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arXiv:2004.05159

On Sylvester solution for degenerate eigenvalues

Dawit Hiluf Hailu

Published Apr 9, 2020• 0 citations
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Abstract

In this paper we introduce the use of Sylvester's formula for systems with degenerate eigenvalues in relation to obtaining their analytical solutions. To appreciate the use we include two other forms of analytical solutions namely adiabatic and Magnus approximations. In quantum mechanics, the Schrödinger equation is a mathematical equation that describes the evolution over time of a physical system in which quantum effects, such as wave--particle duality, are significant. The equation is a mathematical formulation for studying quantum mechanical systems. Just like Newtons's laws govern the motion of objects, Schrödinger equations of motion also govern the motion of quantum objects. Unlike the classical motion of objects the equation of motions of quantum phenomenon deals with the likelihood of the trajectories.

DOI
10.48550/arxiv.2004.05159

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In this paper we introduce the use of Sylvester's formula for systems with degenerate eigenvalues in relation to obtaining their analytical solutions. To appreciate the use we include two other forms of analytical solutions namely adiabatic and Magnus approximations. In quantum mechanics, the Schrödinger equation is a mathematical equation that describes the evolution over time of a physical system in which quantum effects, such as wave--particle duality, are significant. The equation is a mathematical formulation for studying quantum mechanical systems. Just like Newtons's laws govern the motion of objects, Schrödinger equations of motion also govern the motion of quantum objects. Unlike the classical motion of objects the equation of motions of quantum phenomenon deals with the likelihood of the trajectories.
In this chapter, we introduce the application of Sylvester’s formula for systems with degenerate eigenvalues to obtain their analytical solutions. To illustrate its utility, we include two additional methods for analytical solutions: the adiabatic approximation and the Magnus expansion. In quantum mechanics, the Schrödinger equation is a mathematical expression that describes the time evolution of a physical system where quantum effects, such as wave-particle duality, are significant. This equation serves as a framework for analyzing quantum mechanical systems. Similar to how Newton’s laws govern the motion of classical objects, the Schrödinger equation governs the behavior of quantum entities. However, unlike classical motion, quantum mechanics deals with the probabilities of different trajectories. The ability to obtain analytical solutions to such equations provides an in-depth understanding of the dynamics and also allows for the identification of controlling parameters, as analytical functions can readily reveal these insights.
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