The hindered rotations are of crucial relevance in quantum mechanics due to the ability to describe phenomena ranging from the behavior of molecules, through the inversion of chirality and arriving at the coupling of angular momenta and their applications in quantum composition [1] - [11] . Despite the fact that the rotation operator, or the second derivative operator on a cyclic coordinate, is closely linked to the principles of physics and their interpretation, the only analytical solutions known in the literature are those with constant potential [12] - [19] . In other words, we have a single basis set to expand the solutions of physics equations such as Schrödinger’s. Consequently, for non-constant rotation potentials, the only possible approach is that which leads to numerical solutions of the problem addressed.

As is well known, numerical solutions make the process of understanding the physics of the problem an even greater challenge and, when the solutions represent the starting point of other problems as in the case of the study of chemical reactions, the complexity of the numerical treatment, together with the propagation of the error, with a certain frequency they give results without significance. It is with this problem in mind that, in this article, the analytical solutions of the rotational quantum mechanical problem with divergent potential at +∞ for a specific value of the angle that describes the rotation are described.

The article is divided as it follows. In the next section, the problem is mathematically formalized and the solutions are calculated by dividing them into even and odd types. The section “Remarks and perspectives” continues the article presenting some important observations. Appendixes “c coefficients” and “ $s_{i,m}$” conclude the paper offering useful information for the reader that wants to build up the solutions of the problem.

The Schrödinger equation solved here is defined mathematically by the free rotor operator

$-\frac{{\text{d}}^2{\Phi }_i\left(\varphi \right)}{\text{d}{\varphi }^2}$ (1)

plus an interaction potential $V\left(\varphi \right)$ appropriately chosen to represent the hindered rotation. Following, this potential is presented and the eigenfunctions analytically calculated.

Analytic eigenfunctions for a hindered potential applied to a quantum rotor problem were found. Even functions are expressed by finite weighted sums of $\text{cos}\left(m\varphi \right)$ functions while for the odd ones the weighted sums are over infinite terms $\text{sin}\left(m\varphi \right)$. In both cases, the coefficients of the expansions are here determined analytically and the eigenvalues ${\lambda }_i$ for the given potential are calculated as $i\left(i+2\right)$, where i is not negative integer or half-integer for even and odd solutions, respectively.

The eigenproblem described here represents the first step for a generalization of the free rotation quantum problem to a hindered one. The second step is represented by the introduction of a more general potential given by the expression

$V\left(\varphi \right)=\frac{a\cos \left(n\varphi \right)}{1-a\cos \left(n\varphi \right)}.$ (24)

The behavior of the Schrödinger equation under the action of this potential is under study and more investigations are, at the present moment, necessary. However, it is possible to anticipate that, for values of $a=\pm 1$ and n positive integer, ${\lambda }_i=i\left(i+2n\right)$.

But the main challenge is to make a vary in the open interval $\left(-1,1\right)$ because the resulting potential presents one or more finite barriers (It depends on the value of n) and eigenvalues with energies below the barriers. Consequently, the eigenfunctions represent a set of orthogonal functions completely new compared to those encountered in the literature that are never solutions of potentials with finite barriers.

Such a cyclic potential can be used to describe a variety of physical systems ranging from the roto-torsions of molecules, through the inversion of chirality, and arriving at the description of confined atoms and entangled coordinates predicted in field theories with more than four dimensions.

In this appendix, Eq.(5) is used to find the coefficients of the expansion given at the beginning of section 2.2. Using Eq. (3) and considering the case of $M_{\max }=i=4$, the condition to be respected is expressed by

$\begin{array}{l}{c}_{4,4}\left({\lambda }_4-24\right)\cos \left(5\varphi \right)\\ +\left(\left(c_{4,3}-2c_{4,4}\right){\lambda }_4+32c_{4,4}-15c_{4,3}\right)\cos \left(4\varphi \right)\\ +\left(\left(c_{4,4}-2c_{4,3}+c_{4,2}\right){\lambda }_4-8c_{4,4}+18c_{4,3}-8c_{4,2}\right)\cos \left(3\varphi \right)\\ +\left(\left(c_{4,3}-2c_{4,2}+c_{4,1}\right){\lambda }_4-3c_{4,3}+8c_{4,2}-3c_{4,1}\right)\cos \left(2\varphi \right)\\ +\left(\left(c_{4,2}-2c_{4,1}+2c_{4,0}\right){\lambda }_4+2c_{4,1}\right)\cos \left(\varphi \right)\\ +\left(c_{4,1}-2c_{4,0}\right){\lambda }_4+c_{4,1}=0.\end{array}$ (A1)

Observe that the coefficient of $\text{cos}\left(5\varphi \right)$ is automatically zero because ${\lambda }_4=4\left(4+2\right)=24$. Consequently the equation above is reduced to

$\begin{array}{l}\left(9c_{4,3}-16c_{4,4}\right)\cos \left(4\varphi \right)\\ +\left(16c_{4,4}-30c_{4,3}+16c_{4,2}\right)\cos \left(3\varphi \right)\\ +\left(21c_{4,3}-40c_{4,2}+21c_{4,1}\right)\cos \left(2\varphi \right)\\ +\left(24c_{4,2}-46c_{4,1}+48c_{4,0}\right)\cos \left(\varphi \right)\\ +25c_{4,1}-48c_{4,0}=0\end{array}$ (A2)

that is verified if

$\left(9c_{4,3}-16c_{4,4}\right)=0$

$\left(16c_{4,4}-30c_{4,3}+16c_{4,2}\right)=0$

$\left(21c_{4,3}-40c_{4,2}+21c_{4,1}\right)=0$

$\left(24c_{4,2}-46c_{4,1}+48c_{4,0}\right)=0$

$25c_{4,1}-48c_{4,0}=0.$ (A3)

Apparently, it is a system of five linear equations and five unknown coefficients $c_{4,m=0,\cdots ,4}$, but this is not correct because one of these is redundant. This can be demonstrated imposing (for convenience) $c_{4,0}=25$ so that, to respect the last of Eq.(A3), $c_{4,2}=48$. Consequently

$\left(9c_{4,3}-16c_{4,4}\right)=0$

$\left(16c_{4,4}-30c_{4,3}+16c_{4,2}\right)=0$

$\left(21c_{4,3}-40c_{4,2}+1008\right)=0$

$\left(24c_{4,2}-1008\right)=0.$ (A4)

Doing $c_{4,2}=42$:

$\left(9c_{4,3}-16c_{4,4}\right)=0$

$\left(16c_{4,4}-30c_{4,3}+672\right)=0$

$\left(21c_{4,3}-40c_{4,2}-672\right)=0.$ (A5)

Doing $c_{4,2}=42$:

$\left(9c_{4,3}-16c_{4,4}\right)=0$

$\left(16c_{4,4}-30c_{4,3}+672\right)=0$

$\left(21c_{4,3}-672\right)=0.$ (A6)

Doing $c_{4,3}=32$:

$\left(288-16c_{4,4}\right)=0$

$\left(16c_{4,4}-288\right)=0$ (A7)

and this demonstrates the redundancy because the two conditions are identical and $c_{4,4}=18$. To uniquely define ${\Phi }_4\left(\varphi \right)$ the just found coefficients $c_{4,m=0,\cdots ,4}$ have to been scaled by a normalization factor so that ${\displaystyle {\int }_0^{2\pi }{\left|{\Phi }_4\left(\varphi \right)\right|}^2\text{d}\varphi }=1$.

Taking $s_{i,1}=1/2$, they follow the expressions of the firsts eleven coefficients $s_{i,m}$.

$s_{i,2}=\frac{{\lambda }_i-1}{{{\displaystyle \prod }}_{j=0}^0\left({\lambda }_i-j\left(j+2\right)\right)},$ (B8)

$s_{i,3}=\frac{3{\lambda }_i^2-17{\lambda }_i+16}{2{{\displaystyle \prod }}_{j=0}^1\left({\lambda }_i-j\left(j+2\right)\right)},$ (B9)

$s_{i,4}=\frac{2\left({\lambda }_i-5\right)\left({\lambda }_i^2-11{\lambda }_i+12\right)}{{{\displaystyle \prod }}_{j=0}^2\left({\lambda }_i-j\left(j+2\right)\right)},$ (B10)

$s_{i,5}=\frac{5{\lambda }_i^4-170{\lambda }_i^3+1817{\lambda }_i^2-6648{\lambda }_i+5760}{2{{\displaystyle \prod }}_{j=0}^3\left({\lambda }_i-j\left(j+2\right)\right)},$ (B11)

$s_{i,6}=\frac{3{\lambda }_i^5-185{\lambda }_i^4+3965{\lambda }_i^3-35207{\lambda }_i^2+119040{\lambda }_i-100800}{{{\displaystyle \prod }}_{j=0}^4\left({\lambda }_i-j\left(j+2\right)\right)},$ (B12)

$s_{i,7}=\frac{7{\lambda }_i^6-707{\lambda }_i^5+26453{\lambda }_i^4-455137{\lambda }_i^3+3621696{\lambda }_i^2-11620800{\lambda }_i+9676800}{2{{\displaystyle \prod }}_{j=0}^5\left({\lambda }_i-j\left(j+2\right)\right)},$(B13)

$\begin{array}{c}{s}_{i,8}=\frac{4\left({\lambda }_i^7-154{\lambda }_i^6+9184{\lambda }_i^5-269058{\lambda }_i^4+4055247{\lambda }_i^3\right)}{{{\displaystyle \prod }}_{j=0}^6\left({\lambda }_i-j\left(j+2\right)\right)}\\ +\frac{4\left(-29988756{\lambda }_i^2+92685600{\lambda }_i-76204800\right)}{{{\displaystyle \prod }}_{j=0}^6\left({\lambda }_i-j\left(j+2\right)\right)},\end{array}$ (B14)

$\begin{array}{c}{s}_{i,9}=\frac{3\left(3{\lambda }_i^8-668{\lambda }_i^7+59514{\lambda }_i^6-2726652{\lambda }_i^5+68754523{\lambda }_i^4\right)}{2{{\displaystyle \prod }}_{j=0}^7\left({\lambda }_i-j\left(j+2\right)\right)}\\ +\frac{3\left(-947476560{\lambda }_i^3+6646620480{\lambda }_i^2-19971302400{\lambda }_i+16257024000\right)}{2{{\displaystyle \prod }}_{j=0}^7\left({\lambda }_i-j\left(j+2\right)\right)},\end{array}$(B15)

$\begin{array}{c}{s}_{i,10}=\frac{5{\lambda }_i^9-1545{\lambda }_i^8+195882{\lambda }_i^7-13207738{\lambda }_i^6+513568401{\lambda }_i^5}{{{\displaystyle \prod }}_{j=0}^8\left({\lambda }_i-j\left(j+2\right)\right)}\\ +\frac{-11686013181{\lambda }_i^4+150898575168{\lambda }_i^3-1017148380480{\lambda }_i^2}{{{\displaystyle \prod }}_{j=0}^8\left({\lambda }_i-j\left(j+2\right)\right)}\\ +\frac{2989666713600{\lambda }_i-2414168064000}{{{\displaystyle \prod }}_{j=0}^8\left({\lambda }_i-j\left(j+2\right)\right)},\end{array}$ (B16)

$\begin{array}{c}{s}_{i,11}=\frac{11{\lambda }_i^{10}-4565{\lambda }_i^9+792990{\lambda }_i^8-75173098{\lambda }_i^7+4252840471{\lambda }_i^6}{2{{\displaystyle \prod }}_{j=0}^9\left({\lambda }_i-j\left(j+2\right)\right)}\\ +\frac{-147622553073{\lambda }_i^5+3115657332432{\lambda }_i^4-38286831418560{\lambda }_i^3}{2{{\displaystyle \prod }}_{j=0}^9\left({\lambda }_i-j\left(j+2\right)\right)}\end{array}$ (B17)

$+\frac{250120955980800{\lambda }_i^2-722275190784000{\lambda }_i+579400335360000}{2{{\displaystyle \prod }}_{j=0}^9\left({\lambda }_i-j\left(j+2\right)\right)}.$