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Series-termination quantization of the Coulomb radial equation

Writing

R = e^{- γ r} \sum_{n \geq 0} a_{n} r^{n}

gives

a_{n} = \frac{2 γn - β}{n ( n + 1 )} a_{n - 1} .

Normalizability forces the series to terminate, so

2 γ N = β

for a positive integer

N

and

E_{N} = - m q^{4} / (2 ℏ^{2} N^{2})

.

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