By Brain Judd (Auth.)
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Additional resources for Angular Momentum Theory for Diatomic Molecules
2 27 Generators for R(4) very useful for us t o introduce some of t h e more i m p o r t a n t concepts of t h e subject, if only t o provide a working vocabulary for t h e analysis. 2 GENERATORS FOR R(4) T h e operators D(ca) a r e t h e elements of R(3). T h e y a r e formed from t h e components of J a n d t h e Euler angles ω. T h e former a r e t e r m e d t h e generators of Λ ( 3 ) : t h e y satisfy certain c o m m u t a t i o n relations a m o n g themselves a n d can be t h o u g h t of as carrying o u t t h e a c t of r o t a t i o n .
8 THE LENZ VECTOR In the previous chapter, we examined in some detail the properties of the four-dimensional spherical harmonics ΓΛι**(Ω). N o w that Eq. 31) has been established, it becomes interesting to see what physical signifi cance can be attached to generators of the group 72(4) associated with these harmonics. The connection between mathematics and physics is made in Eqs. 24). 4, and we can use Eqs. 34) to express these functions in terms of ρ and θ/θρ. The latter can be removed 58 3 R(4) in Physical Systems by making the replacement d/dp —> — tr.
2). 1 The Rigid Rotator 47 T o assign q u a n t u m n u m b e r s t o t h e eigenfunctions of Γ, we seek oper ators t h a t commute with it. Now, we h a v e already determined t h a t /, defined in E q s . 22), commutes with every component of λ. Hence t h e eigenvalues J(J + 1) a n d Μ of t h e t w o m u t u a l l y c o m m u t i n g operators 2 Ρ and lt can b e used t o label t h e eigenfunctions. Of course, Ρ = λ , a n d t h e t w o q u a n t u m n u m b e r s J a n d Μ correspond t o t h e t o t a l angular m o m e n t u m of t h e rigid r o t a t o r a n d its ζ projection in t h e laboratory frame.
Angular Momentum Theory for Diatomic Molecules by Brain Judd (Auth.)