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Unformatted text preview: Chem 120A Approximation Examples, part I 04/12/06 Spring 2006 Lecture 31 1 Spin Coupling in Hydrogenic Atoms We looked at spin in lectures 25 through 27, here is a brief review. Elementary particles and composite particles carry an intrinsic angular momentum called spin = ~ S , which is related to the intrinsic magnetic moment by ~ =- ge 2 m ~ S . Here g , the g-factor, is a unitless factor derived from relativistic QM theory (which can be thought of as a correction factor). For electrons, g 2. For protons, g 5 . 6. You should also note that m proton m electron 2000, so we conclude that ~ proton ~ electron . We learned that while orbital angular momen- tum has a classical analog where the magnetic moment ~ comes from the particle spinning about some axis, the intrinsic angular momentum of a particle has nothing to do with its orbital motion. Again, and we cant stress this enough, spin is not orbital angular momentum in the classical sense. Experiments tell us, however, that we can take most general properties we derive for the QM operator ~ L = ~ r ~ p = L x i + L y j + L z k and we can simply apply them to the operator ~ S = ~ S x i + S y j + S z k . There are two quantum numbers that label the eigenstates of angular momentum (both orbital and spin). These come from the two eigenvalue equations for the angular momentum operator squared, ~ J 2 , and its z-component, ~ J z . ~ J 2 j , m j = h 2 j ( j + 1 ) j , m j ~ J z j , m j = hm j j , m j (1) we will drop the h s in what follows. For spin angular momentum ~ J = ~ S , j = s and m j = m s . The quantum number s can be an integer or half-integer and m s can take values which range from- s to s in integer steps. Thus, the degeneracy of m s is ( 2 s + 1 ) . If s is an integer = 1, 2, 3,.... then the particle is a boson. Identical bosons are symmetric under pairwise exchange: ( 1 , 2 ) = ( 2 , 1 ) . If s is a half-integer = 1/2, 3/2, 5/2... then the particle is a fermion. Identical fermions are antisymmetric under pairwise exchange: ( 1 , 2 ) =- ( 2 , 1 ) ....
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