Many scientists have speculated that the electron and proton charges might not be exactly equal in the magnitude. Experiments shows that the magnitude of the proton and electron charges are equal to within one part in 10²¹.
We can assume the proton and electron to be point masses those interaction is gives by Coulmb's law. In discussing molecules and atom's, we can usually be considering isolated systems, ignoring interactions and intermolecular interactions.
Instead of treat, just the hydrogenlike atom, which is consist of one electron and a nucleus of charge Ze.
For Z= 1, we have the hydrogen atom for Z=2, then the He⁺ ion, For Z=3, the Li²⁺ ion and so on.
The Hydrogenlike atoms the most important systems in quantum chemistry. This equation for atoms with more than one electron can't be Obtained because the interelectronic repulsion. If, as a first approximations, we can ignore these repulsion, then the electrons can be treating independently. The atomic wave functions will be approximate by a product of one-elecyron wave function (whether or it's not hydrogenlike) it's called an orbital.
Most precisely, orbital is a one - electron spatially wave function, where the word spatial means, that the wave function depends on the electrons three spatial coordinate is, x, y, and z or, r, θ and ø. The electrons spin adds a 4th coordinate to a one-electron wave function, gives it's called a spin-orbital. An orbitals for electron in atom is called as atomic orbitals. We can use atomic orbitals to contrast approximate wave functions for atoms with the many electrons. The orbitals are also used to contrast approximate wave functions for molecules. For the hydrogenlike atoms, let (x, y, z) be the coordinate of the electron relatives to the nucleus and let r= ix + jy + kʐ. The Coulmb's law of the force on the electron in the hydrogenlike atom is,
............ (1)
Where r/r is a units of vector in the r direction. The (-) minus sign indicating an attractive forces.
The possibility of small deviation from Coulmb's law have been considered. The Experiments have shows that if the Coulmb's - law force is written as a being proportional to r⁻²⁺ˢ, then lsl < 10⁻¹⁶. A deviations from Coulomb's law can be shows to implying a nonzero photon rest mass. There's no evidence exists for a nonzero photon mass, and data indicating that any such must lesser than 10⁻⁵¹
The force in equation (1) is central, and comparison with giving dV (r) /dr =Ze²/4πε₀r². The integration gives,
........ (2)
Where the integration constant has been taking as 0 to make V= 0, at infinite separation between the charges. For any two charges are Q1 and Q2 separating by distance is r₁₂,......... (3)
.......... (4)
Since the potential energy of this two types of particles systems depends only on the relative coordinate of the particles, and we can applying the results to reducing the problem to two one- particle problems. The translational motions of the atom as a whole simply added some to the total energy, and we can't concern ourselves,
with it. To deal with the internal motions of the system, we introduced a fictitious particle of the mass,
............. (5)
Where mₑ and mN are the electronic and nuclear masses. The particles of the reducing mass moves subject to the potential energy function.. Equation (4) and it's coordination (r, θ, ø ) are the spherical coordinate of one particle relative to the others.
The Hamiltonian for the internal motion is usually
......(6)
But, since V is a function of the r, coordinate only, and we have a one- particle central - force problem, and we can apply the results. We have for the wave functions,
Equation.. (7) and (8)
To save the time in writing, we can define the constant a as,
Equation.... (9)
And equation (8) becomes,
Equation... (10)
Good information
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