Lecture 13 Highlights The eigenfunctions of can be expressed as linear combinations of states with different values of and using the world-famous Clebsch-Gordan coefficients ( ) as: 2JlmsmjsmmmjsCll smmmjsmmmjmsmCmjjsjs∑=+=lllll Where the ket ll mrepresents the spherical harmonics . The C-G coefficient values are given in Table 4.8 on page 188 of Griffiths. Remember that the all of the coefficients should appear under a square root, with the minus sign (if any) out front. Also note that we have dropped the radial part of the wavefunction ( ) because it plays no role in combining angular momenta. Don’t forget to put it back later. llmYlnR Where do these coefficients come from? Consider starting with a product wavefunction at the top of the ladder (it is a product of the wavefunctions with maximum values of and ). Now apply the lowering operator, and construct orthonormal states on lower rungs of the ladder. The coefficients on the terms of those wavefunctions are the C-G coefficients. jmlmsm−J We did a specific example of a hydrogen atom with 1=land spin . In this case the angular momentum vector and spin vector can either be “parallel” or “anti-parallel.” Consider the two cases: 2/1=s 1) “Parallel” LrandSr: The maximum value of is +1, while the value of is +1/2 for the “parallel” case. This means thatlmsm2/3=+=sjmmml. This is the state at the top of the ladder. There must also be states with2/3,2/1,2/1−−+=jm. This is a set of 4 states on the ladder of . Thus the eigenvalues of for this ladder must be 2/3=j2J22415)123(23hh =+. Note that 0>• SLrris this case, giving a positive spin-orbit Hamiltonian perturbation. 2) “Anti-Parallel” LrandSr: The maximum value of is +1, while the value of is -1/2 for the “anti-parallel” case. This means thatlmsm2/1=+=sjmmml. This is the state at the top of the ladder. There must also be a state with2/1−=jm. This is a set of 2 states on the ladder of . Thus the eigenvalues of for this ladder must be 2/1=j2J2243)121(21hh =+. Note that 0<• SLrris this case, giving a negative spin-orbit Hamiltonian perturbation. There are a total of 6 states possible by simply combining the orbital angular momentum with and spin angular momentum with1=l 2/1=s! Just imagine what happens when you combine 3 or more angular momentum vectors!Now for an example of how to construct states that are simultaneous eigenfunctions of , , and . Take the case again of hydrogen with and spin . How do we find the state with 2L2S2JzJ1=l2/1=s2/3=jand 2/1−=jmin terms of the and spinors? Look at the llmY211×Table on page 188. We are led to this table because we are combining an angular momentum vector with1=land spin vector with . Now look under the column labeled “ “. It says: 2/1=s2/12/3− smmmmmmCss21121232/12/32/112/1∑−=+−=−lll 21211131212101322123−+−=− This can be written in a more familiar way in terms of spherical harmonics and spinors as: +−−+=−χχ110131322123YY One can move back and forth between the coupled and un-coupled representations using the Clebsch-Gordan table on page 188. Here is the schematic layout for the CG table for combining two spins (called21, SSrr) to form a total spin 21SSSrrr+= ( has eigenvalue ): 2S2)1( h+ss21SS ×21ssmm#CGGeneral Schematic of the C-G Tablessm21SSSrrr+=CoupledRepresentationUn-CoupledRepresentationWe considered what happens when two spin-1/2 spins are combined. The lecture followed Griffiths pages 184-188, although not in that order. We considered the 4 naïve product states of the two spins: ↓↓↑↓↓↑↑↑ ,,, where ↑↑represents the product ket 2121212121++, where the first number in each ket represents and , respectively, and the second number represents and . We found that the eigen-kets of the operator (where 1s2s1sm2sm2S21SSSrrr+= ) are in two ladders of states: ↑↑=11 (↓↑+↑↓=2101) This is the s = 1 ladder of 3 states. TRIPLET ↓↓=−11 and (↓↑−↑↓=2100) This is the s = 0 ladder of 1 state. SINGLET The kets on the left are written in the “coupled representation” while those on the right are in the “un-coupled representation.” These are 4 orthonormal states that span the Hilbert space for the two spins. Note that we started with spins that individually live on half-integer ladders, but the combined spin lives on integer ladders. This will have important ramifications for the physics of multi-particle systems, such as multi-electron atoms and the theory of superconductivity, among other
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