page_title: Trigonometric Rosen–Morse potential
section_title: The 
    
      
        
          
            V
            
              t
              R
              M
            
            
              (
              ℓ
              +
              1
              ,
              b
              ,
              1
              )
            
          
          (
          χ
          )
        
      
      {\displaystyle V_{tRM}^{(\ell +1,b,1)}(\chi )}
    
   case: Electric charge confinement on 
    
      
        
          
            S
            
              3
            
          
        
      
      {\displaystyle S^{3}}
    
   and a dipole potential shaped after 
    
      
        
          α
          Z
          cot
          ⁡
          χ
        
      
      {\displaystyle \alpha Z\cot \chi }
hierarchical_section_title: Trigonometric Rosen–Morse potential / The 
    
      
        
          
            V
            
              t
              R
              M
            
            
              (
              ℓ
              +
              1
              ,
              b
              ,
              1
              )
            
          
          (
          χ
          )
        
      
      {\displaystyle V_{tRM}^{(\ell +1,b,1)}(\chi )}
    
   case: Electric charge confinement on 
    
      
        
          
            S
            
              3
            
          
        
      
      {\displaystyle S^{3}}
    
   and a dipole potential shaped after 
    
      
        
          α
          Z
          cot
          ⁡
          χ
        
      
      {\displaystyle \alpha Z\cot \chi }
caption_attribution_description: English: Schematic presentation of the shape of the Charge Dipole potential on S^3. This potential is confining in the sense that with growing distance from the equator, it becomes infinite near the poles, thus preventing the "escape of the charges, and keeping them "confined on S^3. Instead, with the decrease of the distances to the equator, it becomes gradually vanishing, and leaves the charges in this region "asymptotically free".
context_page_description: The trigonometric Rosen–Morse potential, named after the physicists Nathan Rosen and Philip M. Morse, is among the exactly solvable quantum mechanical potentials.
context_section_description: To the amount the cotangent function solves the Laplace–Beltrami equation on ,
  it represents a harmonic function on , a reason for which  Schrödinger considered it as the counterpart to the Coulomb potential in flat space, by itself a harmonic function to the  Laplacian. Due to this analogy, the cotangent function is  frequently termed to as "curved Coulomb" potential. Such an interpretation ascribes the cotangent potential to a single charge source, and here lies a severe problem. Namely, while open spaces, as is ,  support single charges, in closed spaces single charge can not be defined in a consistent way. Closed spaces are necessarily and inevitably charge neutral meaning that the minimal fundamental degrees of freedom allowed on them are charge dipoles (see Fig. 1).
  For this reason, the wave equation
  which  transforms upon the variable change, , into the familiar one-dimensional Schrödinger equation with the  trigonometric Rosen–Morse potential,
  in reality describes quantum motion of a charge dipole perturbed by the field due to another charge dipole, and not the motion of a single charge within the field produced by another charge. Stated differently, the two equations (16) and (17) do not describe strictly speaking a Hydrogen Atom on , but rather quantum motion on  of a light  dipole perturbed by the dipole potential of another very heavy dipole, like the H Atom, so that the reduced mass, , would be of the order of the electron mass and could be neglected in comparison with the energy.
  In order to understand this decisive issue, one needs to focus attention to the necessity of  ensuring validity on  of both the  Gauss law and the superposition principle for the sake of being capable to formulate electrostatic there. With the cotangent function in (15) as a single-source potential, such  can not be achieved. Rather, it is necessary to prove that the cotangent function represents a dipole potential. Such a proof has been delivered in. To understand the line of arguing of  it is necessary to go back to the expression for the Laplace operator in (5) and before considering the hyper-radius as a constant, factorize this space into a time line and . For this purpose, a "time" variable is introduced via the logarithm of the  radius. Introducing this variable change in (7) amounts to the following Laplacian,
  The  parameter is known as "conformal time", and the whole procedure is referred to as "radial quantization". Charge-static is now built up in setting =const in (19) and calculating the harmonic function to the remaining piece, the so-called conformal Laplacian, , on , which is read off from (19) as
  where we have chosen , equivalently, .
  Comparison of the latter expression to (15) shows that the correct operator to be employed in the calculation of the harmonic function is not the regular  Laplace–Beltrami operator,  but the so-called conformal Laplace–Beltrami operator,  in (20). The Green function to  has been calculated for example in. Its values at the respective South and North poles, in turn denoted by , and , are reported as
  From them one can now construct the dipole potential for a fundamental charge  placed, say, on the North pole, and a fundamental charge of opposite sign, , placed on the antipodal South pole of . The associated potentials,  and , are then constructed through multiplication of the respective Green function values by the relevant charges  as
  In now assuming validity of the superposition principle, one encounters a Charge Dipole (CD) potential to emerge at a point  on  according to
  The electric field to this dipole is obtained in the standard way through differentiation as
  and coincides with the precise expression prescribed by the Gauss theorem on , as explained in. Notice that  stands for dimension-less charges. In terms of dimensional charges, , related to  via
  the potential perceived by another charge , is
  For example, in the case of electrostatic, the fundamental charge  is taken the electron charge, , in which case the special notation of
  is introduced for the so-called fundamental coupl
caption: Figure 2: Schematic presentation of the shape of the 
    
      
        
          
            
              C
              _
            
          
        
      
      {\displaystyle {\underline {C}}}
    
  harge 
    
      
        
          
            
              D
              _
            
          
        
      
      {\displaystyle {\underline {D}}}
    
  ipole potential 
    
      
        
          
            V
            
              C
              D
            
          
          (
          χ
          )
        
      
      {\displaystyle V_{CD}(\chi )}
    
   in (25) on 
    
      
        
          
            S
            
              3
            
          
        
      
      {\displaystyle S^{3}}
    
  . This potential is confining in the sense that with growing distance from the equator, it becomes infinite near the poles, thus preventing the "escape" of the charges, and  keeping  them "confined" on 
    
      
        
          
            S
            
              3
            
          
        
      
      {\displaystyle S^{3}}
    
  . Instead, with the decrease of the distances to the equator, it becomes gradually vanishing, and leaves the charges in this region "asymptotically free". In this way, charge-statics on closed spaces provide convenient templates for simulations of confinement phenomena, the most prominent being the confinement of color charges known from the quantum chromodynamics (QCD).
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