Читать книгу Quantum Mechanics for Nuclear Structure, Volume 2 - Professor Kris Heyde - Страница 13

Consider the general SU(2) transformation (cf. Volume 1, chapter 10)

ab−b*a*u1u2=u1′u2′,(1.66)

where the 2 × 2 matrix may, for example, have the form given by equation (1.53) or equation (1.56). Then, defining

q1≔u12,q2≔2u1u2,q3≔u32,(1.67)

under the transformation, equation (1.66), we obtain

q1′=u1′2=(au1+bu2)2=a2u12+2abu1u2+b2u22,(1.68)

∴q1′=a2q1+2abq2+b2q3;(1.69)

and similarly,

q2′=−2ab*q1+(aa*−bb*)q2+2ba*q3,(1.70)

q3′=(b*)2q1−2a*b*q2+(a*)2q3;(1.71)

whence

a22abb2−2ab*(aa*−bb*)2ba*(b*)2−2a*b*(a*)2q1q2q3=q1′q2′q3′.(1.72)

This is still a representation of an SU(2) transformation: there are no new parameters. However, it is a 3 × 3 matrix representation of SU(2). From the Euler angle parameterisation, equation (1.56),

a=exp−i(α+γ)2cosβ2,b=−exp−i(α−γ)2sinβ2;(1.73)

and substitution of these values of a and b into the matrix in equation (1.72) will yield D(1)(α,β,γ), the β-dependent part of which is given by equation (1.65).

The process can be iterated by defining

p1≔u13,p2≔3u12u2,p3≔3u1u22,p4≔u23,(1.74)

which will yield a 4 × 4 matrix representation of SU(2), i.e. an expression for D(32)(R).

Expressions for D(j)(α,β,γ) can be obtained by this process, for any j, together with the values of a and b given in equation (1.73). Likewise, D(j)(nˆ,ϕ) can be obtained using (cf. equation (1.53))

a=cosϕ2−inzsinϕ2,b=(−inx−ny)sinϕ2,(1.75)

where, recall, the constraint nx2+ny2+nz2=1 ensures that (nx,ny,nz,ϕ) corresponds to three free parameters. To reiterate: SU(2) is a three-parameter group.

The representation associated with the two-component spinor (u1,u2) is called the fundamental representation. The representation associated with the three-component entity (q1,q2,q3) is a rank-2 SU(2) tensor, i.e. it is constituted from quadratic combinations of the fundamental representation. In turn, (p1,p2,p3,p4) is a rank-3 SU(2) tensor.