Chapter 1
Mathematical Tools of Quantum Mechanics
Today quantum mechanics forms an important part of our understanding of physical phenom- ena. Its consequences both at the fundamental and practical levels have intrigued mathemati- cians， physicists， chemists， and even philosophers for the past century. A quantum system is usually described in terms of certain Hilbert spaces H and linear operators acting on these spaces. The mathematical properties and structure of Hilbert spaces are essential for a proper understanding of the formalism of quantum mechanics. For this， we are going to review brie°y the properties of Hilbert spaces and those of linear operators. We will then consider Dirac''s bra-ket notation.
Quantum mechanics was formulated in two di.erent ways by Schr.odinger and Heisenberg. Schr.odinger''s wave mechanics and Heisenberg''s matrix mechanics are the representations of the general formalism of quantum mechanics in continuous and discrete basis systems， respectively.So we will also examine the mathematics involved in representing kets， bras， bra-kets， and operators in discrete and continuous bases.
Certain mathematical topics are essential for quantum mechanics， not only as computational tools， but because they form the most e.ective language in terms of which the theory can be formulated. We deal with the mathematical machinery needed to study quantum mechanics in this chapter. Although it is mathematical in scope， no attempt is made to be mathematically complete or rigorous. We limit ourselves to those practical issues that are relevant to the formalism of quantum mechanics. These topics include the theory of linear vector spaces and linear operators. A uniˉed theory based on that mathematical structure was ˉrst formulated by P. A. M. Dirac， and the formulation used in this book is really a modernized version ofDirac''s formalism.
The physical development of quantum mechanics begins in the Chapt.2， and the mathemat- ically sophisticated reader may turn there at once. But since not only the results， but also the concepts and logical framework of this chapter are freely used in developing the physical theory， the reader is advised to at least skim this ˉrst chapter before proceeding to next chapter.
1.1 The Hilbert Space
A linear vector space consists of two sets of elements and two algebraic rules:
1 A set of vectors .; á; .; ￠ ￠ ￠ and a set of scalars a; b; c; ￠ ￠ ￠ ， if the scalars belong to the ˉeld of complex real numbers， we speak of a complex real linear vector space. Henceforth the scalars will be complex numbers unless otherwise stated.
2 A rule for vector addition and a rule for scalar multiplication.
1. Addition rule
The addition rule has the properties and structure of an Abelian groups.
1 If . and á are vectors elements of a space， their sum . + á， is also a vector of the same space.
2 Commutativity: . + á = á +
3 Associativity: . + á + . = á + . + ..
4 Existence of a zero or neutral vector: for each vector .， there must exist a zero vector such that
5 Existence of a symmetric or inverse vector: for each vector .， there must exist a sym- metric vector á such that . + á = #. We write á as later.
2. Multiplication rule
The multiplication rule of vectors by scalars scalars can be real or complex numbers has these properties.
1 The product of a scalar gives another vector. In general， if . and á are two vector of the space， any linear combination a. + bá is also a vector of the space， a and b being scalars.
2 Distributivity with respect to addition:8
a. + á = a. + aá;
a + b. = a. + b.:
3 Associativity with respect to multiplication of scalars: ab. = ab
4 For each element . there must exist a unitary element， 1， and zero， 0， scalar such that: 1 ￠ . = . ￠ 1 = .; 0 ￠ . = . ￠ 0 =
1.1 The Hilbert Space 3
Examples
Among the very many examples of linear vector spaces， there are two classes that are of common
interest:
1 Discrete vectors， which may be represented as columns of complex numbers， a1; a2; ￠ ￠ ￠ T.
2 Spaces of functions of some type， for example the space of all di.erentiable functions. One can readily verify that these examples satisfy the deˉnition of a linear vector space.
1.1.1 Deˉnition of Hilbert Space
A Hilbert space H consists of a set of vectors .; á; .; ￠ ￠ ￠ and a set of scalars a; b; ; ￠ ￠ ￠
which satisfy the following four properties.
1. H is a linear space
The properties of a linear space were considered in the previous section.
2. H has a deˉned scalar inner product that is strict positive
The scalar product of an element . with another element á is scalar， a complex number， denoted by .; á=complex number. The scalar product satisˉed the following properties.
1 The scalar product of . with á is equal to the complex conjugate of the scalar product of á with .: .; á = á; .¤.
2 The scalar product of . with respect to aá1 + bá2 is
.; aá1 + bá2 = a.; á1 + b.;
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