Linear vector spaces

In terms of linear vector space, Buckingham s theorem (Theorem 2) simply states that the null space of the dimensional matrix has a fixed dimension, and Van Driest s rule (Theorem 3) then specifies the nullity of the dimensional matrix. The problem of finding a complete set of B-numbers is equivalent to that of computing a fundamental system of solutions of equation 13 called a complete set of B-vectors. For simplicity, the matrix formed by a complete set of B-vectors will be called a complete B-matrix. It can also be demonstrated that the choice of reference dimensions does not affect the B-numbers (22). 

A Lorentz invariant scalar product can be defined in the linear vector space formed by the positive energy solutions which makes this vector space into a Hilbert space. For two positive energy Klein-... 

This section introduces the basic mathematics of linear vector spaces as an alternative conceptual scheme for quantum-mechanical wave functions. The concept of vector spaces was developed before quantum mechanics, but Dirac applied it to wave functions and introduced a particularly useful and widely accepted notation. Much of the literature on quantum mechanics uses Dirac s ideas and notation. 

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 names vector space, linear vector space and linear space are synonyms. 

Note the similarity of Eqs. 2.43 and 2.44 with Eqs. 2.80 and 2.81 because both the vectors in the former equations and the functions of the latter are all elements of linear vector spaces. The main difference arises in the way in which the inner products are evaluated. Also, as was the case for vectors, if the field functions are non-negative functions, SCar(F, F pj will be non-negative. When this is not the case, however, Sr.ir(F (,F g) may become negative, a situation that also obtains for the other similarity indices discussed in the remainder of this section. Maggiora et al. (43) have treated this case in great detail for continuous field functions, but the arguments can be carried through for finite vectors as well (vide supra). 

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... 

R3, then sa and a +aj are also 3-dimensional vectors, and the vector space is closed under multiplication by scalars, and addition. This is the fundamental property of any linear vector space. Consider the vectors a and... 

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 occupation number vectors are basis vectors in an abstract linear vector space and specify thus only the occupation of the spin orbitals. The occupation number vectors contain no reference to the basis set. The reference to the basis set is built into the operators in the second quantization formalism. Observables are described by expectation values of operators and must be independent of the representation given to the operators and states. The matrix elements of a first quantization operator between two Slater determinants must therefore equal its counterpart of the second quantization formulation. For a given basis set the operators in the Fock space can thus be determined by requiring that the matrix elements between two occupation number vectors of the second quantization operator, must equal the matrix elements between the corresponding two Slater determinants of the corresponding first quantization operators. Operators that are considered in first quantization like the kinetic energy and the coulomb repulsion conserve the number of electrons. In the Fock space these operators must be represented as linear combinations of multipla of the ajaj... 

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. 

Before describing the axioms of quantum mechanics, one needs some mathematical background in linear vector spaces. Since this may be acquired from any of the introductory textbooks on quantum mechanics, we shall just review some of the main points without going into much detail. 

These are nothing but the conditions of orthogonality of the subspace of interest lm/ (lm/ - image P - stands here for the set of vectors of a linear space which are obtained by action of the linear operator P upon all vectors of the linear vector space) and its complementary subspace IrriQ. 

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. 

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