Vector operators, 50 algebra properties

This is as far as we can go with the representation of vector operators without requiring further properties of V in order to obtain the explicit form of the coefficients a and c. The additional properties we shall use are the commutation relations needed to make the six components of J and V into the Lie algebra so(4). 

The mathematical formalism jofitjuantum mechanics is expressed in terms of linear operators, which rep resent the observables of a system, acting on a state vector which is a linear superposition of elements of an infinitedimensional linear vector space called Hilbert space. We require a knowledge of just the basic properties and consequences of the underlying linear algebra, using mostly those postulates and results that have direct physical consequences. Each state of a quantum dynamical system is exhaustively characterized by a state vector denoted by the symbol T >. This vector and its complex conjugate vector Hilbert space. The product clT ), where c is a number which may be complex, describes the same state. 

Anyone who has studied abstract algebra may recognize a way to think about (A8.5) that has attractive mathematical elegance. Think of Ai and as the Cartesian components of a vector Rj in a two-dimensional space . Then think of (A8.5) as embodying an operation T that transforms the vector Rj into the vector Rj +1. Finally think of the cyclic property of the system under study as requiring that six successive performances of the operation carry the vector to identity with the initial vector in other words, T6 = /, where / is the identity operation. Use these ideas to derive the results obtained by the use of finite-difference equations in the appendix. 

Let us now consider an arbitrary set on which the operations summation and scalar multiplication have been introduced, obeying the (so-called) axioms I-III. In the history of Algebra, it turned out that such sets represent a broad class of useful mathematical objects whose properties can be examined solely as consequences of the axioms. For historical reasons, the elements of any such set V are called vectors, and the set is called vector space. Thus... 

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