Using Supersymmetic Quantum Mechanics To Solve Schr(?)dinger Equation

Supersymmetic Quantum Mechanics (SQM) was first introduced by Witten in1981 to analyze the broken supersymmery. From then on, people have paid muchattention to this field. Many questions were discussed and a lot of conceptions wereput forwerd. In1983, Gendenshtein advanced the conception of shape-invariantpotential, and proved in the context of non-relativity, our given potentials are allshape-invariant. Their bound-state energy spectra can be obtained easily.This is abreakthrough in Supersymmetry Quantum Mechanics. In 1985, Sukumar C Vintrodeced the idea to compose a hierarchy of Hamiltonians,which is a greatadvantage in the supersymmetry history again.Then people studied the properties ofSQM from different angles. Such as the broken supersymmetry, the classification ofshape-invariant potentials,etc.In the paper, we mainly introduce the basics knowledge of SQM, and apply it toanalize the properties of potential field. First of all, we inroduce the conception ofsupersymmetric partners, describe the idea of hierarchy of Hamiltonians, and studythe property of shape invarient potential. And then based on SQM, we obtain therelationship that supperpotentials in a hierarchy of Hamiltonians satisfy. Applyingthese theories, we solve Eckart potential: when angular momentum is zero, weillustrate its bound-state spectra and eigenfunctios;when angular momentum is notzero, we use two approximation methods to analize it. One is relating variationalmethod with SQM. We obtain its approximate energy and wave functions;the other isusing Coulomb approximation for the parameters satisty some conditions. Finally, we introduce the SWKB method. At first, we obtain the SWKBquantum condition and BSWKB quantum condition (for the broken supersymmetry).From therotical angle, we prove the bound state energies are exact for shape-invarientpotential by SWKB method. As an example of appling SWKB method, we solve thegeneral ABO potential and discuss the result briefly.