Angular Momentum Operations

GTPack [1,2] contains various modules to handle angular momentum operators and representations.

[1] W. Hergert, R. M. Geilhufe, Group Theory in Solid State Physics and Photonics: Problem Solving with Mathematica, Wiley-VCH, 2018 [2] R. M. Geilhufe, W. Hergert, GTPack: A Mathematica group theory package for applications in solid-state physics and photonics, Frontiers in Physics, 6:86, 2018

GTJx	gives the x component of the total angular momentum operator
GTJy	gives the y component of the total angular momentum operatorXXXX
GTJz	gives the z component of the total angular momentum operator

The components of the total angular momentum operator in terms of matrix representations acting on a finite sub space indexed by the total angular momentum quantum number J.

Components of the total angular momentum operator

Verify commutation relations.

GTJplus	gives the raising operator
GTJminus	gives the lowering operator

The raising operator acts as J₊|j; m> = |j; m+1>

The lowering operator acts as J_-|j; m> = |j; m-1>

J₊ and J_- are related to J_x and J_y:

J₊= J_x + i J_y

J_-= J_x - i J_y

Matrix representations

Define a state vector

Verify action of J₊ and normalization.

Verify action of J₊ as a raising operator:

Verify action of J_- as a lowering operator:

Verify commutation relations

Verify relation to J_x and J_y

GTJTransform	applies a symmetry transformation to the basis functions of an irreducible representation of O(3)
GTJMatrix	gives the representation matrix of a symmetry element for an irreducible representation of O(3)

GTJTransform and GTJMatrix are closely related. While GTJTransform gives the action of a symmetry element on one specific basis function, GTJMatrix gives the transformation matrix of the entire subspace.

GTAngularMomentumRep	applies a symmetry transformation to the basis functions of an irreducible representation of O(3)
GTAngularMomentumChars	gives the representation matrix of a symmetry element for an irreducible representation of O(3)

For the implementation of irreducible representations of O(3), SO(3) and SU(2) we follow [1].

[1] Altman, S. L., Rotations, quaternions, and double groups. Chapter 14. Clarendon, 1986

Install the point group O_h (GTInstallGroup)