Chapter 4

Spin-Orbit Coupling

4.1 Addition of Angular Momenta

4.1.1 Alternate set of angular momentum operators We consider two independent systems that each carries angular momenta (each could be either orbital angular momentum or ). Before the two systems are coupled, each system is described by the angular momentum operators Jˆ1 and Jˆ2 and their eigenstates j1,m1 and j ,m satisfying the conditions | i | 2 2i

ˆ2 2 Ji ji,mi = ~ ji(ji +1)ji,mi , ˆ | i | i Jiz ji,mi = ~mi ji,mi , (4.1) ˆ | i | i Ji ji,mi = ~ ji(ji +1) mi(mi 1) ji,mi 1 , ±| i ± | ± i for i =1, 2. When we consider thep combination of these two systems, the basis of the combined system is created by

j ,j ; m ,m = j ,m j ,m , (4.2) | 1 2 1 2i | 1 1i⌦| 2 2i where indicates the tensor product of two states that describe the two independent systems. ⌦ When the total angular momentum of the two systems is considered, it is described by the sum of the two angular momentum operators Jˆ = Jˆ1 + Jˆ2.Becausethetwosystemsare independent, the operators Jˆ1 and Jˆ2 commute with each other, and the components of each system satisfies the commutation relations of the angular momentum operator. It is easy to verify that the components of the total angular momentum operator Jˆ also satisfies the commutation relationship ˆ ˆ ˆ [Ji, Jj]=i~"ijkJk. (4.3) We now look for four independent operators that commute with each other that includes 2 the operators Jˆ and Jˆz,toprovideanalternativedescriptionofthecombinedsystemas compared to the four commuting operators Jˆ2, Jˆ , Jˆ2, Jˆ . { 1 1z 2 2z} 31 32 CHAPTER 4. SPIN-ORBIT COUPLING

ˆ ˆ ˆ2 ˆ2 ˆ Since J1 and J2 each commute with J1 and J2 ,sodoesJ.Morespecifically,wenotethat ˆ2 ˆ ˆ2 ˆ2 J and Jz both commute with the operators J1 and J2 ;

[Jˆ2, Jˆ2]=[Jˆ2, Jˆ2]=0, 1 2 (4.4) ˆ ˆ2 ˆ ˆ2 [Jz, J1 ]=[Jz, J2 ]=0.

The operators Jˆ1z and Jˆ2z commute with the operator Jˆz = Jˆ1z + Jˆ2z. However, since

ˆ2 ˆ2 ˆ2 ˆ ˆ ˆ2 ˆ2 ˆ ˆ ˆ ˆ ˆ ˆ J = J1 + J2 +2J1 J2 = J1 + J2 +2J1zJ2z + J1+J2 + J1 J2+, (4.5) · ˆ ˆ ˆ2 ˆ2 ˆ2 ˆ2 ˆ the operators J1z and J2z do not commute with J .So,thefouroperators J1 , J2 , J , Jz commute with each other, and forms a natural basis with which to describe the{ total angular} momentum of the combined system.

4.1.2 Addition rule for the angular momentum states

Classically, angular momentum is a vector, so when two angular momenta L~ 1 and L~ 2 are added together, the magnitude of the resulting angular momentum can be between L~ L~ | 1 2| when the two vectors are aligned in opposite directions, and L~ 1 + L~ 2 when the two vectors are aligned in the same direction. Quantum mechanically, the| “magnitude”| of the angular momentum operator Jˆ is roughly represented by its eigenvalue j,sowhenthetwoangular momentum operators Jˆ1 with eigenvalue j1 and Jˆ2 with eigenvalue j2 are added (without loss of generality, we assume j j ), one can speculate that the resulting operator Jˆ will 1 2 have its eigenvalue j between the values j1 j2 and j1 +j2.Itturnsoutthatthisspeculation is exactly true, and the addition rule for angular momentum can be stated as follows: When two systems each with angular momentum of eigenvalues j and j are added, • 1 2 the angular momentum eigenvalue of the added system runs from j1 j2 to j1 + j2 in increments of 1, and | |

For each resulting j value, mj runs from j to j in increments of 1, i.e., mj = • j, j +1,...,j 1,j. Although a formal proof is quite complicated, we provide some examples in the next section to illustrate the results. We note that the total number of states in the two systems is (2j1 +1)(2j2 +1),sincethedegeneracy(totalnumberofmj states) are 2j1 +1forthefirst system and 2j2 +1forthesecondsystem,respectively.Inthedescriptionofthecombined system, the total number of mj states is 2j +1 foreach j value, so the total number of states is given by

j1+j2 j1+j2 j1 j2 1 N