Self-consistent Approximation Method In Quantum Mechanics

Quantum mechanics is the theory to discuss the properties of motion of microscopicparticles (molecules, atoms, nuclei, elementary particles, etc.).The many-body problem in quantum mechanics is more complex, it is unable to give aexact solution, and it is only to find a approximate solution. In a number of approximatemethods (such as the perturbation method, variation method, etc.), the different selectionof appropriate approximation method becomes particularly important. In this paper, weuse the average self-consistent approximation method to calculate the eigenvalue of themany-body problem and give it’s analysis.First, this article analyzes the four-particle T-one-dimensional nonlinear harmonicoscillator with the average self-consistent approximation method and the perturbationmethod. The eigenvalues of the system energy is given. After comparing the results of thetwo methods, we see that more small the perturbation coefficient is, the closer the resultsof two methods are; more small the quantum number is, more similar the two results are.With the quantum number increases, the difference of results coming from two methodsincreases, which is due to that the fluctuations of the operators corresponding to the highquantum number can not be ignored, therefore the average self-consistent approximationmethod is not adapted in this case.Secondly, we use self-consistent average approximation method to investigate theone-dimensional nonlinear harmonic oscillator chain consisting of N particles, theeigenvale of the system is give.Third, we use the average self-consistent approximation method to the Price atomicvalence electron energy levels. The results is compared with results calculated from thehydrogen-like atom method. By comparing the results, it shows that they are wellconsistent. With quantum numbers increase, the results coming from the two methods arebasically consistent.The average self-consistent approximation method has great practical value by theanalysis of several models. It is not only to solve the eigenvalues of the monomer problem,but also an effective analysis of many-particle system eigenvalue problem. The newmethod is a simple and effective one to the nonlinear problem.