Lecture_15

# Lecture_15 - PHYS 342 ll 2011 Fall 2011 Lecture 15...

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Unformatted text preview: PHYS 342 ll 2011 Fall 2011 Lecture 15: Solutions to chrödinger’s Equation for the Schrödinger s Equation for the Hydrogen Atom e- , m U(r ) Lecture 15 Ultimately, Schrödinger’s Equation reduces to three separate equations 2 2 2 2 ( ) ( ) 1 in ( ) ( 1) ( ) d m d d d m 2 2 2 2 2 sin ( ) ( 1) ( ) sin sin 2 ( 1 ) ( ) ( ) ( ) e d d m d rR r U r E rR r 2 2 2 2 ( ) ( ) ( ) 2 w here ( + 1) and m are separation constants e dr m r • The first equation restricts: |m|=0,1,2 . . . . ℓ-1, ℓ • The solution of 2 nd equation restricts the values of m and ℓ ( ℓ +1) where ℓ is a positive integer • . The solution of the third equation provides quantized values of E quation for ( 2 2 2 ( ) 1 ( . ) ) m constant indep of r and d Equation for Φ ( φ ): 2 2 ) ( ) ( ) im i try Ae d 2 2 2 2 ( ) ( ) ( ) ( 2 ) im im A im e Am e m require d d φ +2 2 2 2 ) ( 1 cos(2 ) sin(2 ) 1 0, 1, 2,... i m i m i m im im Ae m i m m A e e e A e φ always 1 always 0 3 m is known as the magnetic quantum number The Θ equation is harder to solve 2 2 1 sin ( ) ( 1) ( ) sin sin d d m d d • The Θ equation is known historically as the associated Legendre (1752- 33) Equation 1833) Equation • There are solutions for ALL values of ℓ ( ℓ +1) • Most solutions are infinite when θ =0 or θ = π ; not good! • There are unique solutions that do NOT diverge when ℓ =+integer AND ℓ ≥ |m| • These acceptable solutions are written as Θ ℓ m ( θ ) • The solutions to the Θ equation are often combined with the solutions for the Φ equation and written in terms of the spherical harmonics Y ℓ m ( , ) ( ) ( ) ( ) im m m m Y c o n s t c o n s t e 2 P dP Math Note 1: Legendre’s Equation: 2 2 (1 ) 2 ( 1) m d P dP x x P dx dx P P P L e gendre function ℓ P ℓ (x) 1 1 2 x − 1 2 6 x 2 − 6x + 1 0x 30x 12x 1 3...
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Lecture_15 - PHYS 342 ll 2011 Fall 2011 Lecture 15...

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