Interaction of Nonrelativistic Particles with a Quantized Scalar Field
We demonstrate the mathematical existence of a meson theory with nonrelativistic nucleons. A system of Schrödinger particles is coupled to a quantized relativistic scalar field. If a cutoff is pu...
We demonstrate the mathematical existence of a meson theory with nonrelativistic nucleons. A system of Schrödinger particles is coupled to a quantized relativistic scalar field. If a cutoff is pu...
Exact Solution of the Schrödinger Equation for Inverse Fourth-Power Potential
J. Math. Phys. 5, 1185 (1964); doi:10.1063/1.1704224
Issue Date: September 1964
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The solution of the Schrödinger equation for the potential V(r)=V/(µr)4 and all angular momenta is given in terms of known functions, namely, the solutions to the modified Mathieu equation with complex parameters and complex argument. Scattering solutions for both attractive and repulsive potentials are given and in the latter case the explicit expression for the S matrix is given.
©1964 The American Institute of Physics
| History: | Received 6 March 1964 |
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http://link.aip.org/link/?JMAPAQ/5/1185/1 |
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REFERENCES (16)
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- A. Bhattacharjie and E. C. G. Sudarshan,
Nuovo Cimento 25, 864 (1962) . - A. K. Bose,
Phys. Letters 7, 245 (1963) . - V. Bargmann, Rev. Mod. Phys. 21, 488 (1949).
- K. M. Case,
Phys. Rev. 80, 797 (1950) . - Moreover, the Coulomb potential behaves very badly at infinity in terms of the standard phase shift analysis. The only reason we use it is because we must (it's physical).
- E. Predazzi and T. Regge,
Nuovo Cimento 24, 518 (1962) . - A. Bastai, L. Bertocchi, S. Fubini, G. Furlan, and M. Tonin,
Nuovo Cimento 30, 1512 (1963) . - A. Bastai, L. Bertocchi, G. Furlan, and M. Tonin,
Nuovo Cimento 30, 1532 (1963) . - P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill Book Company, Inc., New York, 1953), mainly Chaps. 5 and 11.
- N. W. McLachlan, Theory and Applications of Mathieu Functions (Oxford University Press, Oxford, 1947). The reader is cautioned that various passages in this book are incorrect, for example, Secs. 6.20 and 12.30 do not state the regions of validity of the expressions contained therein (these expressions are completely invalid for certain regions). Similarly parts 12 and 15 of the Additional Results are completely wrong. In all cases the correct expressions are found in Ref. 11.
- J. Meixner and F. W. Schäfke, Mathieusche Funktionen und Sphäroidfunktionen (Springer-Verlag, Berlin, 1954).
- The two independent solutions are wrongly given in part 12 of Additional Results of Ref. 10. When
is an integer the form of the solutions and various expressions we use are rather different but lead of course to the same conclusions. - Bateman Manuscript Project (McGraw-Hill Book Company, Inc., New York, 1955), Vol. III.
- It is remarkable that in two books, Ref. 10 and 13, the condition |cosh z|<1 is stated to exclude only real z such that −1
z1. See p. 201 of MS for a diagram of the region in the complex z plane for which |cosh z|1. - More exactly we have
as q
0. See MS, p. 121. - R. Serber,
Phys. Rev. Letters 10, 357 (1963) .
