Hilbert space mechanics

Hilbert Space and Quantum Mechanics.—In this section we shall express the fundamental postulates of quantum mechanics in terms of the concepts developed in the previous sections. 

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... 

Rate of change of observables, 477 Ray in Hilbert space, 427 Rayleigh quotient, 69 Reduction from functional to algebraic form, 97 Regula fold method, 80 Reifien, B., 212 Relative motion of particles, 4 Relative velocity coordinate system and gas coordinate system, 10 Relativistic invariance of quantum electrodynamics, 669 Relativistic particle relation between energy and momentum, 496 Relativistic quantum mechanics, 484 Relaxation interval, 385 method of, 62 oscillations, 383 asymptotic theory, 388 discontinuous theory, 385 Reliability, 284... 

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. 

Our first way of answering the last question will be based on the fundamental theorems on Hilbert space [14], Indeed, the theorem on separability tells us that any subspace of h is also a separable Hilbert space. As a consequence, the inner product defined on, say, the occupied subspace is hermitian irrespectively of the choice of the basis x f (/)], as long as this latter satisfies the fundamental requirements of Quantum Mechanics. One should therefore not have to impose this property as a constraint when counting the number of conditions arising from the constraint CC+ =1 but, on the contrary, can take it for granted. 

Prugovecki, E. (1981) Quantum Mechanics in Hilbert Space (2nd edn.), Academic, New York. 

Potential fluid dynamics, molecular systems, modulus-phase formalism, quantum mechanics and, 265—266 Pragmatic models, Renner-Teller effect, triatomic molecules, 618-621 Probability densities, permutational symmetry, dynamic Jahn-Teller and geometric phase effects, 705-711 Projection operators, geometric phase theory, eigenvector evolution, 16-17 Projective Hilbert space, Berry s phase, 209-210... 

The quantum-mechanical state is represented in abstract Hilbert space on the basis of eigenfunctions of the position operator, by F(q, t). If the eigenvectors of an abstract quantum-mechanical operator are used as a basis, the operator itself is represented by a diagonal square matrix. In wave-mechanical formalism the position and momentum matrices reduce to multiplication by qi and (h/2ni)(d/dqi) respectively. The corresponding expectation values are... 

Limitation to ensembles that allow exchange of energy, but not of matter, with their environment is unnecessarily restrictive and unrealistic. What is required is an ensemble for which the particle numbers, Nj also appear as random variables. As pointed out before, the probability that a system has variable particle numbers N and occurs in a mechanical state (p, q) can not be interpreted as a classical phase density. In quantum statistics the situation is different. Because of second quantization the grand canonical ensemble, like the microcanonical and canonical ensembles, can be represented by means of a density operator in Hilbert space. 

Chaos provides an excellent illustration of this dichotomy of worldviews (A. Peres, 1993). Without question, chaos exists, can be experimentally probed, and is well-described by classical mechanics. But the classical picture does not simply translate to the quantum view attempts to find chaos in the Schrodinger equation for the wave function, or, more generally, the quantum Liouville equation for the density matrix, have all failed. This failure is due not only to the linearity of the equations, but also the Hilbert space structure of quantum mechanics which, via the uncertainty principle, forbids the formation of fine-scale structure in phase space, and thus precludes chaos in the sense of classical trajectories. Consequently, some people have even wondered if quantum mechanics fundamentally cannot describe the (macroscopic) real world. 

Here, for notational convenience, we have assumed that Vnm = We would like to emphasize that the mapping to the continuous Hamiltonian (88) does not involve any approximation, but merely represents the discrete Hamiltonian (1) in an extended Hilbert space. The quantum dynamics generated by both Hamilton operators is thus equivalent. The Hamiltonian (88) describes a general vibronically coupled molecular system, whereby both electronic and nuclear DoF are represented by continuous variables. Contrary to Eq. (1), the quantum-mechanical system described by Eq. (88) therefore has a well-defined classical analog. 

Real space algorithms (section 4) allow for mappings between present day computer programs and strict molecular quantum mechanics [10,11]. It is the separability of base molecular states that permits characterizing molecular states in electronic Hilbert space and molecular species in real space. This feature eliminates one of the shortcomings of the standard BO scheme [6,7,12]. Confining and asymptotic GED states are introduced. In section 5 the concept of conformation states in electronic Hilbert space is qualitatively presented. 

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... 

In principle, the time evolution of a particular linear superposition on the molecular base states will reflect a chemical process via the changes shown by the amplitudes. This represents a complete quantum mechanical representation of the chemical processes in Hilbert space. The problem is that the separability cannot be achieved in a complete and exact manner. One way to introduce a model that is able to keep as much as possible of the linear superposition principle is to use generalized electronic diabatic base functions. 

Physicist P. A. M. Dirac suggested an inspired notation for the Hilbert space of quantum mechanics [essentially, the Euclidean space of (9.20a, b) for / — oo, which introduces some subtleties not required for the finite-dimensional thermodynamic geometry]. Dirac s notation applies equally well to matrix equations [such as (9.7)-(9.19)] and to differential equations [such as Schrodinger s equation] that relate operators (mathematical objects that change functions or vectors of the space) and wavefunctions in quantum theory. Dirac s notation shows explicitly that the disparate-looking matrix mechanical vs. wave mechanical representations of quantum theory are actually equivalent, by exhibiting them in unified symbols that are free of the extraneous details of a particular mathematical representation. Dirac s notation can also help us to recognize such commonality in alternative mathematical representations of equilibrium thermodynamics. 

Consider a quantum mechanical many-body system, e.g., a gas. There is an enormous number of coordinates q of all molecules, and the state of the system is described by a wave function p(q, t) in an enormous Hilbert space H. The evolution is governed by a Hamilton operator H whose spectrum is discrete but inordinately dense. The number of eigenvalues of if in a range of the order of... 

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,... 

It is clear that the various density functional schemes for molecular applications rely on physical aiguments pertaining to specific systems, such as an electron gas, and fitting of parameters to produce eneigy functionals, which are certainly not universal. By focusing on the energy functional one has given up the connection to established quantum mechanics, which employs Hamiltonians and Hilbert spaces. One has then also abandoned the tradition of quantum chemistry of the development of hierarchies of approximations, which allows for step-wise systematic improvements of the description of electronic properties. 

To summarize, we have considered a quantum mechanical N-body system with dilation analytic potentials, Ey, and its dependence on the scaling parameter i] = t] (for some 0 < < 0, depending on V). To be more detailed, we need to restrict Hilbert space to a dense subspace [54], the so-called Nelson class N, which provides the domain over which the unbounded complex scaling is well-defined. Closing the subset < - D T) in ft, see [9] and references therein for a more detailed expose, one obtains the scaled version of the original partial differential equation... 

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 formulation of quantum mechanics requires a representation of the Heisenberg group on the Hilbert space L2 (R") spanned by the functions tp ( ) where the variable indicates a n—dimensional vector = ( 1 , n) whose elelments have physical units of a length [/]. Let us first introduce the set of operators, generators of the Lie group H", I, Xj, and hDj (j = 1,..., n) satisfying the commutation relation... 

In the formalism of quantum mechanics, observables are associated to hermitian operators that act on the Hilbert space of square integrable functions representing the state of the quantum system. In the following, for the sake of definiteness, we shall consider hermitian operators B which can be written as hermitain combinations of position and momentum operators,... 

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

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