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International Conference on Natural Computation (ICNC)

Quantum Interpretation of Frequency Operator

Peng Wang, Jianping Li

Published Jan 1, 2007• 4 citations
Publisher

Abstract

The frequency operator of signal is obtained in this paper. It reflects the quantum characteristics of signal. It is a linear and Hermitian operator. Eigenfunctions, eigenvalues and superposition principle of signal are researched by using the frequency operator. The frequency operator is used to prove the uncertainty principle of signal. The uncertainty principle of signal indicates that the signal can be considered as a kind of the quasi quantum system with wave-particle dualism. The frequency operator is a powerful tool to research the quantum characteristics of signal. The quantum interpretation of frequency operator explains the essence of signal and provides the theory support for applying the framework of quantum mechanics to the signal processing.

DOI
10.1109/icnc.2007.590

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The frequency operator of signal is obtained in this paper. It reflects the quantum characteristics of signal. It is a linear and Hermitian operator. Eigenfunctions, eigenvalues and superposition principle of signal are researched by using the frequency operator. The frequency operator is used to prove the uncertainty principle of signal. The uncertainty principle of signal indicates that the signal can be considered as a kind of the quasi quantum system with wave-particle dualism. The frequency operator is a powerful tool to research the quantum characteristics of signal. The quantum interpretation of frequency operator explains the essence of signal and provides the theory support for applying the framework of quantum mechanics to the signal processing.
The uncertainty principle of signal is proved according to the uncertainty principle of quantum mechanics,and is further interpreted by quantum mechanics.The signal can be considered as a kind of the quasi quantum system with wave-particle dualism.Because of the wave-particle dualism of the signal,time and frequency can not be measured at the same time.The frequency operator(FO) of signal is defined in this study,and it can be widely used in signal theory and,especially in wavelet research.The quantum interpretation for the signal explains the essence of the signal and provides the theory support for applying the framework of quantum mechanics to the signal processing.
This chapter explains that a central feature of quantum theory is its representation of observables by ‘operators’, which act on the wavefunction and extract the information it contains. It shows how operators are constructed and used. Observables are represented by hermitian operators, which have real eigenvalues and orthogonal eigenfunctions. When the system is not described by a single eigenfunction of an operator, it may be expressed as a superposition of such eigenfunctions. One consequence of the use of operators is the ‘uncertainty principle’, one of the most profound departures of quantum mechanics from classical mechanics. The uncertainty principle restricts the precision with which complementary observables may be specified and measured simultaneously.
A basic quantity in quantum mechanics is the wave function ψ. Given a quantum mechanical system we write the Schrödinger equation for it. Its solution is the wave function. When introducing the wave function we stated that the maximum information about the state of a system can be obtained only from ψ. How do we extract observable properties of the system from the wave function? In quantum mechanics for every classical dynamical variable there exists an operator. For example, the operator of momentum p is −i~∇ while the operator of energy variable H = p2/(2m)+V (X) is −(~2/2m)∇2+V (X). Operation of an operator on ψ gives another function. If the new function is λψ, then λ is the eigenvalue of the operator. For example, the energy of a particle in a particular state represented by ψ can be obtained by operating H on ψ. That is, we have Hψ = Eψ where E is the energy eigenvalue. The expectation value of A(X,p) can be obtained from the formula 〈A〉 = ∫ ∞ −∞ ψ∗Aψ dτ , (3.1) where ψ is the solution of the equation Hψ = Eψ. Operators in quantum mechanics which represent observables are of a spe- cial type: They are linear and Hermitian. Such operators not only form computation tools, but also form the most effective language in terms of which the theory can be formulated. The fascinating point is that using linear operators we can understand the quantum mechanics. Therefore, it is important to study the basic theory of linear operators. In this chapter, we present important properties of linear operators and their eigenfunctions. We discuss the features of various operators which are necessary for studying quantum and ematical structure of quantum mechanics is.
Abstract Mathematical formalism of quantum mechanics is closely connected with the theory of linear operators. A key quantum-mechanical principle is that physical quantities (observables) are represented by Hermitian linear operators that act on state vectors that belong to a Hilbert vector space. This chapter analyzes problems that deal with the basic concepts of the theory of linear operators; eigenfunctions, eigenvalues, mean values; the projection operators; quantum-mechanical representations of operators and wave-functions; and unitary operators.
This article establishes the proof of the Schrödinger equation for numerous quantum systems, utilizing Heisenberg's uncertainty principle. The Fourier transform connects functions in the time and frequency domains, resulting in the mathematical inequality that is the foundation of the uncertainty principle. In the part of Methods and Theory, the article derives the uncertainty principle through Fourier transforms by defining the mean and variance of angular frequency and time, and subsequently expanding the integral. This establishes the fundamental connection between time and frequency domains, illustrating the constraints imposed by quantum mechanics. In the part of Results and Application, the article applies the uncertainty principle to derive the Schrödinger equation under different conditions: free particle, particle in a box, harmonic oscillator, and hydrogen atom. For each case, the article assumes wave function solutions, uses the uncertainty in position and momentum to estimate kinetic and potential energies, and shows that the total energy matches the ground state energy derived from the Schrödinger equation. The results highlight the critical role of Heisenberg's uncertainty principle in understanding key aspects of quantum mechanics, providing a unified framework for these diverse systems.