Vectors in Hilbert Space

Any quantum dynamical state of a physical system may be represented by a vector bra or kef) with a unitary norm (see below) within the so-called space of the quantum states or the Hilbert space  

The Hilbert space is a vectorial space with scalar product, which is complete. A metrical space is called complete (or Banach space) if any convergent sequence of space elements has its limits within the space. On the other side, the scalar product is defined as the functional constructed on an abelian (commutative) vectorial space V 

The norm the positive definite self scalar product. 

Proof using the above rules one can consider the successive equivalences  

Quantum Nanochemistry-Volume I Quantum Theory and Observability 

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Quantum Mechanical Generalities.—It will be recalled that in nonrelativistic quantum mechanics the state of a particle at a given instant t is represented by a vector in Hilbert space (f)>. The evolution of the system in time is governed by the Schrodinger equation... 

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. 

Heisenberg representation (matrix mechanics) the position and momentum are represented by matrices which satisfy this commutation relation, and ilr by a constant vector in Hilbert space, the eigenvalues E being the same in two cases,... 

Because the product (3.8) is a probability, it must integrate to 1.0 when all possible outcomes are taken into account. Consequently, the wave functions are multiplied by arbitrary constants (n ),/2 chosen to make this integral come out to 1.0 over the complete range of motion. These are called normalization constants. It is legitimate to multiply solutions to the Schroedinger equation by an arbitrary constant because they are elements of a closed binary vector space. Multiplication of a solution by any scalar yields another element in the space, hence the product of the normalization constant and the wave function (or any other state vector in Hilbert space) also is a solution. 

Figube 1. Graphs of two functions (both normalized) which are very different analytically but whose vectors in Hilbert space are almost parallel. The cosine of the angle between the two vectors differs from 1 by one half of the integral over the square of the difference of the two fimctions. 

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 quantum mechanics, dynamical variables are represented by linear Hermitian operators 0 that operate on state vectors in Hilbert space. The spectra of these operators determine possible values of the physical quantities that they represent. Unlike classical systems, specifying the state ) of a quantum system does not necessarily imply exact knowledge of the value of a dynarttical variable. Only for cases in which the system is in an eigenstate of a dynamical variable will the knowledge of that state IV ) provide an exact value. Otherwise, we can only determine the quantum average of the dynamical variable. 

Linear Manifolds in Hilbert Space.—Any sequence of m vectors, l/iXl/aV called a linearly dependent sequence if... 

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

The general vector in Fock space may have components in some or all of the Hilbert subspaces, which means that it is now possible to consider states in which there is a superposition of different populations. Thus, we may represent the Fock space vector at an arbitrary time t by a symbol and expand this state in terms of its components in each subspace ... 

Transformations in Hilbert space, 433 Transition probabilities of negatons in, external fields, 626 Transport theory, 1 Transportation problems, 261,296 Transversal amplitude, 552 Transversal vector, 554 Transverse gauge, 643 Triangular factorization, 65 Tridiagonal form, 73 Triple product ensemble, 218 Truncation error, 52 Truncation of differential equations/ 388... 

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... 

An eigenstate corresponds to a row vector with zero amplitude everywhere except at the base state function that is the case. It is then defined in Hilbert space and does not stand for an individual molecular state. The corresponding base state cannot be a dynamically unstable state as it is time independent. This is an important difference with standard approaches [15]. 

If the symmetry is different, then of course iL /, > can be nonzero. In this article we assume that 0t,..., VN have definite albeit different time reversal symmetries. The properties can be represented by vectors t/j >... t/jy>... in Hilbert space with scalar product defined above. It is a simple matter to demonstrate that L is Hermitian in this Hilbert Space. Define the time correlation function... 

Likewise, the three columns of the matrix A2 above represent three mutually perpendicular, normalized vectors in 3D space. A better name for an orthogonal matrix would be an orthonormal matrix. Orthogonal matrices are important in computational chemistry because molecular orbitals can be regarded as orthonormal vectors in a generalized -dimensional space (Hilbert space, after the mathematician David Hilbert). We extract information about molecular orbitals from matrices with the aid of matrix diagonalization. 

In quantum mechanics, the state of the system is represented by a vector ip t)) in Hilbert space. 

It is convenient to introduce a Liouville space, or double space, that is a direct product of cap and tilde spaces. In Liouville space, operators are considered to be vectors and Hilbert-space commutators are considered to be operators. Equation (48) is then expressed as... 

The mathematical formalism jofitjuantum mechanics is expressed in terms of linear operators, which rep resent the observables of a system, acting on a state vector which is a linear superposition of elements of an infinitedimensional linear vector space called Hilbert space. We require a knowledge of just the basic properties and consequences of the underlying linear algebra, using mostly those postulates and results that have direct physical consequences. Each state of a quantum dynamical system is exhaustively characterized by a state vector denoted by the symbol T >. This vector and its complex conjugate vector Hilbert space. The product clT ), where c is a number which may be complex, describes the same state. 

Having expressed time-correlation functions in the language of Hilbert space, we can give a geometrical interpretation of these functions. Let A be a vector in Liouville space representing the initial value A(0) then A(t) = eiLtA is another vector representing the property at time t. The operator elLt is unitary it preserves the norm of A. We can regard the time evolution of A, therefore, as a simple rotation in Liouville space. This is illustrated in Fig. 11.3.1a. 

Toward this end, we have attempted to reduce exposition using vector terminology. Occasionally, this was not possible, so an appendix was developed to provide a review of elementary principles for vector-matrices operation. Linear operators in Hilbert space was briefly mentioned, mainly as a means of coping with the case of infinite eigenfunctions. Last, but not least, a careful... 

This special feature of wave-function assures the background on which the postulates of quantum mechanics (especially the integrability of the wave-function, its continuity, asymptotic continuity and representation by means of the scalar product in Hilbert spaces of vectors and operators) may be appropriately formulated and applied. 

M. A. Naimark, On some criteria of completeness of the system of eigen and associated vectors of a linear operator in Hilbert space, Dokl. Akad. Nauk SSSR (N.S.) vol. 98 (1954) pp. 727-730 (in Russian). 

Tlie mechanical quantities tliat describe the particle (energy, the components of vectors of position, momentum, angular momentum, etc.) are represented by linear operators acting in Hilbert space (see Appendix B). There are two important examples of the operators the operator of the particle s position x = x (i.e. multiplication by x, or x = x-. Fig. 1.5), as well as the operator of the (x-component) momentum px = —where i stands foi the imaginary unit. 

Example 7. Rotation of an atomic orbited. Let us construct a single spherically symmetric Gaussian orbital /(r) = exp[— r — rop] in Hilbert space for one electron. Let the atomic orbital be centred on the point indicated by vector tq. Operator R(a z) has to perform the rotation of a function by angle a about axis z... 

Up to this point we have tailored the second-quantization formalism in close connection to the independent-particle picture introduced before. However, the formalism can be generalized in an even more abstract fashion. For this we introduce so-called occupation number vectors, which are state vectors in Fock space. Fock space is a mathematical concept that allows us to treat variable particle numbers (although this is hardly exploited in quantum chemistry see for an exception the Fock-space coupled-cluster approach mentioned in section 8.9). Accordingly, it represents loosely speaking all Hilbert spaces for different but fixed particle numbers and can therefore be formally written as a direct sum of N-electron Hilbert spaces. 

This series can be expressed in a more compact form by using the so-called superoperator formalism (Goscinski and Lukman, 1970). We introduce this formalism here, as we had introduced the interaction picture in Section 3.8, in order to facilitate our derivations. The final equations will, however, be written without any superoperators. The superoperator formalism is one level of abstraction higher than the Hilbert vector space of quantum mechanics. In the infinite-dimensional Hilbert space the vectors of the vector space are given as quantum mechanical wavefunctions and the transformations performed on the vectors in the vector space are given by the quantum mechanical operators. The binary product defined in Hilbert space is the overlap integral /) between two wavefunctions, and 4 . In the superoperator formalism we now have an infinite-dimensional vector space, where the quantmn... 

In quantum mechanics, the state of an A -particle system is described by a state vector ir) in Hilbert space. The wave function ik(q ) corresponding to the state tk) is defined by f(q ) = q f), where is the adjoint of the collective coordinate state vector ), with q = q, ..., q ) denoting the collective coordinate for the N particles. 

State vectors and Hilbert spaces. The state of a physical system is represented by its state vector, which is a vector in a complex Hilbert space 1-1. The notation which is used in the following is due to Paul Dirac. According to this notation the state vectors are represented as follows... 

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