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angular momentum operator

angular momentum operator - Angular Momentum Operator I...

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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 0 = ( V 0 x V 0 y V 0 z ) T , where V 0 x V 0 y V 0 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 0 ( r ) where ψ is a function of ( r ). Obviously, we shall have ψ 0 ( r + δ r ) = ψ ( r ), or ψ 0 ( 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 0 sin ε cos ε 0 0 0 1 1 - ε 2 / 2 - ε 0 ε 1 - ε 2 / 2 0 0 0 1
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2 where we have kept terms up to second order in ε . By a cyclic substitution, we have R ε x = 1 0 0 0 cos ε - sin ε 0 sin ε cos ε 1 0 0 0 1 - ε 2 / 2 - ε 0 ε 1 - ε 2 / 2 R ε y = cos ε 0 sin ε 0 1 0 - sin ε 0 cos ε 1 - ε 2 / 2 0 ε 0 1 0 - ε 0 1 - ε 2 / 2 from which we have R ε x R ε y - R ε y R ε x = 0 - ε 2 0 ε 2 0 0 0 0 0 = 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 a minus sign: ˆ D ˆ n (2 π ) | ψ i = -| ψ i An arbitrary rotation can be represented by ˆ D ˆ n ( φ ) = e - i ˆ S · ˆ n φ/ ~ = e - i σ · ˆ n φ/ 2 = cos φ 2 - i ( σ · ˆ n ) sin φ 2 In terms of Euler angles, we have under the S z -representation ˆ D (
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