Hydrogen Atom

In 1897 J. J. Thomson discovered the electron, a negatively charged particle more than two thousand times lighter than a hydrogen atom.  In 1906 Thomson suggested that each atom contained a number of electrons roughly equal to its atomic number.  Since atoms are neutral, the charge of these electrons must be balanced by some kind of positive charge.  Thomson proposed a 'plum pudding' model, with positive and negative charge filling a sphere only one ten billionth of a meter across.  This plum pudding model was generally accepted.  Even Thomson's student Rutherford, who would later prove the model incorrect, believed in it at the time.
In 1911 Ernest Rutherford proposed that each atom has a massive nucleus containing all of its positive charge, and that the much lighter electrons are outside this nucleus.  The nucleus has a radius about ten to one hundred thousand times smaller than the radius of the atom.  Rutherford arrived at this model by doing experiments.  He scattered alpha particles off fixed targets and observed some of them scattering through very large angles.  Scattering at large angles occurs when the alpha particles come close to a nucleus.  The reason that most alpha particles are not scattered at all is that they are passing through the relatively large 'gaps' between nuclei.Links: The Rutherford Experiment
Rutherford revised Thomson's 'plum pudding' model, proposing that electrons orbit a positively charged nucleus, like planets orbit a star.  But orbiting particles continuously accelerate, and accelerating charges produce electromagnetic radiation.  According to classical physics the planetary atom cannot exist.  Electrons quickly radiate away their energy and spiral into the nucleus. In 1915 Niels Bohr adapted Rutherford's model by saying that the orbits of the electrons were quantized, meaning that they could exist only at certain distances from the nucleus.  Bohr proposed that electrons did not emit EM radiation when moving in those quantized orbits.

Quantum mechanics now predicts what measurements can reveal about atoms.  The hydrogen atom represents the simplest possible atom, since it consists of only one proton and one electron.  The electron is bound, or confined. Its potential energy function U(r) expresses its electrostatic potential energy as a function of its distance r from the proton. 

U(r) = -q2/(4πε0r) = -e2/r. Here e2 is defined as q2/(4πε0).  In SI unit 1/(4πε0) = 9*109 Nm2/C2, and q = 1.6*10-19 C. The figure on the right shows the shape of U(r) in a plane containing the origin.  The potential energy is chosen to be zero at infinity.  The electron in the hydrogen atom is confined in the potential well, and its total energy is negative.

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The energy levels in a hydrogen atom can be obtained by solving Schrödinger’s equation in three dimensions.  We have to solve the radial equation  

(-ħ²/(2m))∂²(rR))/∂r²  + (l(l + 1)ħ²/(2mr²))(rR) - (E - e²/r)(rR) = 0

or
∂²(rR))/∂r² + [(2m/ħ²)(E + e²/r) - l(l + 1)/r²](rR) = 0,
or
∂²(rR))/∂r² + k²(r)(rR) = 0,

with k²(r) = (2m/ħ²)(E + e²/r) - l(l + 1)/r².

To continue reading click on the following link:

http://electron6.phys.utk.edu/phys250/modules/module%203/hydrogen_atom.htm