This paper constitutes a detailed study of the nine−parameter symmetry group of the time−dependent free particle Schrödinger equation in two space dimensions. It is shown that this equation separates in exactly 26 coordinate systems and that each system corresponds to an orbit consisting of a commuting pair of first− and second−order symmetry operators. The study yields a unified treatment of the (attractive and repulsive) harmonic oscillator, linear potential and free particle Hamiltonians in a time−dependent formalism. Use of representation theory for the symmetry group permits simple derivations of addition and expansion theorems relating various solutions of the Schrödinger equation, many of which are new.
REFERENCES
1.
2.
W. Miller Jr., “Lie Theory and Separation of Variables II. Parabolic Coordinates,” SIAM J. Math. Anal. (to appear).
3.
4.
5.
6.
7.
8.
9.
10.
E. G. Kalnins, “On the separation of variables for the Laplace equation in two and three dimensional Minkowski space,” CRM‐319, SIAM J. Math. Anal. (to appear).
11.
12.
A. A.
Makarov
, Ya. A.
Smorodinskii
, Kh. V.
Valiev
, and P.
Winternitz
, Nuovo Cimento A
52
, 1061
(1967
).13.
E. G. Kalnins, W. Miller Jr., and P. Winternitz, “The Group Separation of Variables and the Hydrogen Atom,” CRM‐416 (submitted).
14.
Ya. A.
Smordinskii
and I. I.
Tugov
, Zh. Eksp. Teor. Fiz. [Sov. Phys. JETP
23
, 434
(1966
)].15.
C. P. Boyer, “The maximal kinematical invariance group for an arbitrary potential,” (to appear in Helv. Phys. Acta, 1974).
16.
U. Niederer, “The group theoretical equivalence of the free particle, the harmonic oscillator and the free fall,” Proceedings of the 2nd International Colloquium on Group Theoretical Methods in Physics, 1973, University of Nijmegen, The Netherlands.
17.
18.
A. Erdelyi, et al., Higher Transcental Functions (McGraw‐Hill, New York, 1953), Vols. 1, 2.
19.
20.
21.
W. Miller, Jr., Symmetry Groups and Their Applications (Academic, New York, 1973).
22.
23.
24.
T. Kato, Perturbation Theory for Linear Operators (Springer‐Verlag, Berlin and New York, 1966), p. 493.
25.
F. M. Arscott, Periodic Differential Equations (Pergamon, Oxford, 1964).
26.
27.
M. A. Naimark, Linear Differential Operators, Part II (Ungar, New York, 1968), p. 250.
28.
K. M.
Urwin
and F. M.
Arscott
, “Theory of the Whittaker Hill Equation
,” Proc. Roy. Soc. Edinb. A
69
, 28
(1970
).29.
P. Kramer, M. Moshinsky, and T. H. Seligman, “Complex Extensions of Canonical Transformations and Quantum Mechanics,” Institut für Theoretische Physik der Universität Tübingen (preprint), 1973.
30.
This content is only available via PDF.
© 1975 American Institute of Physics.
1975
American Institute of Physics
You do not currently have access to this content.
