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You are deal with action variables so the integrations extrem are always give by the classical trajectory. So for the hidrogen atom in bound state they are 2 times the distance between R1 and R2.

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How do I transform equation (1) to equation (2) plug $R(r)=u(r)/r$ into (1), you'll get (2) immediately, where $k(x)$ would the expression before $R(r)$ in the second term of (2) and what do I use for the bounds of integration in equation (4) to get the energy eigenvalue? $\int_0^\infty$

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The bounds for r should still be the classical turning points, as you mentioned for the harmonic oscillator. Presumably you're in a bound state of Hydrogen, i.e. have an energy of the form $\frac{-13.6 eV}{n^2}$ for some integer n. The problem then reduces to finding the zeros of the equation \frac{-13.6 eV}{n^2} = -\frac{e}{r^2} - \frac{l(l+1) ...

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