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UNIVERSITY NEWS NORTH-CAUCASIAN REGION NATURAL SCIENCES SERIES

On the Extended Hilbert Space

A. N. Hopersky, Конеев Рустам Викторович, A. M. Nadolinsky

Published Mar 29, 2024• 0 citations
Publisher

Abstract

The construction of the extended Hilbert space is presented in the form of a direct sum of the spaces of vectors of finite and infinite norms as the main space in the mathematical formalism of quantum mechanics of a many-electron atom. On the example of constructing the analytical structure of the probability amplitude 1s-3p of photoexcitation of a neon atom, the implementation of the EHS-construct for solving the equations of the self-consistent Hartree-Fock field is given.

DOI
10.18522/1026-2237-2024-1-38-42

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The construction of the extended Hilbert space is presented in the form of a direct sum of the spaces of vectors of finite and infinite norms as the main space in the mathematical formalism of quantum mechanics of a many-electron atom. On the example of constructing the analytical structure of the probability amplitude 1s-3p of photoexcitation of a neon atom, the implementation of the EHS-construct for solving the equations of the self-consistent Hartree-Fock field is given.
Abstract In this chapter we present the mathematical framework of describing wave functions as vectors in a Hilbert space and observables as Hermitian operators on such a vector space. We describe the mathematical formulation of these ideas for finite and infinite-dimensional cases via the Dirac notation, and discuss the example of the Hilbert space of photon polarization. Important classes of linear operators on Hilbert spaces are discussed, including projectors, Hermitian and self-adjoint operators, as well as unitary operators. As an important centre piece for the mathematical language of quantum mechanics we present the spectral theorem. We discuss the subtleties of this framework in infinite dimensions, giving a brief overview of the distributional aspects of quantum mechanics and the conceptually important problem of finding self-adjoint extensions via the Weyl-von Neumann theorem
We present a version of the formalism of Dirac based on nonstandard analysis, allowing us to deal with state vectors and operators using the resources of finite-dimensional linear algebra. The space of state vectors is a nonstandard Hilbert space with hyperfinite dimension, which includes all square-integrable functions, together with vectors representing states of definite position or momentum. Every vector is normalizable, even when its norm is infinite. Observables are represented by Hermitian operators, which are always (hyper)bounded and defined on the whole space. The connection with the standard theory is established by postulating the existence of “hyper-observables” and nonstandard states. Each observable in the usual sense appears as a kind of standard-scale approximation of some hyper-observable. We show that the probabilistic predictions are consistent with those of the standard theory. Consistency extends to time evolution, in the sense that if an initial nonstandard state is “near-standard,” then the state after a finite time shall be infinitely near the standard state obtained through the Schrödinger equation.
In Chapter 4 we saw that the canonical commutation relations force us to use a space of states of infinite dimension, in which rigor would require the use of advanced mathematical tools. Fortunately, physicists generally need only to carry the results for finite dimension over to infinite dimension with some simple modifications which we shall indicate here, without embarking on sophisticated mathematics. Nevertheless, it is useful to be aware of the lapses in rigor which are customarily made in physics in order to avoid possible unpleasant surprises. The objective of this chapter is, on the one hand, to present some concrete examples illustrating the new features which arise in infinite dimension and, on the other, to give the rules for practical calculations, in particular to write down the spectral decomposition of Hermitian and unitary operators. The mathematics we use is a bit more detailed than commonly found in most quantum mechanics textbooks. The reader interested purely in the practical aspects can proceed directly to Section 7.3, where the results essential for later on are summarized. Hilbert spaces Definitions The space of states of quantum mechanics is a Hilbert space ℌ, which in general is of infinite dimension. The axiomatic definition of a Hilbert space is the following. It is a vector space which, for the needs of quantum mechanics, is defined on complex numbers. The vectors of this space are denoted |ϕ〉. […]
The superposition principle is a founding principle of quantum mechanics; we have already made use of it in interpreting the Young's slit experiment. Quantum mechanics is a linear theory, and so it is natural that vector spaces play an important role in it. We shall see that a physical state is represented mathematically by a vector in a space whose characteristics we shall define; this is called the space of states . A second founding principle, which can also be deduced from the Young's slit experiment, is the existence of probability amplitudes. These probability amplitudes will be represented mathematically by scalar products defined on the space of states. In the theory of waves, the use of complex numbers is just a convenience, but in quantum mechanics the probability amplitudes are fundamentally complex numbers – the scalar product will a priori be a complex number. Physical properties like momentum, position, energy, and so on will be represented by operators acting in the space of states. In this chapter we shall introduce the essential properties of Hilbert spaces, that is, vector spaces on which a positive-definite scalar product is defined, and we shall limit ourselves to the case of finite dimension. This restriction will be lifted later on, because the space of states is in general of infinite dimension. The mathematical theory of Hilbert spaces of infinite dimension is much more complicated than that of spaces of finite dimension, and we shall put off studying them until Chapter 7. The reader familiar with vector spaces of finite dimension and operators in such spaces can proceed directly to Chapter 3 after reviewing the notation.
Quantum mechanics is formulated using a nonstandard Hilbert space. The concept of an eigen vector of a linear operator, which applies to standard as well as nonstandard Hilbert spaces, is replaced by the more general concept of an ultra eigen vector, which applies to nonstandard Hilbert spaces alone. Ultra eigen vectors corresponding to all spectral points of internal self−adjoint operators are proved to exist. This result enables us to set up a formalism which is equally valid for the discrete, as well as the continuous spectrum. Finally, Dirac’s formalism is reproduced, in a rigorous form within the nonstandard Hilbert space structure.