The operator difference equations that arise in the finite-element treatment of a quantum theory are implicit and therefore difficult to solve. By introducing a matrix formulation it is possible to circumvent the implicit character of these equations and obtain explicit closed-form solutions for arbitrary matrix elements of any operator.
The transfer-matrix method was proposed in the 1940s to solve linear ordinary differential equations.It is broadly studied due to its efficiency and simplicity for solving one-dimensional physical problems.In this work, the method is applied to solve the timeindependent Schrödinger equation, in order to improve both the formalism and the computational treatment.Closed-form solutions are derived for quantum wells and periodic potentials of arbitrary profiles.Numerical results are given for symmetric finite-barrier quantum wells and sinusoidal potentials.
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.
Basically, there are two approaches to the formulation of quantum mechanics, i.e., the operator and the Feynman path integral approach. The main object in quantum mechanics is referred to as the transition amplitude between those states, matrix elements of physical operators and, in particular, the S- matrix in the problem of quantum scattering. On the basis of quantum mechanics, the operator method is closed, well formulated mathematically, and so is an appropriate universal language describing quantum field theory, the quantum physics of condensed states and quantum relativistic statistical physics. The differential equation defining the path of the system in the configuration space is called the equation of motion of the system. The aim of classical as well as quantum mechanics is to predict the time evolution of the system, starting from knowledge of the state of the system at a given point in time.
The original version of matrix mechanics is little used in applications nowadays. In that early theory position, momentum and other quantities were represented by infinite matrices, whereas many modern applications of quantum mechanics represent such observables as operators. Infinite matrices are unwieldy to use and have some disturbing properties, e.g. they do not obey the associative rule. By far the most commonly appearing matrix in quantum mechanics is the matrix of the energy operator as set up in some finite basis of trial functions. The procedure has been used for perturbed oscillator problems; the recurrence relation approach is particularly simple when the matrix has only one non-zero element beyond the diagonal element in each row. It seems to the author that there are still many hybrid methods which need investigation.
We study the application of the "leapfrog" method of finite differencing to the approximate solution of operator equations of motion in quantum theory. We show that, for a wide class of linear and nonlinear systems, the leapfrog differencing scheme is exactly unitary. The method is sufficiently general to apply to many-particle systems with arbitrary potential forces and to lattice-regulated nonlinear $\ensuremath{\sigma}$ models and non-Abelian gauge theories. In contrast to the recent proposal of Bender and Sharp (which is based on the finite-elements method) our approach is explicit rather than implicit and, in the case of lattice-regulated field theories, has a lattice analog of microcausality. For systems with finitely many degrees of freedom and self-adjoint Hamiltonians, we show that our approximate solutions converge to the exact solutions in the limit in which the time step tends to zero.
The convergence issues caused by the improper treatment of the ensemble N -representability constraint have severely affected the applicability and reproducibility of reduced density matrix functional theory (RDMFT). Unlike the commonly used Lagrange methods explicitly bringing the constraint into the objective functions, we present a different idea to handle the constraint in an implicit manner, which is achieved by introducing implicit functions to exactly consider the nonlinear unclear connection embedded in the explicit constraint. This explicit-by-implicit idea, denoted as EBI, thus transforms the constrained optimization problem into an unconstrained minimization problem. The tests on different systems, initial guesses, and functionals demonstrate the superiority of EBI in the treatment of the ensemble N -representability constraint. Therefore, EBI solves the convergence issues of the Lagrange methods, which is essential for further development and application of RDMFT. Besides, the idea of EBI is helpful for the treatment of different constrained problems in modern physics and chemistry.