APPLICATION OF CRYSTAL-FIELD SPLITTING IN TEACHING THE PERTURBATION THEORY OF DEGENERATE STATES IN QUANTUM MECHANICS COURSE

To address the scarcity of examples for degenerate perturbation in quantum mechanics, this paper proposes crystal-field splitting as a new example for teaching degenerate perturbations. Starting from the construction of the octahedral crystal field, this part of the content first calculates the perturbation potential based on Coulomb interaction. Then, it selects the single d-electron wave function to calculate the perturbation matrix elements, thereby constructing the secular matrix. The calculation process involves integrating the angular and radial parts separately, which relates to the knowledge of angular momentum. Finally, through diagonalization, the splitting of energy levels after perturbation and the linear combination of zero-order wave functions are obtained. Furthermore, based on the fundamental properties of degenerate states, their real function forms are derived. This example will help students understand the degenerate perturbation at a deeper level.