The Foundations of Quantum Theory by Sol Wieder (Auth.)

By Sol Wieder (Auth.)

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By Sol Wieder (Auth.)

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K = j k = 0. If the space is linear, then an arbitrary vector may be represented as a linear combination of the basis vectors, that is, a = ax \ 4- ay) + azk. (3-1) The quantities ax, ay, and az are called the components of the vector a in the basis i, j , k. It should be stressed that while a is unique, ax, ay, and az are arbitrary varying with the basis used. Using the properties of the basis vectors we find ax = a · i, ay = a · j , and az = a · k. In another basis the components have the form ax = a · i', ay = a · j ' , and The length of the vector is defined by 2 |a| = {a 2 + a„ + α ζ ψ 2 az = a · k'.

These rules conform to the well-known results for matrix multiplication. The simplest operator is the unit or identity operator Î , which has the property î | * > = \à> for all vectors of the space. / if it exists,î is defined by the operation = Î. (3-23) In effect, the inverse A' undoes the original operation A. The adjoint of an operator A^ may be defined in a variety of ways. The X An operator whose determinant [see (3-29)] vanishes is said to be singular and does not have an inverse. 50 3 TH E F O R M A L I S M O F Q U A N T U M M E C H A N I C S definition use d her e define s b y th e relatio n \b} = A\a} J ...

It will become clear that there are Ν linearly independent vectors which satisfy (3-31). The vectors \aty and the scalars at are termed respectively the eigenvectors (or eigenkets) and eigenvalues of the operator A. The prefix " eigen " means " self" or " own " in German. Hence these quanti­ ties are self or characteristic vectors and values of the operator. The set of eigenvalues is called the spectrum of the operator. Since (3-31) does not restrict the magnitudes of the eigenvectors, it is convenient to normalize them, that is, to require < α (| β>ί = 1.

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