Quantum Mechanics for Nuclear Structure, Volume 2. Professor Kris Heyde. Читать онлайн. Newlib. NEWLIB.NET

Автор: Professor Kris Heyde
Издательство: Ingram
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Жанр произведения: Физика
Год издания: 0
isbn: 9780750321716
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The Pauli spin matrices

      The Pauli spin matrices, σx,σy,σz, possess a number of useful properties. We redefine them by σj,σk,σl,(j,k,l)=(x,y,z). Then

      σj2=σk2=σl2=Iˆ,(1.20)

      σjσk+σkσj=0,forj≠k,(1.21)

      i.e.

      where ‘{, }’ is an anticommutator bracket (also written ‘ [,]+’). Further,

      where

      εjkl≡εklj≡εljk≡1;εkjl≡εjlk≡εlkj≡−1.(1.24)

      From equations (1.22) and (1.23)

      σjσk=−σkσj=iσl.(1.25)

      Also,

      σj†=σj,(1.26)

      det(σj)=−1,(1.27)

      tr(σj)=0.(1.28)

      For the three-dimensional Cartesian vector a⃗, σ⃗·a⃗ is a 2 × 2 matrix2:

      σ⃗·a⃗≔σxax+σyay+σzaz,(1.29)

      ∴σ⃗·a⃗=azax−iayax+iay−az.(1.30)

      This leads to the important identity:

      (σ⃗·a⃗)(σ⃗·b⃗)=a⃗·b⃗Iˆ+iσ⃗·(a⃗×b⃗).(1.31)

      This can be obtained from equations (1.22) and (1.23):

      (σ⃗·a⃗)(σ⃗·b⃗)=∑jσjaj∑kσkbk.(1.32)

      ∴(σ⃗·a⃗)(σ⃗·b⃗)=∑j∑k12{σj,σk}+12[σj,σk]ajbk,(1.33)

      ∴(σ⃗·a⃗)(σ⃗·b⃗)=∑j∑k(δjk+iεjklσl)ajbk,(1.34)

      ∴(σ⃗·a⃗)(σ⃗·b⃗)=a⃗·b⃗Iˆ+iσ⃗·(a⃗×b⃗).(1.35)

      If the components of a⃗ are real then

      where ∣a⃗∣ is the magnitude of the vector a⃗.

      We are now in a position to obtain matrix representations of rotation operators in ket space. For a rotation about an axis nˆ through an angle ϕ (cf. Volume 1, chapter 10),

      D(R)=D(nˆ,ϕ)=exp−iJ⃗·nˆℏϕ,(1.37)

      the matrix elements of D(nˆ,ϕ) are

      〈jm′∣exp−iJ⃗·nˆℏϕ∣jm〉≔Dm′m(j)(nˆ,ϕ),(1.38)

      where j′=j is explicitly incorporated: 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.

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

      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