Читать книгу Quantum Mechanics for Nuclear Structure, Volume 2 - Professor Kris Heyde - Страница 23

 1.1. Explore the commutator properties ofT1=00000−i0i0,T2=00i000−i00,T3=0−i0i00000,(1.258)in comparison with SO(3) and SU(2), (3,R) and (2,C).

 1.2. Show thatd32(β)=c3−3c2s3cs2−s33c2sc3−2cs2s3−2c2s3cs23cs2−s3+2c2sc3−2cs2−3c2ss33cs23c2sc3,(1.259)where c≔cosβ2, s≔sinβ2.

 1.3. Show that the results of equation (1.167) agree with equation (1.229) for ϕ=α=0 and j=l=1,2, and 3.

 1.4. Show thatdm′m(j)(β)=(−1)m′−mdmm′(j)(β).(1.260)[Hint: in the binomial expansion of equation (1.160), which results in equation (1.162) and eventually equation (1.167), reverse the positions of cosβ2a+†, sinβ2a−† and −sinβ2a+†, cosβ2a−†, i.e. express the expansion so that it contains a+†cosβ2j+m−l, etc.]

 1.5. Show that for R=(0,β,0) the Y1μ(θ,ϕ), μ=0,±1 obey equation (1.256). [Hint: express the Y1μ(θ,ϕ) in terms of x, y and z (cf. equations (1.188), (1.191), (1.193), (1.202) and (1.203)), obtain (x,y,z)R using Ry(β), and show that d(1)(β) (equation (1.65)) transforms the Y1μ(θ,ϕ) into the Y1μ(θR,ϕR).]