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

This preview shows pages 1–5. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 ....
View Full Document

## 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.

### Page1 / 20

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

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online