一維的狄拉克 勢量子系統在很多量子力學參考書已經有討論。在束縛態下，該系統只有一個束縛態；在散射態下，波函數是連續性且有限的。但是在更高維的狄拉克 勢量子系統中，束縛能是無窮大的，而且在散射態中無法計算散射波在原點的值。這篇論文只會討論二維的狄拉克 勢量子系統，我們將 勢的微分形式套入系統中去進行運算，推導得出束縛能、散射幅度、微分散射截面和第零階的相位移。
在論文的最後，我們將計算結果和另外兩個正規化方法——實空間正規化和一般化不確定性關係來進行比較。

One-dimensional Dirac delta potential has been widely considered in quantum mechanicstextbooks. The system has only one bound state and its scttering state the wave function is continuous and finite. However, in higher dimensions, the bound state energy is infinite, and the scattered wave is undetermined at the origin. In this thesis, we apply differential regularization to two-dimensional Dirac delta potential to obtain the ground state energy, the scattering amplitude, the differential scattering cross section and the zeroth partial wave shift. Finally, we compare our result with other regularization methods such as real space regularization and generalized uncertainty relation.

[1] R. Jackiw, Diverse Topics in Theoretical and Mathematic Physics (World Scienti c, Singapore,1995).
[2] M. L. Boas, Mathematical Methods in the Physical Sciences, 2nd Editon (John Wiley & Sons, Inc., New York, 1983).
[3] I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products, 6th Edition (Academic Press, San Diego, 2000).
[4] G. Arfken and H. Weber, Mathematical Methods for Physicists, 6th Edition (Elsevier Academic Press, San Diego, 2005).
[5] Su-Long Nyeo, Regularization methods for delta-function potential in twodimensional quantum mechanics," Am. J. Phys. 68, 571 (2000).
[6] H. E. Camblong and Carlos R. Ord o~nez, Renormalized path integral for the two dimensional-function interaction," Phys. Rev. A 65, 052123 (2002).
[7] H. E. Camblong, L. N. Epele, H. Fanchiotti, and C. A. Garc a Canal, Dimensional transmutation and dimensional regularization in quantum mechanics," Ann. Phys. 287, 1(2000).
[8] H. E. Camblong, L. N. Epele, H. Fanchiotti, and C. A. Garc a Canal, Dimensional transmutation and dimensional regularization in quantum mechanics," Ann. Phys. 287, 57 (2000).
[9] L. R. Mead and J. Godines, An analytical example of renormalization in twodimensional quantum mechanics," Am. J. Phys. 59, 935 (1991).
[10] B. R. Holstein, Anomalies for pedestrians," Am. J. Phys. 61, 142 (1993).
[11] D. Z. Freedman, K. Johnson, J. I. Latorre, Di erential regularization and renormalization: a new method of calculation in quantum eld theory," Nucl. Phys. B 371, 353 (1992).
[12] C. Thorn, Quark con nement in in nite-momentum frame," Phys. Rev D 19 (2),639 (1979).
[13] C. Manuel, R. Tarrach, Perturbative renormalization in quantum mechanics," Phys. Lett. 328, 113 (1994).
[14] N. Ferkous, Regularization of the Dirac potential with minimal length," Phys. Rev. A 88, 064101 (2013).
[15] I. Mitra, A. DasGupta, and B. Dutta-Roy, Regularization and renormalization in scattering from Dirac delta potentials," Am. J. Phys. 66, 1101 (1998).
[16] P. Gosdzinsky and R. Tarrach, Learning quantum eld theory from elementary quantum mechanics," Am. J. Phys. 59, 70 (1991)
[17] M. de Llano, A. Salazar, M.A. Sol s, Two-dimensional delta potential welss and condensed-matter physics," Rev. Mex. Fis. 51, 626 (2005)