The quantum mechanics of a bound particle in the delta-function potential in two dimensions is studied with a discussion of its regularization and renormalization. A simple regularization approach is considered with the introduction of several regularizing functions for defining the quantum system. More systematic regularization is introduced from the mathematical viewpoint of the theory of distributions. The renormalization scheme independence of the physical observable is demonstrated.
REFERENCES
1.
S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden, Solvable Models in Quantum Mechanics (Springer-Verlag, New York, 1988).
2.
K. S.
Gupta
and S. G.
Rajeev
, “Renormalization in quantum mechanics
,” Phys. Rev. D
48
(12
), 5940
–5945
(1993
).3.
J. Fernando
Perez
and F. A. B.
Coutinho
, “Schrödinger equation in two dimensions for a zero-range potential and a uniform magnetic field: An exactly solvable model
,” Am. J. Phys.
59
(1
), 52
–54
(1991
);D. K.
Park
, “Green’s-function approach to two- and three-dimensional delta-function potentials and application to the spin-1/2 Aharonov–Bohm problem
,” J. Math. Phys.
36
(10
), 5453
–5464
(1995
).4.
C.
Thorn
, “Quark confinement in infinite-momentum frame
,” Phys. Rev. D
19
(2
), 639
–651
(1979
).5.
K. Huang, Quarks, Leptons and Gauge Fields (World Scientific, Singapore, 1992), 2nd ed., pp. 225–227.
6.
R. Jackiw, “Delta-function potentials in two- and three-dimensional quantum mechanics,” in M.A.B. Bég Memorial Volume, edited by A. Ali and P. Hoodbhoy (World Scientific, Singapore, 1991), pp. 25–42;
L. R.
Mead
and J.
Godines
, “An analytical example of renormalization in two-dimensional quantum mechanics
,” Am. J. Phys.
59
(10
), 935
–937
(1991
);P.
Gosdzinsky
and R.
Tarrach
, “Learning quantum field theory from elementary quantum mechanics
,” Am. J. Phys.
59
(1
), 70
–74
(1991
);B. R.
Holstein
, “Anomalies for pedestrians
,” Am. J. Phys.
61
(2
), 142
–147
(1993
);C.
Manuel
and R.
Tarrach
, “Perturbative renormalization in quantum mechanics
,” Phys. Lett. B
328
, 113
–118
(1994
);A.
Cabo
, J. L.
Lucio
, and H.
Mercado
, “On scale invariance and anomalies in quantum mechanics
,” Am. J. Phys.
66
(3
), 240
–246
(1998
);R. M. Cavalcanti, “Exact Green’s functions for delta-function potentials and renormalization in quantum mechanics,” http://xxx.lanl.gov:quant-ph/9801033.
7.
M. Abramowitz and I. A. Stegun (Eds.), Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1972).
8.
G. E. Shilov, Generalized Functions and Partial Differential Equations (Gordon and Breach, New York, 1968).
9.
M.
Hans
, “An electrostatic example to illustrate dimensional regularization and renormalization group technique
,” Am. J. Phys.
51
(8
), 694
–698
(1983
).
This content is only available via PDF.
© 2000 American Association of Physics Teachers.
2000
American Association of Physics Teachers
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.
