There are recursive expressions for calculating the

Info icon This preview shows pages 40–48. Sign up to view the full content.

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
Image of page 40

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

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
Image of page 41
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
Image of page 42

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

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
Image of page 43
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
Image of page 44

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

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
Image of page 45
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
Image of page 46

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

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.
Image of page 47
Image of page 48
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern