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Introduction A Euclidean group consists of 2 types of transformations: Uniform translations T(b). Uniform rotations R n ( ). Goals: Introduce techniques for IRs of non-compact groups. Pave way for Lorentz & Poincare groups. Bessel functions. Definition 9.1: Euclidean Groups E n E n consists of all continuous linear transformations on the n–D Euclidean space R n which leave the length of all vectors invariant. 9. Euclidean Groups in 2- & 3-D Space

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General linear transformation:with Length l of vector from x to y : Point in R n : Length preserving :( R orthogonal ) E n = Group of motion in R n Homogeneity of space → V contains only relative coordinates of particles. → Conservation of total linear momentum. Isotropy of space → V is invariant under rotations. 2-particle system: V = V(r) General system → Conservation of total angular momentum.

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9. Euclidean Groups in 2- & 3-D Space 9.1 The Euclidean Group in Two-Dimensional Space 9.2 Unitary Irreducible Representations of E 2 — the Angular- Momentum Basis 9.3 The Induced Representation Method and the Plane-Wave Basis 9.4 Differential Equations, Recursion Formulas, and Addition Theorem of the Bessel Function 9.5 Group Contraction — SO(3) and E 2 9.6 The Euclidean Group in Three Dimensions 9.7 Unitary Irreducible Representations of E 3 by the Induced Representation Method 9.8 Angular Momentum Basis and the Spherical Bessel Function

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9.1. The Euclidean Group in 2-D Space

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