QuantumPhysics.dvi

groupSO(n). The element−Ialways belongs toO(n), and belongs toSO(n) forneven,

but not fornodd. Therefore, one may define a parity transformationPfor allnas reflecting

only the last coordinate, leaving the others unchanged. AnyM with detM=−1 may then

be decomposed asM=PR, whereRis a proper rotation.

All rotations inSO(3) may be parametrized by 3 real parameters. A convenient way

of choosing those is to pick a direction around which to rotate by specifying a unit vector

n= (n 1 ,n 2 ,n 3 ), and then further specify the angleωby which to rotate aroundn. One may

write down the rotation matrix explicitly,

R(n,ω) = exp

{

ωn·T

}

(8.4)

where the matricesT 1 ,T 2 ,T 3 are given by

T 1 =







0 0 0

0 0 − 1

0 1 0





 T 2 =







0 0 1

0 0 0

−1 0 0





 T 3 =







0 −1 0

1 0 0

0 0 0





 (8.5)

The matricesTare real antisymmetric. The rotationR(n,ω) applied to an arbitrary vector

vis given by

R(n,ω)v = vcosω+v(n·v) (1−cosω) +n×vsinω

= v+ωn×v+O(ω^2 ) (8.6)

Note that, because the matricesT 1 ,T 2 ,T 3 are traceless, we automatically have detR(v,ω) = 1

for allv,ω. Thus, the operation of space parity reversalR=−Icannot be represented by

the above exponential parametrization. The reason is that the element with detR= 1 and

detR=−1 are disconnected from one another, and not continuously connected by rotations.

The matricesT 1 ,T 2 ,T 3 satisfy the following commutation relations,

[T 1 ,T 2 ] =T 3 [T 2 ,T 3 ] =T 1 [T 3 ,T 1 ] =T 2 (8.7)

It is convenient to summarize these relations by using the totally antisymmetric tensorεabc,

[Ta,Tb] =

∑^3

c=1

εabcTc ε 123 = 1 (8.8)

wherea,b,c= 1, 2 ,3. It is more usual to work with Hermitean or self-adjoint operators, and

to include also a unit of ̄hfor proper units of angular momentum,

[La,Lb] =i ̄h

∑^3

c=1

εabcLc La=i ̄hTa (8.9)

and the rotations are now represented by

R(n,ω) = exp

{

−

i

h ̄

ωn·L

}

(8.10)

It is in this form that we shall most often use rotations.