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International Journal of Innovative Science and Research Technology (IJISRT)

Cairo Techniques Solution of Schrödinger's Equation – Time Dependence

Ismail Abbas

Published Mar 27, 2024• 4 citations
Publisher

Abstract

We have shown in previous papers that B- matrix string mechanics, which is the product of a numerical statistical method called the Cairo technique, can be successfully applied to solve the most general form of the diffusion equation of heat as well as the time- independent equation. Schrödinger equation. In this paper we examine the extension of matrix chains from the Cairo technique to the solution of the time-dependent Schrödinger equation. Numerical results for a quantum particle in a three- dimensional box are presented and are intended to be strikingly precise.

DOI
10.38124/ijisrt/ijisrt24mar911

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We have shown in previous papers that B- matrix string mechanics, which is the product of a numerical statistical method called the Cairo technique, can be successfully applied to solve the most general form of the diffusion equation of heat as well as the time- independent equation. Schrödinger equation. In this paper we examine the extension of matrix chains from the Cairo technique to the solution of the time-dependent Schrödinger equation. Numerical results for a quantum particle in a three- dimensional box are presented and are intended to be strikingly precise.
The modern theory of quantum mechanics is incomplete. It is capable of describing the quantum energy field on the microscopic scale via the Schrödinger equation and its derivatives but is not capable of describing the energy field on the macroscopic scale such as the domain of thermal diffusion and sound intensity in audio rooms. . etc. On the other hand, in previous articles we have shown that the so-called theory of Cairo techniques and its chains of B matrices are more complete. They can numerically resolve both the macroscopic energy field on the thermodynamic scale, such as the energy field in thermal diffusion PDEs, and the sound energy field of PDEs in audio rooms. . etc. In a precise and revolutionary way. Additionally, they are also capable of describing and resolving the quantum energy distribution at the microscopic scale initially described via the Schrödinger equation and its derivatives. Considering that they can describe and solve pure mathematical problems such as numerical integration and infinite integer series in more detail, we better conclude that the numerical statistical methods of Cairo techniques and its B matrix chains are capable of describing almost all fields with the exception of the gravitational field (of general relativity) which will be the subject of the next article. Therefore, we propose that the Cairo techniques and their B-matrix chains constitute the required foundations of a unified field theory. It's logical and it makes sense. In this paper, we present detailed theoretical and numerical studies of six diverse physical and mathematical studies where the numerical results are surprisingly accurate. In conclusion, B-matrix strings and numerical statistical theory of Cairo techniques provide a framework for a unified energy density field theory. Schrödinger's equation can be considered as a diffusion equation with a diffusion coefficient β 2 = ħ / 2 m . D=(6.65E-34/2 Pi/2.9.31E-31=E-34/18.6 E-31=E- 3/18.6=5.6E-4. . !! SQRT Mue 0/Eps 0)=377 Ohm = Z01/Z0 =2.65 E-3 mho
The time-dependent and time-independent Schrödinger equations have been numerically solved for several quantum mechanical problems with known analytical solutions using the basis-spline collocation algorithm. This algorithm has been demonstrated to be efficient and versatile. The results from these calculations illustrate the necessary numerical considerations required for solving both time-independent and time-dependent Schrödinger equations and form a basis that could be used for solving these and similar problems.
A method for the solution of the time-dependent Schrödinger equation in two dimensions is presented which is based on the hydrodynamic analogy to quantum mechanics. The continuum introduced in that analogy is approximated by a finite number of particles, whose trajectories are computed. The procedure is applied here to the dynamics of a Gaussian wave packet on a two-dimensional quadratic potential surface containing a saddle point. Comparison with the known analytical solution indicates that the method is capable of accurate results with substantially less computer time required than in methods previously employed.
A method for the solution of the time-dependent Schrödinger equation in two dimensions is presented which is based on the hydrodynamic analogy to quantum mechanics. The continuum introduced in that analogy is approximated by a finite number of particles, whose trajectories are computed. The procedure is applied here to the dynamics of a Gaussian wave packet on a two-dimensional quadratic potential surface containing a saddle point. Comparison with the known analytical solution indicates that the method is capable of accurate results with substantially less computer time required than in methods previously employed. This particular check problem was chosen because of its close relationship to the wave-packet dynamics of chemical reaction rates.
We have analyzed Dyson's UUU-matrix theory of solving the Schr\"odinger equation in the interaction picture and are able to express the UUU matrix as a dominant term plus an infinite series involving multiple integrals of time. For a certain rather restrictive class of Hamiltonians, our theory is exact for a general time-dependent problem. For other Hamiltonians, we can only obtain approximate expressions for our UUU matrix and hence the wave function. Treating a time-independent problem as a special case of the time-dependent situation with a sudden-switching process, we have shown that our UUU matrix is exact. To demonstrate the working procedures of our theory, we apply it to study the well-known time-independent charged harmonic-oscillator problem and the more general harmonic oscillator with a time-dependent driving force. Compared with other methods, our new theory appears to lead to a result which contains more information than others due to the inclusion of noncommutability properties of operators in the operator Schr\"odinger equation. It has been shown that the classical Feynman path-integral formalism can be deduced from quantum mechanics with the use of the Green's-function operator. It is interesting to note that apart from a step function, the Green's-function operator is the same as that of our U(s){U}^{(s)}U(s) matrix, which is the UUU matrix obtained within the regime of the Schr\"odinger picture for a time-independent Hamiltonian, as a special case of our general time-dependent treatment.