Tensors for Physics

250 13 Spin Operators operators given in (13.11). Thus the relative difference of the occupation numbers is a measure for the de ...

13.6 Rotational Angular Momentum of Linear Molecules, Tensor Operators 251 is perpendicular tou. This also applies to the rotati ...

252 13 Spin Operators ThediagonalpartofO(u)canbeexpressedasafunctionoftheangularmomentum operator, viz. ∑ j PjO(u)Pj ′ =O(J). (1 ...

13.6 Rotational Angular Momentum of Linear Molecules, Tensor Operators 253 are used. Thus the diagonal part(umuuν)diagofuμuνis g ...

254 13 Spin Operators Notice that tr{Pj}= 2 j+ 1. The summation overjhas to be taken with the allowed values, which may be all i ...

13.6 Rotational Angular Momentum of Linear Molecules, Tensor Operators 255 13.6.5 Anisotropic Dielectric Tensor of a Gas of Rota ...

256 13 Spin Operators ∑ m 1 ∑ m 2 |j 1 m 1 〉|j 2 m 2 〉(j 1 m 1 ,j 2 m 2 |jm)=|jm〉. The nondiagonal elements of the spherical har ...

13.6 Rotational Angular Momentum of Linear Molecules, Tensor Operators 257 Both coefficients approach^12 √ 3 2 for large values ...

Chapter 14 Rotation of Tensors Abstract This chapter is concerned with the active rotation of tensors. Firstly, infinitesimal an ...

260 14 Rotation of Tensors The rotation by an finite angleφ=nδφis given by( 1 +δφH)n. Withδφ=φ/n, the limitn→∞leads to a′μ=(exp[ ...

14.1 Rotation of Vectors 261 reflects that thePμν(m)are ‘eigen-tensors’ of the tensorHμν. On the other hand,Hμν can be represent ...

262 14 Rotation of Tensors In hindsight, this result is not unexpected for the rotation of a vector. However, the formal conside ...

14.2 Rotation of Second Rank Tensors 263 The fourth rank tensorH is defined by Hμν,μ′ν′≡εμλμ′hλδνν′+ενλν′hλδμμ′=Hμμ′δνν′+Hνν′δμμ ...

264 14 Rotation of Tensors Notice thatm 1 +m 2 assumes the five valuesm=− 2 ,− 1 , 0 , 1 ,2. The fourth rank tensor obeys the ei ...

14.3 Rotation of Tensors of Rank 265 14.3 Rotation of Tensors of Rank‘....................... The obvious generalization of the ...

266 14 Rotation of Tensors and R···(),···(φ)=P···(^0 ),···+ ∑ m= 1 [ (cos(mφ) ( P···(m,)···+P···(−,m···) ) +sin(mφ)i ( P(···m, ...

14.4 Solution of Tensor Equations 267 A simple application, for=1, is the computation of the electrical conductivity in the pre ...

268 14 Rotation of Tensors with the longitudinal, perpendicular and transverse conductivity coefficients determined by σ‖=σ 0 ≡n ...

14.5 Additional Formulas Involving Projectors 269 bμνPμν,μ(±^2 )′ν′aμ′ν′= 1 2 bμνaμν−hμbμνaνκhκ+ 1 4 (hμbμνhν)(hμ′aμ′ν′hν′) ∓ i ...

270 14 Rotation of Tensors ( Pμν,μ(^2 ) ′ν′+P(μν,μ−^2 )′ν′ ) eμ′uν′ =eμuν + 1 2 hμhν (h·e)(h·u) −[hμeν(h·u)+hμuν(h·e)], 2 i ( P( ...