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 $$\upvarphi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>φ</mml:mi> </mml:math> -motion, the QA is not applied: instead, the exact solution is used. The QA is then applied to both the $$\uptheta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>θ</mml:mi> </mml:math> -motion and the r-motion. The resulting QA wave function for the $$\uptheta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>θ</mml:mi> </mml:math> -motion is valid for some, but not all values of $$\uptheta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>θ</mml:mi> </mml:math> . Besides, the resulting QA wave function for the $$\uptheta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>θ</mml:mi> </mml:math> -motion is valid only with the substitution of L(L $$+$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>+</mml:mo> </mml:math> 1) by (L $$+$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>+</mml:mo> </mml:math> 1/ $$2)^{\textrm{2}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mtext>2</mml:mtext> </mml:msup> </mml:math> , 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 $$+$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>+</mml:mo> </mml:math> 1) by (L $$+$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>+</mml:mo> </mml:math> 1/ $$2)^{\textrm{2}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mtext>2</mml:mtext> </mml:msup> </mml:math> in the centrifugal energy term. As a result of the above standard procedure, the obtained QA wave functions are valid only for (L $$+$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>+</mml:mo> </mml:math> 1/2) greater or of the order of $$\hbox {N}^{\mathrm {1/2}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mtext>N</mml:mtext> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:math> , where N is the principal quantum number. Besides, for potentials having the pole of the order 3 or higher at r $$=$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>=</mml:mo> </mml:math> 0 or for relatively large r, it would be preferable to have L(L $$+$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>+</mml:mo> </mml:math> 1) without the substitution by (L $$+$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>+</mml:mo> </mml:math> 1/ $$2)^{\textrm{2}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mtext>2</mml:mtext> </mml:msup> </mml:math> . 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) $$= -\hbox {a}_{\textrm{n}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>=</mml:mo> <mml:mo>-</mml:mo> <mml:msub> <mml:mtext>a</mml:mtext> <mml:mtext>n</mml:mtext> </mml:msub> </mml:mrow> </mml:math> / $$\hbox {r}^{\textrm{n}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mtext>r</mml:mtext> <mml:mtext>n</mml:mtext> </mml:msup> </mml:math> , where $$\hbox {a}_{\textrm{n}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mtext>a</mml:mtext> <mml:mtext>n</mml:mtext> </mml:msub> </mml:math> &gt; 0 and the integer n &gt; 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 e