Classical physics began to be superseded by quantum physics because it could not explain microscopic phenomena. Quantum physics employs mathematical approaches such as wave mechanics and matrix mechanics, one of which is the eigenvalue problem involving matrix operators. There are many types of matrix operators; however, not all are suitable for measuring physical quantities in quantum physics. Operators in quantum physics must be Hermitian to ensure that their eigenvalues are real. Eigenvalue problems for matrix operators can be solved either analytically or numerically. Therefore, the objective of this research is to analyze symmetric and antisymmetric matrix operators in the measurement of physical quantities through eigenvalue problems. The research method utilized is a theoretical computational approach. The results demonstrate that symmetric matrix operators are Hermitian and yield real eigenvalues. In contrast, antisymmetric matrix operators are non Hermitian, yielding imaginary or complex eigenvalues. The results of solving eigen problems using Python obtained relative errors between 0 and orders., while the absolute error remained below 10%, thus confirming the validity of the results. Based on the findings, symmetric matrix operators satisfy the requirements for use in physical problems through eigenvalue analysis. The eigenvalues of antisymmetric matrices cannot represent the measurement of physical quantities directly and serve only as rotation operators. However, antisymmetric matrices can be utilized in quantum physics for physical measurements if they are transformed into Hermitian matrices. Furthermore, solving eigenvalue problems with Python is effective due to its high accuracy and small error margins.
Classical physics began to be superseded by quantum physics because it could not explain microscopic phenomena. Quantum physics employs mathematical approaches such as wave mechanics and matrix mechanics, one of which is the eigenvalue problem involving matrix operators. There are many types of matrix operators; however, not all are suitable for measuring physical quantities in quantum physics. Operators in quantum physics must be Hermitian to ensure that their eigenvalues are real. Eigenvalue problems for matrix operators can be solved either analytically or numerically. Therefore, the objective of this research is to analyze symmetric and antisymmetric matrix operators in the measurement of physical quantities through eigenvalue problems. The research method utilized is a theoretical computational approach. The results demonstrate that symmetric matrix operators are Hermitian and yield real eigenvalues. In contrast, antisymmetric matrix operators are non Hermitian, yielding imaginary or complex eigenvalues. The results of solving eigen problems using Python obtained relative errors between 0 and orders., while the absolute error remained below 10%, thus confirming the validity of the results. Based on the findings, symmetric matrix operators satisfy the requirements for use in physical problems through eigenvalue analysis. The eigenvalues of antisymmetric matrices cannot represent the measurement of physical quantities directly and serve only as rotation operators. However, antisymmetric matrices can be utilized in quantum physics for physical measurements if they are transformed into Hermitian matrices. Furthermore, solving eigenvalue problems with Python is effective due to its high accuracy and small error margins.
The traditional framework of quantum mechanics and many classical wave phenomena is underpinned by Hermitian operators, which guarantee real eigenvalues corresponding to observable quantities and orthogonal eigenvectors. However, a growing number of physical systems exhibit non-reciprocal interactions, leading to Hamiltonians or system matrices that are inherently non-Hermitian. This paper delves into the spectral theory of such non-reciprocal systems, exploring the unique mathematical and physical characteristics that emerge when the constraint of Hermiticity is relaxed. We discuss phenomena like complex energy eigenvalues, exceptional points where both eigenvalues and eigenvectors coalesce, and the breakdown of standard perturbation theory. The implications of non-Hermitian spectral properties extend to diverse fields, including parity-time (PT) symmetric photonics, active matter, open quantum systems, and even classical wave propagation in active or lossy media. By extending spectral analysis beyond the Hermitian paradigm, we aim to provide a comprehensive overview of the current understanding, challenges, and future directions in characterizing the dynamics and phases of non-reciprocal systems. This exploration is crucial for advancing both fundamental theoretical understanding and practical applications in emerging technologies that leverage tailored non-reciprocity.
Operators in quantum mechanics which represent observables are of a special 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. This chapter presents important properties of linear operators and their eigenfunctions. It discusses the features of various operators which are necessary for studying quantum mechanical systems. The chapter shows how rich and beautiful the mathematical structure of quantum mechanics is. There exists a class of eigenvalue problems, the so-called parity-time reversal (PT) -symmetric non-Hermitian Hamiltonian problems. They have arisen in recent years in a number of physical contexts such as localization transitions, vortex pinning and population biology. Such non-Hermitian Hamiltonians can be used to describe certain experimental phenomena. PT-symmetry has been noticed in certain optical systems, such as optical couplers, microwave billiard and large-scale temporal lattices.
Abstract The Eigen problem is a problem that exists when there is an operator that is affected by the function, then that function does not change except being multipled by the constant in the form of an eigenvalue. In general, the eigenvalue problem uses the Hermitian operators by using observable variable in the form of mathematical operations. Hermitian operator is an operator that happens when the Hermit conjugation is performed, then the operator will return to its original condition. This research aims to determine the base matrix of Hermitian operators with order n ≤ 4 (size 2 × 2, 3 × 3 and 4 × 4). Entry element of the hermitian matrix operator uses complex numbers of which each order consists of few Hermitian operators. The method used is an analytical method by inserting problems into the eigen equation then normalized to obtain a diagonal matrix, so as to obtain a complete solution of the eigen problem on form of a matrix (including eigenvalues, normalized eigenvectors, and base matrix). Base matrix that has been obtained will be a diagonal matrix with diagonal elements of the eigenvalue. The results of this research indicate that the base matrix obtained in each order forms a square matrix that has entry elements in the form of complex numbers.
Quantum mechanics can be formulated either in terms of matrix mechanics or in terms of wave mechanics. This chapter centers on a discussion of the historical origins, mathematical structure, and extension to non-Hermitian quantum theory of matrix mechanics. The development of atomic discrete spectra and the old quantum theory showed that classical mechanics could not account for the discreteness of atomic energy levels and spectral lines, which motivated Heisenberg, Born, Jordan, and others to establish matrix mechanics centered on noncommuting physical quantities. Combining the Bohr atomic model, angular momentum quantization, and radiative transitions, one can see how matrix mechanics describes the energy level structure and quantum transitions of atomic systems, and how the Heisenberg equation of motion gives the fundamental form of quantum dynamics. The example of the harmonic oscillator further shows that the discreteness of the energy spectrum can be obtained directly from the matrix method, thereby establishing a clear connection among transition frequencies, matrix elements, and radiation-induced transitions. The matrix representation corresponds to the operator theory in Hilbert space, where state vectors, inner products, complete bases, and spectral decomposition constitute the fundamental mathematical framework of quantum mechanics. For closed quantum systems, physical observables are generally required to be represented by Hermitian or self-adjoint operators, so as to ensure that their eigenvalues possess the physically acceptable property of being real numbers. The Hermiticity of an operator is also closely related to its domain and boundary conditions, which provides a mathematical possibility for the emergence of non-Hermitian Hamiltonians. Based on these discussions, this chapter further explains how non-Hermitian dynamics is connected with open systems, dissipative processes, and special boundary conditions, thereby demonstrating the natural connection between traditional matrix mechanics and modern non-Hermitian quantum theory.
The mechanical quantity is expressed by a Hermite operator is one of the basic hypotheses of quantum mechanics.All eigenvalues solved by the mechanical quantity operator eigenvalue equation are the possible values of the corresponding mechanical quantities.If one with a measuring instrument to measure the amount of value,eigenvalues can only be measured.In this paper,a Hermite matrix being similar to a diagonal matrix is proved in mathematics,and then combined with examples to explain its physical meaning.
The quantum theoretical description of microscopic systems is based on differential operators and the solution of the underlying eigenvalue equations. This approach is called Schrödinger approach or wave mechanics. Heisenberg developed another equivalent approach called matrix mechanics. In matrix mechanics all observables are represented by matrices. All possible eigenvalues of a matrix representing an observable are the possible outcomes of experimental measurement on that observable. This chapter gives the preliminary concepts relative to the use of the matrix method. It presents the various elements of matrix mechanics and discusses the matrix representation of operators of certain systems. The chapter introduces Dirac's notation of state vectors, Hilbert space, projection operator, displacement operators and the generators of translations. The theory of quaternions has found applications in classical mechanics, quantum mechanics and relativity. It has been used in aerospace applications, flight simulators, computer graphics, animation, visualization, fractals and virtual reality.