Klein-Gordon and Schrödinger equations for a free particle in the rest frame

A system of two equations is found that has solutions which coincide with the solutions of the Klein-Gordon equation in the rest frame. This system includes the Schrödinger equation for a free neutral spinless particle. Using the Schrödinger equation as an additional condition for solving the Klein-Gordon equation in the rest frame leads to two Helmholtz equations. Helmholtz equations can be solved by specifying a particle model and boundary conditions. One of the Helmholtz equations leads to discreteness of the rest masses of relativistic particles.本研究构建得到一组二元方程组，其解与静止参考系下的克莱因-戈登方程（Klein-Gordon equation）的解完全一致。该方程组涵盖描述自由中性无自旋粒子的薛定谔方程（Schrödinger equation）。若将薛定谔方程作为附加条件，用于求解静止参考系中的克莱因-戈登方程，则可推导出两个亥姆霍兹方程（Helmholtz equation）。亥姆霍兹方程可通过指定粒子模型与边界条件进行求解。其中一个亥姆霍兹方程可导出相对论粒子静止质量的离散性。