Tensors for Physics

Appendix: Exercises... 411

13.2 Verify a Relation Peculiar for Spin 1 /2 (p.243)
Fo r s p i n s= 1 /2,the peculiar relation

sμsν=

i

2

εμνλsλ+

1

4

δμν

holds true. To prove it, start fromsμsν =0,for s= 1 /2,and use the commutation
relation.

For spins= 1 /2, one has, by definition,

2 sμsν =sμsν+sνsμ−

1

2

δμν.

The general commutation relation (13.1) impliessνsμ=sμsν−iεμμλsλ. Then, from

sμsν =0, the desired relation for spin 1/2 is obtained.

Exercise Chapter 14

14.1 Scalar Product of two Rotated Vectors(p.262)
Leta ̃μ=Rμν(φ)aνandb ̃μ=Rμκ(φ)aκbe the cartesian components of the vectors
aandbwhich have been rotated by the same angleφabout the same axis. Prove
that the scalar productsa ̃·b ̃is equal toa·b.

Application of (14.15) leads to

a ̃μb ̃μ=

∑^1

m=− 1

∑^1

m′=− 1

exp[imφ]exp[im′φ]Pμν(m)P(m

′)

μκ aνbκ.

Due toPμν(m) = Pνμ(−m),cf.(14.9), and with the help of the orthogonality rela-

tionPνμ(−m)P(m

′)

μκ =δ−mm

′

P(m

′)
νκ the exponential functions cancel each other. The
completeness relation

∑

mP

(m)
νκ =δνκ,cf.(14.10), then impliesa ̃μb ̃μ=aνbν,as
expected.

Exercises Chapter 15

15.1 Derivation of the Landau-de Gennes Potential(p.282)
In general, the free energy F is related to the internal energy U and the entropy S
by F=U−TS.Thus the contributions to these thermodynamic functions which
are associated with the alignment obey the relation

Fa=Ua−TSa.