wang
(Wang)
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8 Theory of Angular Momentum
In a subsequent chapter, we shall study symmetry transformations in quantum systems in
a systematic manner. In this chapter, we shall concentrate on rotations, which is one of
the most important cases, both practically and conceptually, and work out the addition of
angular momentum.
8.1 Rotations
Rotations in 3-dimensional real space may be defined as follows. LetXbe the column matrix
of the real coordinatesx 1 ,x 2 ,x 3 of 3-dimensional space. The length squaredℓ^2 (X), or norm,
ofXis defined by,
ℓ^2 (X) =x^21 +x^22 +x^23 =XtX (8.1)
Anorthogonal transformation is defined to be a real linear mapX →X′ =MX which
preserves the length of every vectorX. This requires (X′)t(X′) = (MX)t(MX) =XtX, for
allX, or equivalently
MtM=I (8.2)
The space of allMforms a group under multiplication, which is denoted byO(3).
Actually, the groupO(3) consists of two disconnected components. To see this, we take
the determinant ofMtM =I, which yields detM =±1. The component with detM = 1
corresponds torotations, which we shall henceforth denote by the letterRinstead ofM.
Thus, the group of rotationsRin 3-dimensional space is defined by,
RtR=I detR= 1 (8.3)
and is usually referred to asSO(3). TheOstands for orthogonal (expressedRtR=I), while
theSindicates the determinant condition. The elementM =−Ibelongs toO(3), but not
toSO(3), since det(−I) =−1. Thus,M =−I is NOT a rotation. Instead, it is a space
parity transformation, usually denoted byP. As a result of the multiplicative property of
the determinant, any elementM ofO(3) with detM =−1 may be expressed asM=−R
whereRis a rotation.
Rotations inn-dimensional real space may be defined analogously. LetXbe the column
matrix of the real coordinatesx 1 ,x 2 ,···,xn, andℓ^2 (X) its norm, defined byℓ^2 (X) =XtX.
Orthogonal transformations are defined as the linear mapsX → X′ = MX which leave
ℓ^2 (X) invariant for allX, and form a groupO(n) ofn×nmatricesMsatisfyingMtM=I.
RotationsRinn-dimensional space correspond to the component detR= 1, and form the
