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2. Ground State |
Hydrogen Atom |
The solution of the hydrogen atom in quantum mechanics is one of its earliest triumphs, both because it demonstrates the validity of the quantum approach and because it provides a stepping-stone for solving more difficult quantum problems. The hydrogen atom is one of the few quantum solutions that can be obtained exactly in analytical form. Other more difficult problems require approximation methods or numerical simulations.
The solution to the hydrogen atom begins with the insertion
of Coulomb’s Law into the radial
wave equation. The solution of this equation shows that the energy
is quantized according to the equation 
, where k is Coulomb’s constant, e is the
charge on an electron, Ze is the charge on the nucleus of the
hydrogen-like atom, a0 is the Bohr radius of the hydrogen
atom, and n is the principle quantum number. The radial wavefunctions
depend upon both the principle quantum number n and the
orbital quantum number 
.
When the radial wavefunctions are combined with the orbital wave functions, one is able to obtain the complete ground state wavefunction of the hydrogen atom. This allows one to determine the probability of finding the electron in various positions around the hydrogen atom. A plot of the probability density of the electron in the ground state of the hydrogen atom shows that it sits like an invisible symmetrical cloud surrounding the nucleus of the atom.
In a similar manner, the wavefunctions of various excited states of the hydrogen atom can be determined along with their probability densities. A plot of the various probability densities show how the excited hydrogen atoms exhibit cylindrical symmetry with varying amounts of angular momentum present.
Radial Wave Equation See Serway pp. 280-281 or Tipler p.
299.
Ground State See Serway pp. 284-286 or Tipler p. 302.
Excited States See Serway pp. 287-294 or Tipler p. 304-6.
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