| Budget Amount *help |
¥3,580,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥180,000)
Fiscal Year 2007: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2006: ¥1,700,000 (Direct Cost: ¥1,700,000)
Fiscal Year 2005: ¥1,100,000 (Direct Cost: ¥1,100,000)
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| Research Abstract |
This study presents the new method to solve Schroedinger equations for some electron orbits using genetic algorithm. The Schroedinger equation is represented as a partial differential equation of wave function, and it also includes an eigen value as unknown variable.
Our approach to solve this equation is based on an eigen-function expansion of a wave function. Firstly, this partial differential equation is transformed to the integral equation including a fundamental solution. When we impose a particular condition to the set of functions to expand, the integration with fundamental solution can be given by recurrence formula without integration In these processes, since both gridding to solve a partial equation and numerical integrations are not required, any approximations are not involved.
To determine coefficients of the eigen-function we apply a genetic algorithm. The genetic algorithm is a method to emulate evolution of species. The coefficients, a set of coefficients, and the Schroedinger equation are, respectively, considered as genes a individual, and environment. In the system there are many individuals at a time. If some individuals have better fitness to environment they produce children having parents' gene. The system evolves gradually to have many better individuals.
Using these combination, we succeeded to solve the Schroedinger equation for the spherical symmetrical potential such as hydrogen atom. For more complex model with unsymmetrical potential, the convergence of the genetic algorithm was not good. However, we found the integral equation can be transformed to algebraic simultaneous equations, which does not need help of the genetic algorithm. And we succeeded to solve the orbit of hydrogen atom under electric field.
In addition, the genetic algorithm was improved. This improvement gives us not only the method to solve a equation that is the purpose of this study, but also the solution of fitting problems with discontinuous functions.
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