Hydrogen-like atom Bohr radius

As examples of radial and angular wave functions, those for values of the principal quantum number, , up to 3 are given, respectively, in Tables 2.3 and 2.4. Z represents the atomic number (1 for the hydrogen atom, but the formulae shown represent hydrogen-like atoms such as He for which Z = 2), and the term is the atomic unit of distance, explained below, and known as the Bohr radius. It has the value 52.9177 pm. 

A positronium atom is a hydrogen-like atom consisting of an electron and a positron (an antielectron with charge -1-6 and mass equal to the electron mass). Find the energy of a positronium atom in the Is state. Describe the classical motion of the two particles about the center of mass. Find the value of the Bohr radius for positronium. 

Hg. 3.12 Tlie orbital for a Is electron in the hydrogen atom. The radius of the sphere in which it is 90% likely that the electron will be found, is about 100 pm. The single radius at which the Is electron is most likely to be found is a distance 52.9 pm from the nucleus. This may be compared with the Bohr theory of the atom, where it was assumed that the electron was certain to be found at a radius of 52.9 pm. 

Consider what the H2 wave function would look like for large values of the inter-nuclear separation R. When the electron is neeu nucleus a, nucleus b is so far away that we essentially have a hydrogen atom with origin at a. Thus, when r is small, the ground-state electronic wave function should resemble the ground-state hydrogen-atom wave function of Eq. (6.104). We have Z = 1, and the Bohr radius Oq has the numerical value 1 in atomic units hence (6.104) becomes... 

To a rough approximation, an electron in the ground state of a hydrogen atom can be considered to be like a particle in a onedimensional box with a length equal to twice the Bohr radius,... 

Table 3.3 The unnormalized radial functions in atomic hydrogen-like orbitals (equation 3.26). The distance r is measured in units of the Bohr radius, 0.529 A...

Bohr supposed that the electron in the hydrogen atom revolves around the proton in an orbit just like a planet revolving around the sun. He assumed that only orbits of certain definite sizes are possible. The possible orbits were determined by the assumption that the product of the momentum of the electron and the radius of the orbit must be an exact multiple of Planck s constant divided by 2tt. He calculated the energy of the atom for each of these supposed possible orbits, and so... 

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