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Unformatted text preview: Angular Momentum Operator I. GENERATOR OF ROTATIONS In 3D real space, a vector is represented by a column matrix V = ( V x V y V z ) T . After a rotation, it is changed to another column matrix V = ( V x V y V z ) T , where V x V y V z = R V x V y V z where R is a 3 × 3 matrix representing the rotation. R must be orthogonal, i.e., R T R = 1. • The group of all orthogonal matrices in 3D is O (3). Orthogonal matrices have determinant equal to ± 1. • The group of all rotation matrices is SO (3) which is a subgroup of O (3) and determinant equal to 1. • Rotation matrix must possess an eigenvalue 1, the associated eigenvector points toward the axis of rotation. Consider an infinitesimal rotation in real space along the direction of a unit vector ˆ n by a small angle δφ . After rotation, we have r → r + δ r = r + δφ ˆ n × r , ψ ( r ) → ψ ( r ) where ψ is a function of ( r ). Obviously, we shall have ψ ( r + δ r ) = ψ ( r ), or ψ ( r ) = ψ ( r- δ r ) ≈ (1- δ r · ∇ ) ψ ( r ) = (1- δφ · r × ∇ ) ψ ( r ) = ˆ 1- i ˆ J · ˆ n ~ δφ ! ψ ( r ) where we have defined an angular momentum operator ˆ J ≡ - i ~ r × ∇ and we shall call ˆ D ˆ n ( δφ ) ≡ 1- i ˆ J · ˆ n ~ δφ an infinitesimal rotation operator. Once we have established the infinitesimal rotation, we can carry out a rotation by any finite angle φ by compounding an infinite number of infinitesimal rotations with δφ = lim N →∞ φ N The finite rotation ˆ D ˆ n ( φ ) is then given by ˆ D ˆ n ( φ ) = lim N →∞ ‡ ˆ D ˆ n ( δφ ) · N = e- i ˆ J · ˆ n φ/ ~ Therefore, angular momentum is the generator of rotation. It is easy to see that rotations do not commute, even for infinitesimal ones. Consider an infinitesimal rotation in real space along z-axis by an angle ε . The matrix describing this rotation is R ε z = cos ε- sin ε sin ε cos ε 1 ≈ 1- ε 2 / 2- ε ε 1- ε 2 / 2 0 1 2 where we have kept terms up to second order in ε . By a cyclic substitution, we have R ε x = 1 0 cos ε- sin ε 0 sin ε cos ε ≈ 1 0 1- ε 2 / 2- ε ε 1- ε 2 / 2 R ε y = cos ε 0 sin ε 1- sin ε 0 cos ε ≈ 1- ε 2 / 2 0 ε 1- ε 0 1- ε 2 / 2 from which we have R ε x R ε y- R ε y R ε x = - ε 2 ε 2 = R ε 2 z- 1 (1) Quantum mechanically, this means cartesian components of angular momentum operators do not commute. In fact, we have [ ˆ J i , ˆ J j ] = i ~ ² ijk ˆ J k However, [ ˆ J i , ˆ J 2 ] = 0 In terms of the Euler angles, an arbitray rotation can be characterized by three angles: ˆ D ( α,β,γ ) = ˆ D z ( α ) ˆ D y ( β ) ˆ D z ( γ ) II. SPIN-1/2 PARTICLE For spin-1/2 particle, the rotation operators are represented by 2 × 2 matrices which form the group SU (2) (i.e., unimodular unitary 2D matrices). A 2 π rotation along an arbitray axis changes an arbitrary spin state to itself with...
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