Quasiclassical approximation does not have to be applied to all degrees of freedom: the solution for the S-states of singular attractive potentials

Abstract

For the spherically symmetric potentials V(r), the quasiclassical approximation (QA) usually is applied as follows. For the $$\upvarphi $$ φ -motion, the QA is not applied: instead, the exact solution is used. The QA is then applied to both the $$\uptheta $$ θ -motion and the r-motion. The resulting QA wave function for the $$\uptheta $$ θ -motion is valid for some, but not all values of $$\uptheta $$ θ . Besides, the resulting QA wave function for the $$\uptheta $$ θ -motion is valid only with the substitution of L(L$$+$$ + 1) by (L$$+$$ + 1/$$2)^{\textrm{2}}$$ 2 ) 2 , where L is the angular momentum quantum number. Then the QA is applied to the r-motion, but again with the substitution of L(L$$+$$ + 1) by (L$$+$$ + 1/$$2)^{\textrm{2}}$$ 2 ) 2 in the centrifugal energy term. As a result of the above standard procedure, the obtained QA wave functions are valid only for (L $$+$$ + 1/2) greater or of the order of $$\hbox {N}^{\mathrm {1/2}}$$ N 1 / 2 , where N is the principal quantum number. Besides, for potentials having the pole of the order 3 or higher at r $$=$$ = 0 or for relatively large r, it would be preferable to have L(L $$+$$ + 1) without the substitution by (L$$+$$ + 1/$$2)^{\textrm{2}}$$ 2 ) 2 . This is yet another restriction of the QA validity in this procedure. In the first part of the present paper, we provide a better alternative. This alternative can also be applied to some potentials whose geometrical symmetry is not spherical. In the second part of the present paper, we focus on the attractive singular spherically-symmetric potentials V(r) $$= -\hbox {a}_{\textrm{n}}$$ = - a n /$$\hbox {r}^{\textrm{n}}$$ r n , where $$\hbox {a}_{\textrm{n}}$$ a n > 0 and the integer n > 2. They are encountered in many physical systems. There exists a paradigm that “singular attractive potentials do not lead to physically reasonable results” because eigenfunctions depend on the chosen cutoff at small r. In the present paper we disprove this paradigm. We demonstrate that while the unnormalized eigenfunctions of singular attractive potentials do depend on the cutoff at small r, there is no such problem for the corresponding normalized eigenfunctions. Namely, the normalized eigenfunctions, corresponding to any two different sufficiently small cutoffs (all other parameters being the same), are equal to each other or differ just by the sign. The sign difference does not affect the probability density: the latter is invariant to the variations of the cutoff. Thus, we prove that singular attractive potentials actually do lead to physically reasonable results. In doing so, we also demonstrated that a divergent integral can lead to physically meaningful results independent of the cutoff. This is a counterintuitive result of the importance from the fundamental point of view both for physics and mathematics.