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Please answer the question below in a clean and correct manner as quick as you can

. Questions are associated to Introductory Quantum Mechanics.

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Consider electron in the ground state of the hydrogen atom, whose wavefunction takes
the simple form
(F ) =
1
nose r/a
(7)
where a is the Bohr radius.
First, show this state is properly normalized.
Given this state, we can define a probability density function along the r direc-
tion as
(8)
such that [ p(r)dr = 1. Find the radial position r at which p(r) is maximized, in
unit of the Bobr radius.
Derive the formula for the average of Coulomb potential energy (operator)
V(r) =
AREor
(9)
under the ground state, where co is the vacuum permittivity. In this result, replace the
Bohr radius a with 4xEgh?/(me?), where / is the reduced mass of the electron-proton
system.
Numerically, we know the ground state energy of hydrogen atom is -13.6 eV.
What is the value of this average potential energy?

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