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International Journal of Mathematical Archive

ON THE CHANGE OF THE BASIS OF FINITE QUANTUM SYSTEMS

M. Tubani, Amna Gresh Gresh

Published Jun 4, 2021• 0 citations
OpenAlex

Abstract

The analytic functions of finite quantum systems are considered. The zeros of this functions and their evolution time are discussed. As the basis change the quantum state transforms into different quantum state. A brief introduction to the Symplectic Transformationsis given. A map between two toruses where the domain and the range of this map are the analytic functions on toruses is defined.

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The analytic functions of finite quantum systems are considered. The zeros of this functions and their evolution time are discussed. As the basis change the quantum state transforms into different quantum state. A brief introduction to the Symplectic Transformationsis given. A map between two toruses where the domain and the range of this map are the analytic functions on toruses is defined.
The paper contains an investigation on basic problems about the zeros of analytic theta functions. A brief introduction to analytic representation of finite quantum systems is given. The zeros of this function and there evolution time are discussed. Two open problems are introduced. The first problem discusses the cases when the zeros follow the same path. As the basis change the quantum state |f transforms into different quantum state. The second problem is to define a map between two toruses where the domain and the range of this map are the analytic functions on toruses.
Quantum systems with a finite Hilbert space, where position x and momen- \ntum p take values in Z(d) (integers modulo d), are studied. An analytic \nrepresentation of finite quantum systems is considered. Quantum states are \nrepresented by analytic functions on a torus. This function has exactly d \nzeros, which define uniquely the quantum state. The analytic function of a \nstate can be constructed using its zeros. As the system evolves in time, the \nd zeros follow d paths on the torus. Examples of the paths ³n(t) of the zeros, \nfor various Hamiltonians, are given. In addition, for given paths ³n(t) of the \nd zeros, the Hamiltonian is calculated. Furthermore, periodic finite quantum \nsystems are considered. Special cases where M of the zeros follow the same \npath are also studied, and general ideas are demonstrated with several ex- \namples. Examples of the path with multiplicity M = 1; 2; 3; 4; 5 are given. It \nis evidenced within the study that a small perturbation of the initial values \nof the zeros splits a path with multiplicity M into M different paths.
Finite quantum systems with d-dimension Hilbert space, where position x and \nmomentum p take values in Zd(the integers modulo d) are studied. An analytic \nrepresentation of finite quantum systems, using Theta function is considered. \nThe analytic function has exactly d zeros. The d paths of these zeros on the \ntorus describe the time evolution of the systems. The calculation of these \npaths of zeros, is studied. The concepts of path multiplicity, and path winding \nnumber, are introduced. Special cases where two paths join together, are also \nconsidered. A periodic system which has the displacement operator to real \npower t, as time evolution is also studied. \nThe Bargmann analytic representation for infinite dimension systems, with \nvariables in R, is also studied. Mittag-Leffler function are used as examples of \nBargmann function with arbitrary order of growth. The zeros of polynomial \napproximations of the Mittag-Leffler function are studied.
A transform between functions in and functions in is used to define the analogue of number and coherent states in the context of finite d -dimensional quantum systems. The coherent states are used to define an analytic representation in terms of theta functions. All states are represented by entire functions with growth of order 2, which have exactly d zeros in each cell. The analytic function of a state is constructed from its zeros. Results about the completeness of finite sets of coherent states within a cell are derived.