Two-Particle Systems: Central Potentials

  • Alberto Galindo
  • Pedro Pascual
Part of the Texts and Monographs in Physics book series (TMP)


In this chapter, we will study some characteristics of the Schrödinger equation for an isolated system of two spinless particles of masses M1 and M2, which interact through a potential V(r1, r2) dependent only on their positions. The Schrödinger equation for the stationary states of this system is
$$ \left[ { - \frac{{{\hbar^2}}}{{2{M_1}}}\Delta 1 - \frac{{{\hbar^2}}}{{2{M_2}}}\Delta 2 + V\left( {{r_1},{r_2}} \right)} \right]\Psi \left( {{r_1},{r_2}} \right) = E\Psi \left( {{r_1},{r_2}} \right) $$
It is of interest to introduce the center of mass position operator R and the relative position operator r defined by
$$ R = \frac{{{M_r}{r_1} + {M_2}{r_2}}}{{{M_1} + {M_2}}},\,\,\,r = {r_1} - {r_2} $$
The canonically conjugate variables are
$$ P = {p_1} + {p_2},\,\,p = \frac{{{M_2}{p_1} - {M_1}{p_2}}}{{{M_1} + {M_2}}} $$
i.e., the total and relative momenta, respectively.


Essential Spectrum Central Potential SchrOdinger Equation Radial Equation Magnetic Quantum Number 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1990

Authors and Affiliations

  • Alberto Galindo
    • 1
  • Pedro Pascual
    • 2
  1. 1.Departamento de Física TeóricaUniversidad Complutense Facultad de Ciencias FísicasMadridSpain
  2. 2.Departamento de Física TeóricaUniversidad de Barcelona Facultad de FísicaBarcelonaSpain

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