# Nuclear Science and Techniques

《核技术》(英文版) ISSN 1001-8042 CN 31-1559/TL     2019 Impact factor 1.556

Nuclear Science and Techniques ›› 2020, Vol. 31 ›› Issue (2): 15

• NUCLEAR PHYSICS AND INTERDISCIPLINARY RESEARCH •

### Solution to the Dirac equation using the finite difference method

Ji-Yu Fang, Shou-Wan Chen, Tai-Hua Heng

1. School of Physics and Materials Science, Anhui University, Hefei 230601, China
• Received:2019-09-12 Revised:2019-12-26 Accepted:2020-01-05
• Contact: Tai-Hua Heng E-mail:hength@ahu.edu.cn
• Supported by:
This work was partly supported by the National Natural Science Foundation of China (No. 11875070) and the Natural Science Foundation of Anhui Province (No. 1908085MA16).
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Ji-Yu Fang, Shou-Wan Chen, Tai-Hua Heng. Solution to the Dirac equation using the finite difference method.Nuclear Science and Techniques, 2020, 31(2): 15
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Abstract: In this study, single-particle energy was examined using the finite difference method by taking 208Pb as an example. If the first derivative term in the spherical Dirac equation is discretized using a three-point formula, a one-to-one correspondence occurs between the physical and spurious states. Although these energies are exactly the same, the wave functions of the spurious states exhibit a much faster staggering than those of the physical states. Such spurious states can be eliminated when applying the finite difference method by introducing an extra Wilson term into the Hamiltonian. Furthermore, it was also found that the number of spurious states can be reduced if we improve the accuracy of the numerical differential formula. The Dirac equation is then solved in a momentum space in which there is no differential operator, and we found that the spurious states can be completely avoided in the momentum space, even without an extra Wilson term.