Expansions for Quasistationary States

Applications of 1/n expansions to calculations of energies and wave functions are considered, including quasistationary states. As illustrations we have examined the power-law, funnel, and Yukawa potentials. It has been shown that in many cases the method ensures high accuracy, even for small quantum numbers. The connection of these results with the properties of coherent states is briefly discussed. 

Consider a particle of mass M, subject to an attractive spherically symmetric potential characterized by a length parameter R thus, 

To develop systematically the 1/n-expansion, we note that comparison of the Schrodinger equation at r w ro = Rxq with the equation describing the harmonic oscillator shows that n is analogous to M/h. The amplitude of the vacuum fluctuation of this oscillator is proportional to (h/Mufy Accordingly we introduce 

Special consideration is required for the case E = 0 (i.e. for the moment of bound state appearance in a short-range potential). As a rule, the reduced energy, e(u), increases with increasing u, and at some value, u = u i, the level escapes to the continuous spectrum. In the lowest approximation (n — oo). 

With a further increase of u, the frequency u becomes zero at u — u. At this point the classical solution loses its stability and a collapse of two solutions occurs, one corresponding to stable and the other to unstable equilibrium points in the effective potential I7(r), which includes the centrifugal energy. Ai u i/ the potential U(r) possesses no minimum at real r, the equilibrium point escapes to the complex plane, and the coefficients of the 1/n expansion become complex. Such solutions, although they have no physical meaning in terms of classical mechanics, are of special importance in quantum mechanics They are just the solutions determining, within the 1/n expansion, not only the positions but also the widths of the resonance levels. 

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