The Application Of The Approximation Of Self-Consistent Average In Quantum Mechanics

Schrodinger equation, which describes the behavior of microscopic particles, is a basic dynamic equations in quantum physics. It reveals the basic properties of material in the microscopic physical world. In quantum mechanics, solving problems on particles is often boils down to solving the Schrodinger equation, but for complex system of more than two atoms, to get the Schrodinger equation's exact solution is an impossible thing. So it seems especially important to use a approximate calculating method in finding the approximate solution of Schrodinger equation.Draw inspiration from the average field theory for muli-particle system, we make an approximation of self-consistent average to solve eigenvalue for various steady perturbation problems in this paper, The point is by using an average field to take the place of the interaction forces between a particle and other particles and that convert the complicated interaction into a simple system, and eventually get the analytic results by the approximation method.What we have mainly discussed in this article is solving some common problems about the energy eigenvalue of a three-dimensional coupled nonlinear harmonic oscillator and a central force-field system by using the approximation of self-consistent average and Hellmann-Feynman Theorem.In chapter two, we calculate energy eigenvalue of a three-dimensional coupled nonlinear harmonic oscillator by making use of the approximation of self-consistent average and Hellmann-Feynman theorem, and we compare the result with the calculation by using perturbation mode, The comparison shows that the ground state energy of a three-dimensional coupled nonlinear harmonic oscillator under second order approximation is almost the same.In chapter three, we use the approximation of self-consistent average to solve the eigenvalue in hydrogen-like atom in central force field which contains 1/r3 or 1/r4 nonlinear potential, and we compare the result with its exact solution. By analysis, it comes to a conclusion that this method can cause more less error. When nucleus suddenly decay, we use the approximation of self-consistent average to calculate K electron atoms decay rate.In chapter four, we also use the approximation of self-consistent average to solve steady eigenvalue in the central force field of power function which contains ?r2v nonlinear system.And through our research application of the approximation of self-consistent average in the above three models, it was found that this method may be easily extended to solving other problems like that. So it is a simpler method that can solve the problem about energy eigenvalue in quantum mechanics.