Abstract
The exact solutions of the non-relativistic and relativistic equations with a Coulomb field have been the subject both in quantum mechanics and in classical mechanics. The well-known exact solutions in almost all textbooks are important achievements at the beginning stage of quantum mechanics, which provided a strong evidence in favor of the quantum theory being correct. In this Chapter, we shall study its exact solutions, the shift operators, the mapping between the Coulomb potential and harmonic oscillator radial functions, the realization of the dynamic algebra su(1,1) and its generalized case, i.e., the Coulomb potential plus an inverse squared potential.
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Notes
- 1.
To show this formula explicitly, we want to derive it in more detail. It is shown from Eqs. (7.33) and (7.35) that
(7.36)which can be further modified to
(7.37)from which, together with Eq. (7.35) again, we have
(7.38)Moreover, this equation can be rewritten as
(7.39)Using Eq. (7.35) once again, we have
$$ (\alpha-1)L_{n+1}^{\alpha}(y)-(n+\alpha)L_{n+1}^{\alpha-2}(y)-(\alpha+y-1)L_{n}^{\alpha}(y)=0,$$(7.40)which is nothing but Eq. (7.41).
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Dong, SH. (2011). Coulomb Potential. In: Wave Equations in Higher Dimensions. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-1917-0_7
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