The normalized 1s wavefunction (n = 1, l = 0) of any one-electron atom or ion can bewritten as100 (r, 0, 0) = R₁0(r)Y(0,0) =Z3παρ 3where Z is the nuclear charge (number of protons) and a =πmpe2€0h²(a) Construct an integral expression, in r, 0, and that can be used to determine theexpectation value for the hydrogen atom potential energy, (V), and evaluate theintegrals over and p. Be mindful of the integration volume element.En==e(b) Define x = Zr/ao and use this form to simplify the integral over r and evaluate it (usingthe tabulated integral provided above) to obtain an expression for (V). Your resultshould include a factor of V(a). Use the potential energy operatorV(r) =where e represents the electron charge and is the permittivity of a vacuum. Expressyour result in terms of V(a) (that is, express it as something times V(a)).(c) The energy levels of one-electron atoms and ions can be written in terms of a asZZ²e²(2²72²2) V (ao) = (E)8περαon2Using this and your result from part (b), find the expectation value for the kineticenergy, (T), of the electron in 100 (r, 0, $). What is the value of the ratio (T)/(V)?=-Zr/aoZe²4π€oris the Bohr radius. purpose, the integralS xn e-ßxdr :=will be helpful after you have made a suitable change of coordinate.n!Bn+1

Given: ψ100r,θ,ϕ=R10(r)Y00θ,ϕ=Z3πa03 e-Zr/a0a0 = ε0h2πmee2 is Bohr radius

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