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Solution of a Three‐Body Problem in One Dimension
1.H. R. Post, Proc. Phys. Soc. (London) A69, 936 (1956).
2.J. Hurley, J. Math. Phys. 8, 813 (1967).
3.L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Pergamon Press, Inc., New York, 1958), Sec. 35.
4.The factor in the definition of x is convenient for the comparison with the three‐body problem.
5.I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic Press Inc., 1965);
5.Higher Transcendental Functions, A. Erd́lyi, Ed. (McGraw‐Hill Book Co., Inc., 1953), Vol. II. Note that the definition of Laguerre polynomials given here is different from that used in some textbooks, for instance in Ref. 3.
6.A wavefunction is considered physically acceptable if both and are continuous. This condition may be interpreted as deriving from the requirement that both the density and the current of probability (that the particle be found at x) vary continuously with x. Moreover, the wavefunction must be normalizable (for closed problems).
7.In the case of equal but distinguishable particles (Boltzmann statistics), each independent eigenfunction may be chosen to correspond to a definite ordering of the particles, because the singularity of the “centrifugal” pair interaction excludes the possibility that the particles overtake one another. Such an eigenfunction vanishes unless the variable which distinguishes the different orderings of the particles (x in the two‐body case, φ in the three‐body case) lies within the appropriate range.
8.This problem is separable (and easily solvable) also in the x, y variables [see Eq. (3.5)]. (In fact, any problem characterized by equalstrength harmonical potentials acting between every two particles and, in addition, by one arbitrary potential depending only upon the interparticle distance of one pair is completely separable;2 and this statement remains true for any number of particles and spatial dimensions). But the solution of the more complete three‐body problem with can be simply obtained only using the spherical variables r, φ (see below).
9.At the differential equation (3.17) has no singularity, but the mapping between the variable φ and the argument of the hypergeometric function does.
10.It is amusing to recognize that the equality holds both for and It does not, however, hold for
11.An interesting question to ask in this connection is whether the energy spectrum will continue to depend linearly on all quantum numbers. It is expected that it will if only the number of particles is increased, but that it will not if the number of space dimensions is increased. Incidentally, using the approach of this paper, one can immediately solve the one‐dimensional N‐body problem with equal‐strength harmonical forces between every two particles and, in addition, either only one or only three (equal‐strength) centrifugal potentials depending, respectively, either only upon the interparticle distance between two particles or only upon the three interparticle distances between three particles.
12.Also the problem considered in this paper is, however, in some sense equivalent to that with decoupled oscillators, as indicated by the structure of its spectrum. But this correspondence does not appear to be a trivial one.
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