PHYSICAL REVIEW D VOLUME 30, NUMBER 10 Path-integral solution for a Mie-type potential 15 NOVEMBER 1984 S. Erkoq and R. Sever Department of Physics, Middle East Technical University, Ankara, Turkey (Received 14 June 1984) A path-integral solution for a Mie-type potential is obtained. The exact eigensolutions for a spe- cial case of the potential are calculated. I. INTRODUCTION The path-integral formulation of Feynman' offers an alternative approach for solving dynamical problems in quantum mechanics. A great interest has recently been devoted- to the application of this technique to the exactly solvable problems by the Schrodinger equation. To apply this technique one usually follows the point canonical transformation and a change of time parameter to reduce the problem to a proper form that has an exact solution. Recently the H-atom and one-dimensional Morse- potential Green's function have been calculated with the path integral, by converting them to a four-dimensional harmonic oscillator, and a one-dimensional harmonic os- cillator with an additional potential barrier, respectively. In recent years Mie-type potentials have been used to study the dynamical properties of solids, and are given by 'I k V(x) =e k o. l o. (1) I — k x l — k x where e is the interaction energy between two atoms in a solid at x =o, and l & k is always satisfied. In the present study we solve the path integral for the one-dimensional Mie potential with l =2k combination. I Choosing the special case k = 1, corresponding to a Coulombic-type potential with an additional centrifugal potential barrier, we test the validity of our transforma- tions by comparing the, results with the known exact eigensolutions of a Coulombic potential. II. PATH INTEGRAL FOR V(x) The probability amplitude for a particle of mass m traveling from a position x, at time t, =0 to xb at time tb — — T in a Mie-type potential with l =2k, 2k k V(x) = Vo — — — —, Vo — — 2@k (2) 1 o. 1 o. 2k x k x can be written as the phase-space path integral in Carte- sian coordinates: K(xb, T;xe, 0) exp — f dt px — — V(x) 2~ A o 2m I (3) which is understood as the limiting case of its time-graded form n n+1 dp. ~ n+1 K(x, T;x„o)= lim f +dx; + ' e p n~~, . ) ', . & 2m Pi p(x — x )) — e 2' — V(x; ) (4) where e=t; t; &, (n+1— )e=tb t, =T, and xo — —— x„x„+~ xb. — — We define a new coordinate Q E (0, ao ) with the point canonical transformations ~g 1/k p g 1 — I/k P k CT generated from the function k F2(x, P)= — P . X After evaluating the Jacobian at the point b, the path integral in Eq. (3) takes the form '2 k Qi i/k f DQDP i d PQ 1 k Qz~& ~/k~P2 o 1 1 1 2~ '"P m o 2m o k 2 g' Q 30 2117 1984 The American Physical Society