Quantum Mechanics for Nuclear Structure, Volume 2. Professor Kris Heyde

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СКАЧАТЬ this is because

      Jˆ2D(R)∣jm〉=D(R)Jˆ2∣jm〉,(1.39)

      ∴Jˆ2D(R)∣jm〉=j(j+1)ℏ2D(R)∣jm〉,(1.40)

      which follows from the general relationship [Aˆ,exp{Aˆ}]=0. This is sensible because rotations cannot change the length of a vector. Thus, the matrix representation of D(R) has the form:

      D(R)↔D(0)0000D120000D(1)0000D32⋱(1.41)

      and we can discuss the D(j) individually.

      The Euler angle parameterisation leads to a simplification when one considers a matrix representation:

      Dm′m(j)(α,β,γ)=〈jm′∣e−iJˆzαℏe−iJˆyβℏe−iJˆzγℏ∣jm〉;(1.42)

      but,

      〈jm′∣e−iJˆzαℏ=〈jm′∣e−im′α,(1.43)

      ∴Dm′m(j)(α,β,γ)=e−i(m′α+mγ)〈jm′∣e−iJˆyβℏ∣jm〉,(1.44)

      i.e. only the ‘Jˆy’ rotation is non-trivial. We define

      dm′m(j)(β)≔〈jm′∣e−iJˆyβℏ∣jm〉.(1.45)

      The Dm′m(j)(R)′s (R=nˆ,ϕ or α,β,γ) are called Wigner functions. They tell us how much of ∣jm〉 rotates into ∣jm′〉 under the action of R:

      D(R)∣jm〉=∑m′∣jm′〉〈jm′∣D(R)∣jm〉,(1.46)

      where the completeness relation has been used.

      We are now in a position to obtain explicit matrix representations of D(R), the so-called Wigner matrices:

       D(0): This is trivial. It is the 1 × 1 matrix (1).

       D(12): This is a 2 × 2 matrix and can be evaluated from the properties of the Pauli spin matrices. ConsiderD12(nˆ,ϕ)=exp−iJ⃗12·nˆℏϕ=exp−iσ⃗·nˆϕ2.(1.47)Then, expanding the exponential:D12(nˆ,ϕ)=Iˆ−iϕ2σ⃗·nˆ−12!ϕ22(σ⃗·nˆ)2+i3!ϕ23(σ⃗·nˆ)3+⋯.(1.48)But, from equation (1.36),(σ⃗·nˆ)m=Iˆ,meven,(1.49)(σ⃗·nˆ)m=(σ⃗·nˆ),modd,(1.50)∴D12(nˆ,ϕ)=Iˆ1−12!ϕ22+⋯−iσ⃗·nˆϕ2−13!ϕ23+⋯,(1.51)∴D12(nˆ,ϕ)=Iˆcosϕ2−iσ⃗·nˆsinϕ2.(1.52)Explicitly,∴D12(nˆ,ϕ)=cosϕ2−inzsinϕ2(−inx−ny)sinϕ2(−inx+ny)sinϕ2cosϕ2+inzsinϕ2(1.53)for an axis-angle parameterisation.

       For an Euler angle parameterisationD12(α,β,γ)=Dz12(α)Dy12(β)Dz12(γ),(1.54)then using equation (1.52):D12(α,β,γ)=e−iα200eiα2cosβ2−sinβ2sinβ2cosβ2e−iγ200eiγ2,(1.55)∴D12(α,β,γ)=e−i(α+γ)2cosβ2−e−i(α−γ)2sinβ2ei(α−γ)2sinβ2ei(α+γ)2cosβ2.(1.56)Note that D(12)(nˆ,ϕ) and D(12)(α,β,γ) fulfil the unitary unimodular or special unitary form ab−b*a*, cf. Volume 1, equation (10.42), and herein section 5.10.2.

       D(1) : This is a 3 × 3 matrix. It can be evaluated using a series expansion if we use its Euler angle parameterisation. FromDm′m(1)(α,β,γ)=e−i(m′α+mγ)dm′m(1)(β),(1.57)expanding the exponential in d(1):e−iJˆy(1)βℏ=Iˆ−iβℏJˆy(1)−12!β2ℏ2Jˆy(1)2+i3!β3ℏ3Jˆy(1)3+⋯.(1.58)This is greatly simplified by the following identityJˆy(1)3ℏ3=180−2i02i0−2i02i00−2i02i0−2i02i0×0−2i02i0−2i02i0,(1.59)∴Jˆy(1)3ℏ3=180−2i02i0−2i02i020−2040−202,(1.60)∴Jˆy(1)3ℏ3=180−42i042i0−42i042i0,(1.61)∴Jˆy(1)3ℏ3=Jˆy(1)ℏ.(1.62)Then, equation (1.58) reduces toe−iJˆy(1)βℏ=Iˆ+Jˆy(1)ℏ−iβ+iβ33!+⋯+Jˆy(1)2ℏ2−β22!+⋯,(1.63)∴e−iJˆy(1)βℏ=Iˆ−iJˆy(1)ℏsinβ+Jˆy(1)2ℏ2(cosβ−1).(1.64)Thus,d(1)(β)=12(1+cosβ)−12sinβ12(1−cosβ)12sinβcosβ−12sinβ12(1−cosβ)12sinβ12(1+cosβ).(1.65)

      To evaluate D(1)(nˆ,ϕ) and D(j)(nˆ,ϕ) or D(j)(α,β,γ) with j>1, we must develop the theory of tensor bases of representation in ket space.

1.5 Tensor representations for SU(2)

      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.

1.6 Tensor representations for SO(3)

      Consider СКАЧАТЬ