# Group-Theory - trivial representation n = 1 example...

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Definition: group is a set with a product . identity with inverse with associative Example: symmetry operation of square h d non-abelian 1 Group theory

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subgroup: group subset of examples: number of elements: devides 2 Group theory
linear transformations: consider n -dimensional vector space transformations on by unitary n x n -matrices matrices satisfies all properties of a group mapping (homomorphism) of group on n x n -matrices in conserving group structure representation of equivalent representations: representation characters: basis transformation independent of basis 3 Group theory representations

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irreducible representation: independent of basis connects whole

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Unformatted text preview: trivial representation: n = 1 example: transformation of basis function 4 character table Group theory representations symmetry operations of Hamiltonian form a group Hilbertspace is vector space stationary states: and degenerate degenerate states form a vector space with an irred. representation of with dimension m of representation corresponds to the degeneracy of eigenvalues 5 Group theory representations & quantum mechanics symmetry lowering splitting of degeneracy through symmetry lowering A 1 A 2 B 1 B 2 E A 1 ‘ A 1 ‘ B 1 ‘ B 1 ‘ A 2 ‘ B 2 ‘ k x high symmetry point E A,B ε 6 Group theory representations & quantum mechanics...
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Group-Theory - trivial representation n = 1 example...

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