A Linear Vector Space

A linear vector space (LVS) is a mathematical system where the following operations of a vector addition and a scalar multiplication are defined with a scalar 

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A set of complete orthonormal functions ipfx) of a single variable x may be regarded as the basis vectors of a linear vector space of either finite or infinite dimensions, depending on whether the complete set contains a finite or infinite number of members. The situation is analogous to three-dimensional cartesian space formed by three orthogonal unit vectors. In quantum mechanics we usually (see Section 7.2 for an exception) encounter complete sets with an infinite number of members and, therefore, are usually concerned with linear vector spaces of infinite dimensionality. Such a linear vector space is called a Hilbert space. The functions ffx) used as the basis vectors may constitute a discrete set or a continuous set. While a vector space composed of a discrete set of basis vectors is easier to visualize (even if the space is of infinite dimensionality) than one composed of a continuous set, there is no mathematical reason to exclude continuous basis vectors from the concept of Hilbert space. In Dirac notation, the basis vectors in Hilbert space are called ket vectors or just kets and are represented by the symbol tpi) or sometimes simply by /). These ket vectors determine a ket space. 

The concept of quantum states is the basic element of quantum mechanics the set of quantum states I > and the field of complex numbers, C, define a Hilbert space as being a linear vector space the mapping < P P> introduces the dual conjugate space (bra-space) to the ket-space the number C(T )=< I P> is a... 

The definition of a mathematical space begins with the set of objects X, Y, Z,. .. that occupy the space (an intrinsically empty space being a physically problematic concept). Among the simplest algebraic structures that can characterize such objects is that of a linear manifold, also called a linear vector space, affine space, etc. By definition, such a manifold has only two operations— addition (X + Y) and multiplication by a scalar (AX)— resulting in each case in another element of the manifold. These operations have the usual distributive,... 

Exercise. The objects (1.4) form a linear vector space. Let the scalar product be defined with a weight function 1/s , so that (1.5) is the scalar product (A, Q). Write (1.3) and (1.7) as scalar products. 

The occupation number vectors are basis vectors in an m-dimensional abstract linear vector space, the Fock space, F(m). For a given spin orbital basis, there is a one-to-one mapping between a Slater determinant and an occupation number vector in the Fock space. The occupation number vectors are not Slater determinants they do not have any spatial structure, they are just basis vectors in a linear vector space. Much of the terminology which is used for Slater determinants is, however, used for occupation number m... 

The dependence of the used orbital basis is opposite in first and second quantization. In first quantization, the Slater determinants depend on the orbital basis and the operators are independent of the orbital basis. In the second quantization formalism, the occupation number vectors are basis vectors in a linear vector space and contain no reference to the orbitals basis. The reference to the orbital basis is made in the operators. The fact that the second quantization operators are projections on the orbital basis means that a second quantization operator times an occupation number vector is a new vector in the Fock space. In first quantization an operator times a Slater determinant can normally not be expanded as a sum of Slater determinants. In first quantization we work directly with matrix elements. The second quantization formalism represents operators and wave functions in a symmetric way both are expressed in terms of elementary operators. This... 

Theorem 1. 2lT, the totality of molecular species formed from the atomic species, . l38T, constitutes a linear vector space of dimension t over the... 

These properties define a linear vector space over the integers. Also since 98x,...,9Bj belong to SlT and are independent, they form a basis for the space which is therefore of dimension t. ... 

Theorem 3. The totality of reactions r over a given set of species s/s, s = 1, 2,..., s, forms a linear vector space 93s of dimension s over the field of real numbers. 

A physical system, which may be part of a larger system, is associated with a linear vector space whose elements are ket vectors... 

A linear vector. space is a set L containing elements (vectors) which can be related by two operations, addition and scalar multiplication, satisfying the conditions... 

Definition 38 A linear subspace of L is a subset of L that forms a linear vector space under the rules of addition and scalar multiplication defined for L. 

Thus we can see that the linear vector space contains another very important property of the Euclidean space it has a basis. However, there is no distance in a linear vector space. It would be extremely useful if we could combine these two properties of the Euclidean space, a distance and a basis, within one space. This space is called a normed linear space. 

Let us introduce a linear vector space in which there is defined for every pair of elements f, g a functional, the inner product (f, g), with the properties... 

Hilbert space A linear vector space that can have an infinite number of dimensions. The concept is of interest in physics because the state of a system in quantum mechanics is represented by a vector in Hilbert space. The dimension of the Hilbert space has nothing to do with the physical dimension of the system. The Hilbert space formulation of quantum mechanics was put forward by the Hungarian-born US mathematician John von Neumann (1903-57) in 1927. Other formulations of quantum mechanics, such as matrix mechanics and wave mechanics, can be deduced from the Hilbert space formulation. Hilbert space is named after the German mathematician David Hilbert (1862-1943), who Invented the concept early in the 20th century. 

In the first chapter, we saw that if we wanted to rotate the 2px function, we automatically also needed its companion 2py function. If this is extended to out-of-plane rotations, the 2/ function will also be needed. The set of the three p-orbitals forms a prime example of what is called a linear vector space. In general, this is a space that consists of components that can be combined linearly using real or complex numbers as coefficients. An n-dimensional linear vector space consists of a set of n vectors that are linearly independent. The components or basis vectors will be denoted as fi, with I ranging from 1 to n. At this point we shall introduce the Dirac notation [1] and rewrite these functions as / >, which characterizes them as so-called kef-functions. Whenever we have such a set of vectors, we can set up a complementary set of so-called fera-functions, denoted as /t I The scalar product of a bra and a ket yields a number. It is denoted as the bracket fk fi). In other words, when a bra collides with a ket on its right, it yields a scalar number. A bra-vector is completely defined when its scalar product with every ket-vector of the vector space is given. 

The solution of the electronic Schrodinger equation, Eq. (8.76), requires the integration of a complicated partial differential equation depending on 3N electronic coordinates as variables. The complexity of this problem increases with the number of electrons in the molecule. In order to establish a general solution strategy it is mandatory to first study the underlying formal framework of many-particle quantum mechanics, namely that of a linear vector space — the Hilbert space (cf. section 4.1). The N-electron Hilbert space, which hosts the total quantum mechanical state vector, is then constructed by direct multiplication of the one-electron Hilbert spaces. 

Strictly speaking, these vectors should be called geometric vectors since they do not, in all cases, satisfy the properties of algebraic vectors (e.g., algebraic vectors satisfy the axioms of a linear vector space, namely, the addition of two vectors or the multiplication of a vector by a scalar should result in another vector that also lies in the space). Nevertheless, the terminology vector, which is common in chemical informatics, will be used here to include both classes of vectors. 

Besides the apparent similarities. Table 8.1 illustrates also the obvious formal differences between bras and kets and their second quantized counterparts. Namely, the corresponding symbols are mathematically very different. The bra and ket vectors are elements of a linear vector space over which quantum-mechanical operators are defined, while the creation and annihilation operators are defined over the abstract space of particle number represented wave functions serving as their carrier space. This carrier space leads to the concept of the vacuum state, which has no analog in the bra-ket formalism. Moreover, an essential difference is that the effect of second quantized operators depends on the occupancies of the one-electron levels in the wave function, since no annihilation is possible from an empty level and no electron can be created on an occupied spinorbital. At the same time, the occupancies of orbitals play no role in evaluating bra and ket expressions. Of course, both formalisms yield identical results after calculating the values of matrix elements. 

The kind of approximation to be employed is analogous to the truncation of a basis in a linear vector space V truncation to n elements produces a subspace Y , and the truncation of a general vector corresponds to finding its projection onto the subspace. We are therefore concerned with the projection operator, p say, associated with a truncated basis and if the basis (e, say) is orthonormal we may conveniently write... 

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