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

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СКАЧАТЬ This can be generalised to

      Ti′j′k′⋯=∑i∑j∑k⋯Ri′iRj′jRk′k⋯Tijk⋯,(1.77)

      where the T_ijk⋯ are the Cartesian components of a tensor, the rank of which is equal to the number of indices and the Ri′i are, as before, elements of the 3 × 3 matrix R. Details are clarified in the following.

      The simplest Cartesian tensor is of rank 2 and is often called a dyad or dyadic. It is formed from two Cartesian vectors, e.g.

      U⃗=(U1,U2,U3),V⃗=(V1,V2,V3),(1.78)

      Tij=UiVj,(1.79)

      of which there are nine components. Thus,

      Ti′j′=∑i∑jRi′iRj′jTij,(1.80)

      where the T_ij are the nine components of the dyadic and the Ri′iRj′j are the elements of the 9 × 9 matrix that effects the underlying transformation. We note that there are still only three parameters involved in describing this (SO(3)) transformation.

      The nine components of T are reducible, i.e. they can be expressed as linear combinations that form subsets which transform among themselves under rotations. They are:

      U1V1+U2V2+U3V3=U⃗·V⃗≔T,(1.81)

      12(U2V3−U3V2)≔A1,(1.82)

      12(U3V1−U1V3)≔A2,(1.83)

      12(U1V2−U2V1)≔A3,(1.84)

      i.e.

      12(UiVj−UjVi)≔Akεijk.(1.85)

      Further,

      12(U1V2+U2V1)≔S12,(1.86)

      12(U2V3+U3V2)≔S23,(1.87)

      12(U3V1+U1V3)≔S31,(1.88)

      U1V1−13T≔S11,(1.89)

      U2V2−13T≔S22,(1.90)

      i.e.

      12(UiVj+UjVi)−13Tδij=Sij.(1.91)

      The combination U3V3−13T is excluded because

U3V3−13T=−U1V1−13T+U2V2−13T.(1.92)

      T is a scalar product which is invariant under rotations. The A_k are the three independent components of an antisymmetric tensor: by antisymmetric we mean that they change sign under exchange of the indices. The S_ij are the five independent components of a traceless second-rank tensor. The trace, ∑i=13Sii is evidently zero from equation (1.92). Note that

      UiVj=U⃗·V⃗3δij+(UiVj−UjVi)2+UiVj+UjVi2−U⃗·V⃗3δij,(1.93)

      i.e.

      UiVj=T3δij+Akεijk+δij.(1.94)

      Consider a rotation around the z-axis through an angle γ:

      cosγ−sinγ0sinγcosγ0001U1U2U3=U1′U2′U3′,(1.95)

      whence

      U1′=U1cosγ−U2sinγ,U2′=U1sinγ+U2cosγ,U3′=U3;(1.96)

      and similarly

      V1′=V1cosγ−V2sinγ,V2′=V1sinγ+V2cosγ,V3′=V3.(1.97)

      Then for the A_k,

      A1′=12U2′V3′−U3′V2′=12(U1sinγ+U2cosγ)V3−U3(V1sinγ+V2cosγ)=A1cosγ−A2sinγ;(1.98)

      similarly

      A2′=A1sinγ+A2cosγ(1.99)

      and

      A3′=A3,(1.100)

      i.e.

      cosγ−sinγ0sinγcosγ0001A1A2A3=A1′A2′A3′.(1.101)

      Evidently, we can write

      A⃗=12U⃗×V⃗.(1.102)

      Further, for the S_ij,

      S12′=12(U1cosγ−U2sinγ)(V1sinγ+V2cosγ)+(U1sinγ+U2cosγ)(V1cosγ−V2sinγ)=(U1V1−U2V2)sinγcosγ+12(U1V2+U2V1)(cos2γ−sin2γ)=(S11−S22)sinγcosγ+S12(cos2γ−sin2γ)=12(S11−S22)sin2γ+S12cos2γ;(1.103)

      similarly,

      S23′=S23cosγ+S31sinγ,(1.104)

      S31′=−S23sinγ+S31cosγ,(1.105)

      S11′=−S12sin2γ+12S11(1+cos2γ)+12S22(1−cos2γ),(1.106)

      and

      S22′=S12sin2γ+12S11(1−cos2γ)+12S22(1+cos2γ),(1.107)

      i.e.

      cos2γ0012sin2γ−12sin2γ0cosγsinγ000−sinγcosγ00−sin2γ0012(1+cos2γ)12(1−cos2γ)sin2γ0012(1−cos2γ)12(1+cos2γ)S12S23S31S11S22=S12′S23′S31′S11′S22′.(1.108)

      Similar equations can be obtained for S12,S23,S31,S11,S22 for rotations about the x and y axes.

      The {Ak}, T, and the {Sij} transform separately under rotations. Dyadics are said to possess a reducible structure with respect to rotations.

1.7 The Schwinger representations for SU(2)

      Representations of SU(2) can be constructed using a method due to Schwinger. Consider

      j=12,m=12≔a+†∣0〉,j=12,m=−12≔a−†∣0〉;(1.109)

      i.e. a+† creates a state (particle in a state) of spin-12 up and a−† creates a state of spin-12 down by action on the ‘vacuum’ ∣0〉 (which has no particles), where

      ai,aj†=δij,{i,j}={+,−}.(1.110)

      Then, defining

Jˆ+≔a+†a−,Jˆ−≔a−†a+,Jˆ0≔a+†a+−a−†a−2,(1.111)

      it follows that

      [Jˆ0,Jˆ±]=±Jˆ±,(1.112)

      [Jˆ+,Jˆ−]=2Jˆ0,(1.113)

      which define the structure developed for angular momentum and spin (here, ℏ≡1).

      Although the elementary building blocks in the Schwinger representation have spin-12, they should not be regarded as fermions. These spin-12 ‘objects’ are designed to be combined to produce any desired value of total spin: the number of spin-12’s needed to produce СКАЧАТЬ