LectureCh19.pdf

# There are recursive expressions for calculating the

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There are recursive expressions for calculating the Clebsch-Gordon coeﬃcients and tables and calculators are easy to find. P. J. Grandinetti (Chem. 4300) Magnetism, Ang. Mom., & Spin Nov 13, 2017 40 / 62

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Clebsch-Gordon coeﬃcients j 1 j 2 m 1 m 2 | j 1 j 2 JM = ( - 1) J - j 1 - j 2 j 2 j 1 m 2 m 1 | j 2 j 1 JM d Square-root sign is to be understood over every coeﬃcient, e.g., for - 8 15 read - 8 15 . P. J. Grandinetti (Chem. 4300) Magnetism, Ang. Mom., & Spin Nov 13, 2017 41 / 62
Example Construct wave function with j = 1 2 and m j = - 1 2 for p-electron in H-atom in terms of wave functions associated with 𝓁 , m 𝓁 , s , and m s . Solution: Addition of 𝓁 = 1 and s = 1 2 leads to 2 possibilities of j = 1 2 and j = 3 2 . Focus on j = 1 2 states and write Clebsch-Gordon series as ? n , j = 1 2 , m j = 𝓁 m 𝓁 =- 𝓁 s m s =- s C ( j = 1 2 , m j , 𝓁 = 1 , m 𝓁 , s = 1 2 , m s ) ? n , 𝓁 = 1 , m 𝓁 , s = 1 2 , m s Expanding out the m j = - 1 2 case we find six terms: ? 1 2 , - 1 2 = C ( 1 2 , - 1 2 , 1 , - 1 , 1 2 , - 1 2 ) n p - 1 ? + C ( 1 2 , - 1 2 , 1 , 0 , 1 2 , - 1 2 ) n p 0 ? + C ( 1 2 , - 1 2 , 1 , 1 , 1 2 , - 1 2 ) n p 1 ? + C ( 1 2 , - 1 2 , 1 , - 1 , 1 2 , 1 2 ) n p 1 ? + C ( 1 2 , - 1 2 , 1 , 0 , 1 2 , 1 2 ) n p 0 ? + C ( 1 2 , - 1 2 , 1 , 1 , 1 2 , 1 2 ) n p - 1 ? Only terms where m j = m 𝓁 + m s = - 1 2 survive... P. J. Grandinetti (Chem. 4300) Magnetism, Ang. Mom., & Spin Nov 13, 2017 42 / 62

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Example Construct wave function with j = 1 2 and m j = - 1 2 for p-electron in H-atom in terms of wave functions associated with 𝓁 , m 𝓁 , s , and m s . Solution: We are left with ? n , 1 2 , - 1 2 = C ( 1 2 , - 1 2 , 1 , 0 , 1 2 , - 1 2 ) n p 0 ? + C ( 1 2 , - 1 2 , 1 , - 1 , 1 2 , 1 2 ) n p - 1 ? Use 1 × 1 2 Clebsch-Gordon Coeﬃcient table to find ? n , 1 2 , - 1 2 = 1 3 n p 0 ? - 2 3 n p - 1 ? Using orbital diagrams we could represent j = 1 2 , m j = - 1 2 state as P. J. Grandinetti (Chem. 4300) Magnetism, Ang. Mom., & Spin Nov 13, 2017 43 / 62
Addition of three or more angular momenta What about addition of three or more angular momenta? For example, What about 𝓁 = 1 , s = 1 2 , and i = 1 2 ? We have 3 choices on where to start, and all are valid. First add 𝓁 and s to get j = | 𝓁 - s | to 𝓁 + s , then add j and i to get f = | i - j | to i + j . First add i and s to get j = | i - s | to i + s , then add j and 𝓁 to get f = | j - 𝓁 | to j + 𝓁 . First add 𝓁 and i to get j = | i - 𝓁 | to i + 𝓁 , then add j and s to get f = | j - s | to j + s . P. J. Grandinetti (Chem. 4300) Magnetism, Ang. Mom., & Spin Nov 13, 2017 44 / 62

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Example What are total angular momenta values and number of states that result when 𝓁 = 1 , s = 1 2 , and i = 1 2 are added? Solution: Starting with 𝓁 and s we find that j = | 𝓁 - s | to 𝓁 + s gives j = 1 2 , 3 2 for 𝓁 = 1 and s = 1 2 then f = | j - i | to j + i gives f = 0 , 1 for j = 1 2 and i = 1 2 and f = 1 , 2 for j = 3 2 and i = 1 2 Total angular momentum can be f = 2 , 1 , 1 , 0 each with 2 f + 1 values of m f Thus we have ( 2 2 + 1 ) + ( 2 1 + 1 ) + ( 2 1 + 1 ) + ( 2 0 + 1 ) = 12 states P. J. Grandinetti (Chem. 4300) Magnetism, Ang. Mom., & Spin Nov 13, 2017 45 / 62
Fine Structure of the H-Atom 1 Relativistic Correction 2 Spin Orbit Interaction P. J. Grandinetti (Chem. 4300) Magnetism, Ang. Mom., & Spin Nov 13, 2017 46 / 62

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Fine Structure of the H-Atom : Relativistic Correction While classical kinetic energy expression is good starting approximation for H-atom there are slight relativistic effects that can be observed in emission spectra of H-atoms.
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