Tensors for Physics

230 12 Integral Formulae and Distribution Functions withx=exνrνandy=e y νrν, whereexandeyare unit vectors parallel to thex-axis ...

12.4 Anisotropic Pair Correlation Function and Static Structure Factor 231 12.4.7 Cubic Symmetry. When a crystal melts, its high ...

232 12 Integral Formulae and Distribution Functions Fig. 12.3 The cubic pair correlation functiong 4 , variablesr^2 andtare as i ...

12.4 Anisotropic Pair Correlation Function and Static Structure Factor 233 For=2 this relation implies, see also (12.1), Sμν= 1 ...

234 12 Integral Formulae and Distribution Functions The shear flow induced distortion of the pair correlation function implies a ...

12.5 Selection Rules for Electromagnetic Radiation 235 The normalization condition ∫ |Ψ|^2 d^3 r=1 implies ∑∞ = 0 |c|^2 = 1 , ...

236 12 Integral Formulae and Distribution Functions 12.5.2 Electric Dipole Transitions. Electromagnetic waves, induce transition ...

12.5 Selection Rules for Electromagnetic Radiation 237 radiation. According to (12.140) the angle dependent part of the=1 wave ...

238 12 Integral Formulae and Distribution Functions Multiplication of (12.144)byΨ∗′and subsequent integration overd^3 ryields n ...

Chapter 13 Spin Operators Abstract Spin operators are introduced in this chapter. The spin 1/2 and 1 are looked upon explicitly. ...

240 13 Spin Operators Here,[.., ..]−indicates the commutator. Much as (7.87) implies the relation (7.88), the spin commutation r ...

13.2 Magnetic Sub-states 241 13.2 Magnetic Sub-states 13.2.1 Magnetic Quantum Numbers and Hamilton Cayley The spin operator poss ...

242 13 Spin Operators These projectors have the properties P(m)P(m ′) =δmm′P(m), ∑s m=−s P(m)= 1. (13.9) Thus they are orthogona ...

13.3 Irreducible Spin Tensors 243 As stated before, for a spins, the irreducible tensors of ranks≥ 2 s+1 vanish. For = 2 s+1, ...

244 13 Spin Operators In theHeisenberg picture, the time dependence of an operatorOis governed by the commutator with the releva ...

13.3 Irreducible Spin Tensors 245 For=2, (13.21) corresponds to sμsν sμsν = 2 3 S^20 S^21 , S^20 =s(s+ 1 ), S 12 =S^20 − 3 4 . ...

246 13 Spin Operators By analogy to symmetry arguments which lead to (12.1), the trace of the product of two irreducible spin te ...

13.4 Spin Traces 247 13.4.3 Multiple Products of Spin Tensors The trace of the fourfold product of the spin is similar to the cl ...

248 13 Spin Operators The density operator can be expressed as a linear combination of the projection operatorsP(m), introduced ...

13.5 Density Operator 249 ρ(s)=ρ 0 ( 1 +Φ), ρ 0 =( 2 s+ 1 )−^1 ,Φ= ∑2s = 1 bμ 1 μ 2 ···μsμ 1 sμ 2 ···sμ. (13.42) Clearly,Φis ...