Hilbert space basis vectors

Quantum mechanics is generally formulated on the Hilbert space of vectors The functions ij/(q) = and (p) =

are the expansion coefficients of ) in an appropriate complete orthonormal basis, that is,... 

Other postulates required to complete the definition of will not be listed here they are concerned with the existence of a basis set of vectors and we shall discuss that question in some detail in the next section. For the present we may summarize the above defining properties of Hilbert space by saying that it is a linear space with a complex-valued scalar product. 

In any Hilbert space the basis vectors can always be chosen to be orthonormal ... 

Transformations in Hilbert Space.—Consider any vector /> in with components with respect to some orthonormal basis... 

One speaks of Eqs. (9-144) and (9-145) as a representation of the operators a and o satisfying the commutation rules (9-128), (9-124), and (9-125). The states 1, - , ) = 0,1,2,- are the basis vectors spanning the Hilbert space in which the operators a and oj operate. The representation (9-144) and (9-145) is characterized by the fact that a no-particle state 0> exists which is annihilated by a, furthermore this representation is irreducible since in this representation a(a ) operating upon an n-particle state, results in an n — 1 ( + 1) particle state so that there are no invariant subspaces. Besides the above representation there exist other inequivalent irreducible representations of the commutation rules for which neither a no-particle state nor a number operator exists.8... 

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. 

In an n-dimensional Hilbert space, Ln the set of n independent vectors define a complete set in Ln. This set is called a basis, and the vectors are called the basis vectors. Basis vectors can be chosen in an infinite number... 

Concepts in quantum field approach have been usually implemented as a matter of fundamental ingredients a quantum formalism is strongly founded on the basis of algebraic representation (vector space) theory. This suggests that a T / 0 field theory needs a real-time operator structure. Such a theory was presented by Takahashi and Umezawa 30 years ago and they labelled it Thermofield Dynamics (TFD) (Y. Takahashi et.al., 1975). As a consequence of the real-time requirement, a doubling is defined in the original Hilbert space of the system, such that the temperature is introduced by a Bogoliubov transformation. 

Basis functions between these vector spaces are orthogonal because they are contained in Hilbert spaces with different numbers of particles. Hence the four metric matrices that must be constrained to be positive semidefinite for 3-positivity [17] are given by... 

In the Hubbard model, the electron occupation of each site has four possibilities there are four possible local states at each site, v). = 0)y, t) -, i) -, Ti)y The dimensions of the Hilbert space of an L-site system is 4 and IV] V2 Vf,) = vj)j can be used as basis vectors for the system. The entanglement of the jth site with the other sites is given in the previous section by Eq. (65). 

The plan of the paper is the following. In section 2 we introduce the elements at the basis of the Heisenberg group representation representation theory [12-14,20] that are needed to understand the alternative group-theoretical formulation of quantum mechanics. In section 3 the Heisenberg representation of quantum mechanics (with the time dependence transferred from the vectors of the Hilbert space to the operators) is used to introduce quantum observables and quantum Lie brackets within the group-theoretical formalism described in the previous section. In section 4 classical mechanics is obtained by taking the formal limit h —> 0 of quantum observables and brackets to obtain their classical counterparts. Section 5 is devoted to the derivation of... 

The boundary conditions are periodic and the number of allowed values of the wave vector k is equal to the number of unit cells in the crystal. These eigenfunctions constitute the basis for the infinite-dimensional Hilbert space of the crystal Hamiltonian and any function with the same boundary conditions can be expressed as a linear combination of functions in this complete set. 

It is not always feasible to directly measure the ancilla independently from the information system in other words, it is sometimes impossible to perform a projection onto disentangled subspaces of H of the form 7T/0Span o ) in some cases, as for the example proposed in Sec. 3, one can only project onto entangled subspaces of the total Hilbert space H. In such a case the information initially stored in the vector ipi) = J2i=i r< Iu<) G Hi must be transferred into an entangled state of X and A of the form ip) = i r> H) where the I vectors 0) (i = 1.. .., I) which form an orthonormal basis of the information-carrying subspace C, are generally not factorized as earlier but entangled states. Nevertheless the same method as before can be used in that case to protect information, albeit in a different subspace C. 

Here S) denotes the spinup or spindown state of the left or right particle. The two state vectors in the right-hand side of (49) form a basis of the corresponding Hilbert space in which we can define the representation of the SU(2) algebra by the following generators... 

Quantum mechanics can also be formulated in terms of an alternate Hilbert space whose elements are operators, with the density operator p and the typical measurables F among them. A variety of complete sets of basis operators in the space may be constructed, for example, as the tensor product of a basis set of the Hilbert space vectors > and its dual, yielding elements of the form... 

Most assuredly, the individual eigenfunctions py of LQ and p, of Lc differ quantitatively, However, the formal structure set up thus far provides little insight into their qualitative differences since both classical and quantum mechanics have been formulated in terms of formally identical52 Hilbert spaces of L2 functions that are spanned by both the eigendistributions p, of Lc and pij eigendistributions of La. Completeness of both bases ensures that any L2 function p(p,q) may be written as linear combinations of either set of basis vectors. Indeed, it is the auxiliary conditions which arise from the interpretation of distributions in classical and quantum mechanics that are fundamental to the qualitative distinction between classical and quantum mechanics. 

We denote the two-dimensional subspace of the Hilbert space L (5 ) that contains the linear combinations of the basis vectors... 

The transformation law, P = O NO, interprets Nj as occupations of atomic state vectors i>, forming the AIM basis set in the Hilbert space... 

We have obtained the decomposition of the function (i.e.. a vector of the Hilbert space) x on its components Cfc along the basis vectors Jrk) of the Hilbert space. The coefficient Ck = I x) is the corresponding scalar product, and the basis vectors are normalized. This formula says something trivial any vector can be retrieved when adding all its components together. 

F. B.2. A pictorial representation of something that surely cannot be represented. Using poetic license, an orthonormal basis in the Hilbert space looks like a hedgehog of the unit vectors (their number equal to oo), each pair rf them orthogonal. This is in analogy to a 2-D or 3-D basis set, where the hedgehog has two or three orthogonal unit vectors. 

Eaeh veetor (function) can be represented as a linear combination of the hedgehog" functions. It is seen, that we may rotate the hedgehog" (i.e., the basis set) and the completeness of the basis will be preserved i.e., any vector of the Hilbert space can be represented as a linear eombination of the new basis set vectors. 

The next level of generalization is to consider both the discrete and continuum spectra of the eigen-values of an operator that is to woiic with die unity operators on the entirely Hilbert space whose basis is constructed from the reimion of the discrete and continuum eigen-vectors for the given operator ... 

The individual unit of classical information is the bit an object that can take either one of two values, say 0 or 1. The corresponding unit of quantum information is the quantum bit or qubit. It describes a state in the simplest possible quantum system [1,2]. The smallest nontrivial Hilbert space is two-dimensional, and we may denote an orthonormal basis for the vector space as 0> and 11 >. A single bit or qubit can represent at most two numbers, but qubits can be put into infinitely many other states by a superposition ... 

The intermediate normalization means that as a vector of the Hilbert space (see Appendix B available at booksite.elsevier.com/978-0-444-59436-5 on p. e7), has the normalized as the component along the unit basis vector In other words, ifk = terms orthogonal to The intermediate normalization is convenient, but not necessary. Although convenient for derivation of perturbational equations, it leads to some troubles when the mean values of operators are to be calculated. 

Let us imagine an infinite sequence of functions (i.e., vectors) fi, fi, /3, in a unitary space (Fig. B.l). The sequence will be called a Cauchy sequence if for a given > 0, a natural number N can be found, such that for i > N, we will have /i+i — /ill < s. In other words, in a Cauchy sequence, the distances between the consecutive vectors (functions) decrease, when we go to sufficiently large indices i.e., the functions become more and more similar to each other. If the converging Cauchy sequences have their limits (functions) that belong to the unitary space, then such a space is called the Hilbert space A basis in the Hilbert space is a set of the linearly independent functions (vectors) such that any function belonging to the space can be expressed as a linear combination of the basis set... 

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