l16 - Schrödinger of atoms Filling states Angular momentum...

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Unformatted text preview: Schrödinger of atoms Filling states Angular momentum Pauli exclusion principle Transition rates Selection rules Finding ψ for an atom • To solve the Schrödinger equation for an atom, we used U ( r ) = − Ze 2 / ( 4 πǫ r ) and worked in spherical coordinates. • We tried a solution based on separable variables: ψ ( r , θ, φ ) = R n ,ℓ ( r )Θ ℓ, m ℓ ( θ )Φ m ℓ ( φ ) = R n ,ℓ ( r ) Y m ℓ ℓ ( θ, φ ) (1) • We found radial functions R n ,ℓ ( r ) (Serway Table 8.4, p. 280). Here are the first two: R 10 ( r ) = parenleftbigg Z a parenrightbigg 3 / 2 2 e − Zr / a , R 20 ( r ) = parenleftbigg Z 2 a parenrightbigg 3 / 2 parenleftbigg 2 − Zr a parenrightbigg e − Zr / 2 a • We found spherical harmonics Y m ℓ ℓ ( θ, φ ) (Serway Table 8.3, p. 269). Here are the first three: Y = 1 √ 4 π , Y 1 = radicalbigg 3 4 π cos θ, Y ± 1 1 = − radicalbigg 3 8 π sin θ e ± i φ • Both R n ,ℓ ( r ) and Y ℓ, m ℓ ( θ, φ ) are separately normalized. Schrödinger of atoms Filling states Angular momentum Pauli exclusion principle Transition rates Selection rules Quantum numbers for the atom • The nature of polynomial solutions to the recurrence relationships that apply to the solution of the atom’s Schrödinger differential equations leads us to three quantum numbers. n = 1 , 2 , 3 , . . . : the principal quantum number, which mainly determines energy. ℓ = , 1 , . . . , ( n − 1 ) : the orbital angular momentum quantum number. Total angular momentum is L = planckover2pi1 p ℓ ( ℓ + 1 ) , which is different than Bohr’s assumption of L = n planckover2pi1 !!! m ℓ = − ℓ, . . . , , . . . , + ℓ : ˆ z axis angular momentum quantum number, or magnetic quantum number. The ˆ z axis angular momentum is L z = m ℓ planckover2pi1 . • We characterize wavefunction solutions according to n and ℓ , with ℓ = , 1 , 2 , 3 , 4 , . . . corresponding to s , p , d , f , g , . . . . Examples are 1 s , 2 p , 3 f , and so on. • Radial probabilities for the electron go like R n ,ℓ ( r ) 2 r 2 dr because in spherical coordinates the volume of integration goes like r 2 dr in radius. Schrödinger of atoms Filling states Angular momentum Pauli exclusion principle Transition rates Selection rules Filling states • Using our three quantum numbers, we considered a table of successive allowed quantum states: State n ℓ m ℓ Number 1 s 1 1 2 s 2 1 2 p 2 1-1,0,+1 3 3 s 3 1 3 p 3 1-1,0,+1 3 3 d 3 2-2,-1,0,+1,+2 5 4 s 4 1 4 p 4 1-1,0,+1 3 4 d 4 2-2,-1,0,+1,+2 5 4 f 4 3-3,-2,-1,0,+1,+2,+3 7 Schrödinger of atoms Filling states Angular momentum Pauli exclusion principle Transition rates Selection rules Filling states II • Recall that Bohr found E = − E Z 2 / n 2 ....
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This note was uploaded on 05/28/2011 for the course PHY 251 taught by Professor Rijssenbeek during the Fall '01 term at SUNY Stony Brook.

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l16 - Schrödinger of atoms Filling states Angular momentum...

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