26 2Basics
Fig. 2.6 Parity operation:
r→−r
The parity operatorP, when applied on any functionf(r), yieldsf(−r):Pf(r)=f(−r). (2.52)Clearly, one hasPf(−r)=f(r)and consequently
P^2 = 1 , (2.53)or(P− 1 )(P+ 1 )=0. Thus the eigenvalues of the parity operator are
P=± 1. (2.54)Usually eigenfunctions are referred to as havingpositiveornegativeparity, when
P=1 andP=−1, respectively, applies.
2.6.2 Parity of Vectors and Tensors.
In most applications tensors, and this includes vectors, are eigenfunction of the parity
operator. Tensors of rankwith
P=(− 1 ) (2.55)are calledproper tensors, those with
P=−(− 1 )=(− 1 )+^1 (2.56)arereferredtoaspseudo tensors.
For vectors (=1), also the termspolar vectorandaxial vectorare used to
distinguish between proper and pseudo vectors. Examples for polar vectors are the
linearmomentumpofaparticleandtheelectricfield,whereastheangularmomentum
and the magnetic field are axial vectors, as will be discussed later.