Rnl (r )Ylm (θ , φ ) . Normalization: These wave functions form and orthonormal set ∞ 2π π ∫ ∫ ∫ [ψ (r ,θ , φ )] ψ nlm (r ,θ , φ )r 2 sin θdrdθdφ ∗ n 'l 'm ' 0 00 ∞ 2π π = ∫ R (r ) Rnl (r )r dr ∫ ∫ [Yl 'm ' (θ , φ )] Ylm (θ , φ ) sin θdθdφ = δ n 'nδ ll 'δ mm ' ∗ n 'l ' ∗ 2 0 00 2π π ∫ ∫ [Y (θ , φ )] Ylm (θ , φ ) sin θdθdφ = δ ll 'δ mm ' ∗ l 'm ' 00 ∞ ∫R ∗ n 'l (r ) Rnl (r )r 2 dr = δ n 'n 0 Ground State: For the ground state n = 0, l = 0, and m = 0. E1 = − 1 α 2 mc2 ≈ −13.6eV , R10 (r ) = 2 2 e −r / r0 , Y00 (θ , φ ) = 1 / 4π (r0 )3 / 2 and ψ 100 (r ,θ ,φ ) = 1 πr 3 0 e − r / r0 . The <r> in the ground state is ∞ 2π π < r >100 = ∫ ∫ ∫ [ψ (r ,θ , φ )] rψ 100 (r ,θ , φ )r 2 sin θdrdθdφ ∗ 100 0 00 ∞ ∞ ( 4 r r = 3 ∫ r 3e −2 r / r0 dr == 0 ∫ y 3e − y dy = 0 (− y 3 − 3 y 2 − 6 y − 6)e − y 40 4 r0 0 Department of Physics Chapter4_11.doc ) ∞ 0 = 3r0 2 University of Florida ...
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