## 11. Coulomb potential.

If a particle P with mass m is attracted by a body S at the origin O with a force inversely proportional to the square of its distance x (x > 0) from S, its potential energy is expressed by a function having form where C is a positive proportionality constant depending on P, S and on medium between them. In this case the potential well has an outline like the following If the total energy of the particle is negative, the graph of energy is like this The Schrödinger equation in this case is If we let we have and after, from (11.1.1), To simplify the notation we let x→∞ ⇒ ψ→0 , so we may assume where φ is a function to determine later, with a singularity for ξ=0 but finite for every other ξ.

By differentiating twice the ψ(x) in (11.6) with respect to ξ we have and this expression of the second derivative in (11.5) gives Now, like in the previous section, we assume that and therefore By using these values in (11.7) we obtain Since φ is not defined for x=0, we do not consider the terms of the expansion with negative exponents of ξ The sum in (11.8) is zero only if all the coefficients of the powers of ξ are zero: Like in the previous section, we have obtained a recursive relationship between the coefficients of the series expansion. Furthermore, the expansion reduces to a polynomial, and therefore is finite for every ξ, if a not zero coefficient is followed by a coefficient equal to zero. Let k be the index of the last not zero coefficient. From (11.9) we obtain that is A must be a not zero even natural number (in (11.10) n represents a natural number ≥ 1).

By recovering the definition of A from (11.4), from (11.10) we obtain Once more we reach the conclusion that the energy of a particle can have only discrete, quantized, values. The minimum value is If the particle is an electron in the field of a proton, we have and, from (11.1), The equation (11.13) coincides with that obtained by N. Bohr to explain the hydrogen spectrum, under the hypothesis that the electron has circular orbits such that its angular momentum is an integer multiple of the reduced Planck constant In fact, if r is the orbital radius and v the electron speed, from (11.14) Coulomb's force is a centripetal force By dividing the equation (11.16) by the equation (11.15) we obtain The total energy of an electron with circular orbit in the field of a proton is Finally, from (11.17) and (11.18) we obtain (11.13)

In conclusion, the Schrödinger equation provides a general theoretical foundation to explain results previously obtained on the basis of ad hoc hypotheses, like these of Bohr, assumed to justify specific experimental data.