(Originais Teachy 2023) - Question Easy of Physics

In Bohr's model for the hydrogen atom, electrons orbit the nucleus in stable orbits, with radii determined by an equation that takes into account Planck's constant, the electron's charge, the vacuum permittivity, and the electrostatic force between the electron and the proton. From quantum theory, we know that electronic energy levels are discrete and quantized, represented by positive integers n, where n = 1, 2, 3,..., and correspond to the different layers of the atom. Consider an electron in the hydrogen atom in its outermost allowed orbit, i.e., in the orbit with the highest possible value of n. Determine the radius of the electron's orbit as a function of n and the necessary physical constants. To do this, use Bohr's equation to calculate the radius of the electronic orbit and present the complete steps of your reasoning, including the expression of Planck's constant and the values of the relevant physical constants. Then, briefly discuss how quantum mechanics generalizes Bohr's model for the hydrogen atom, considering the wave nature of the electron and Heisenberg's uncertainty principle.