On the Three Dimensional Reduction of the Kustaanheimo‐Stiefel Transformation

Abstract

The Kustaanheimo-Stiefel [l] symmetry transformation x = {xi) -+ u = {u,], definetl by = Uf -u; -. : + 4 , 93 = 2(u,u2 -u3u4), 5 3 = 2(u,u3 + U274)r where j = 1, 2, 3 and iy = I, 2, 3, 4, has received much a of (1) the Hamiltoniaii ---= E ,P 1 -P ten ion 2-51. WI h he help of the three dimensional ( N = N1 = 3 ) hydrogen atom has been converted into the one of a related four-dimensional ( N = N , = 4) harmonic oscillator: 2F2 = 212 -4u2E = 48.( 3 ) The Corresponding eigenavlues have been displayed to the right. This conversion is implied by the, interconnection [ 2, 41 T A , = r A , + G R 2 , 1 1 in which ~~_ ( u , a ; -u , a ; ; + u , a~-u , a~) y = o .This latter equation has the meaning of a superselection rule relying on the singlevaluedness of the state function y ( u ) . A generalization of (I) for N l = 5 and N , = 8 has also been proposed [B, 71. Above r = 1x1 = u2, ai = a/axi = i& and a: = a/au, = iGa. The corresponding Laplace-operators read A , = Ziat and A , = Za a:', whereas the conjugated momenta have been denoted by p i and vx, respectively. I n this paper we shall analyse (1) by considering, this time, constant u,-values such that aiu4 = 0. We shall then prove that (1) enables us to convert (2) into the Hamiltoniaii of a related three-dimensional harmonic oscillator, now with a constraint generated by the selection of constant u,-values. Note that conversions concerning power-potentials [8-121 as well as other cases have also been discussed before.