GROUP THEORY TUTORIAL

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

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Verify commutation relations.

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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

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Define a state vector

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Verify action of J₊ and normalization.

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Verify action of J₊ as a raising operator:

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Verify action of J_- as a lowering operator:

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Verify commutation relations

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Verify relation to J_x and J_y

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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.

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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)

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