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The method of the Hypervirial Theorems is used to obtain analytically the ground-state· Schroedinger eigenfunction for a wide class of potentials in non-relativistic Quantum Mechanics.The whole scheme, despite its simplicity, yields in many cases a good approximation. The method of the Hypervirial Theorems (HVT)[1 — 7] is a powerful technique of Perturbation Theory without eigenfunctions. The energy eigenvalues in nonrelativistic Quantum Mechanics are obtained in the form of an expansion, the first terms of which are, in many cases, sufficient to achieve a reasonable approximation. The question arises whether the same technique can be used to obtain the corresponding eigenfunctions in a fair approximation. Here we address this question for quite a general class of potentials of the form : V(r)=-V0ffy 0 < r < o o (1) where V0 > 0, R > 0. The class of potentials we are interested in, is further specified by assuming that / is an even analytic function of χ = £ , with -jg |a ; =o< 0. Thus, such potentials behave like an harmonic oscillator potential near the origin. It will be also assumed in this note that the function / tends to zero at large r. Typical examples of that oscillator-like class of potentials is the Gaussian and the Poeschl-Teller-type potential V(r) = —Vocosh" i^J . Another example is the potential of the form: V(r) = ^ (2) 1 + eU; There has been shown [7] that the application of the Hypervirial Theorems technique enables one to study the class of potentials (1) in quite a general way and obtain approximate expressions not only for the energy eigenvalues but also for other quantities of physical interest, such as the expectation values of the kinetic and the potential energy operators of a particle moving nonrelativistically in such a potential and of the mean-square radius of its orbit in a given energy eigenstate. In the present study use is made of those results by considering , in the interest of simplicity, the ground-state of a particle ,of mass μ , moving in a potential of the above mentioned class. In an effort to obtain an approximate gföund-state eigenfuriction for a potential belonging to the class specified atUhe beginning, using the above mentioned approximate expressions for Elisala < r >n/(with η = 0 and / = 0),we proceed as follows: We break up the desired edgenfunction into two parts : uin(r) (XrKRv woo (r) = { (3) Uex{r) Ro < r < oo As long as Ro is sufficiently large, the "external" wave function uex (r) is: uex(r) = Cekr (4) where k = f£\Eœ\ (5) On the other hand, the solution in the internal region, u tn (r) , can be approximated by the groundstate edgenfunction of an appropriately specified harmonic oscillator (HO): uin (r) = Nre ^ (6) A plausible requirement, for the determination of the H.O. parameter 6, is that the mean square radius (m.s.r) of the H.O. ground state orbit equals the corresponding one obtained through the HVT scheme: < r > S ° = < r >oo (7)

Approximate Determination of the ground-state Schroedinger Eigenfunction by Means of the Hypervirial Theorems Technique