For the hydrogen atom in its ground state calculate (a) the probability density w?(r) and (b) the radial probability density P(r) for r = 2.37a, where a is the Bohr radius. (a) Number Units (b) Number Units
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- ground state wave function of Hydrogen (with l=m=0), calculate the electron’s average distance from the proton in terms of the Bohr radius, a ∼0.5 ×10−10m.The quantum-mechanical treatment of the hydrogen atom gives an expression for the wave function ψ, , of the 1s orbital:where ris the distance from the nucleus and a₀ is 52.92 pm. The electron probability density is the probability of finding the elec-tron in a tiny volume at distance rfrom the nucleus and is pro-portional to ψ² . The radial probability distribution is the total probability of finding the electron at all points at distance rfromthe nucleus and is proportional to 4πr² ψ² . Calculate the values(to three significant figures) of ψ, ψ² , and 4πr2² ψ² to fill in the fol-lowing table, and sketch plots of these quantities versus r.(1) Find the average orbital radius for the electron in the 3p state of hydrogen. Compare your answer with the radius of the Bohr orbit for n=3. (2) What is the probability that this electron is outside the radius given by the Bohr model?
- Compute and compare the electrostatic and gravitational forces in the classical hydrogen atom, assuming a radius 5.3 x 10-11 m.(b) A photon is emitted by a doubly ionised lithium atom (Li²+) when an electron makes a transition to the ground state. The wavelength of the photon is measured to be 10.83 nanometres. Determine the principal quantum number and the energy of the initial state The atomic number of lithium is Z = 3.