Rotations, Quaternions, and Double Groups
This self-contained text presents a consistent description of the geometric and quaternionic treatment of rotation operators, employing methods that lead to a rigorous formulation and offering complete solutions to many illustrative problems.
Geared toward upper-level undergraduates and graduate students, the book begins with chapters covering the fundamentals of symmetries, matrices, and groups, and it presents a primer on rotations and rotation matrices. Subsequent chapters explore rotations and angular momentum, tensor bases, the bilinear transformation, projective representations, and the geometry, topology, and algebra of rotations. Some familiarity with the basics of group theory is assumed, but the text assists students in developing the requisite mathematical tools as necessary.
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algebra angular momentum axes bases basis bilinear transformation binary rotation called Chapter column commute complex numbers components conﬁguration space conjugate corresponding deﬁned deﬁnition denote double group eigenvalues eigenvectors entails equal equation Euler angles Euler—Rodrigues parameters factor system ﬁgure ﬁnd ﬁrst ﬁxed follows functions geometrical given Hamilton Hermitian identiﬁed identity imaginary unit improper rotations inﬁnitesimal rotations introduced invariant inversion irreducible representations mapping mathematical multiplication rules multiplication table negative normal notation Notice obtained operation g orthogonal parametric ball parametric point path Pauli matrices phase factors plane point groups pole Problem projective factors projective representations proper rotations Prove pseudovector pure quaternion quantum mechanics quaternion parameters quaternion units reﬂection representation of 80(3 respectively result Rodrigues rotation angle rotation group rotation matrix satisﬁed scalar sin2 spherical spinor representations standard subgroup symbol symmetry operation tensor theorem theory tion transformation unit sphere unitary unitary matrix vector representations