Materials in Electronics/The Hydrogen Atom
The hydrogen atom is the simplest possible atom, comprising a single proton and a single electron.
To determine the wavefunction of the atom's electron, we must solve the 3D Schrödinger wave equation with the Coulomb potential of the single proton:
We reexpress the wavefunction in spherical coordinates:
If there is no angular dependence, we can simple solve for the radial component, r.

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The Schrödinger equation for the radial solution of the hydrogen atom is then:
Ground State[edit]
For the lowest energy state, this differential equation can be solved by the following trial solution:
Substituting in, we get:
The coefficents of r^{0} and r^{−1} must be separately satisfied:
We therefore have values for a and E:
Higher states[edit]
Higher energy levels have a more complex wavefunction:
where L_{n}(r) is the Laguerre polynomial of order n. It is no important here how this result was reached  for a more thorough explanation, see the Quantum Mechanics Wikibook. The associated energies are:
The energy levels become closer and closer as n increases and the energy approaches zero. n = ∞ corresponds to the vacuum level at which the electron is free of the proton. Electrons can move between the states by absorbing (to move up) or emitting (to move down) the correct amount of energy.
Spherical orbitals[edit]
For every n, a spherical solution exists, in which the wavefunction is invariant with angle. These spherical orbitals are known as the sorbitals.
Nonspherical orbitals[edit]
The complete solution for the wavefunction has four quantum numbers:
For example, when n = 2 and l = 1 there are three solutions corresponding to m = −1,0,1. These are known as the porbitals. Up to six electrons can be held in the porbitals  two for each solution, one spinup and one spindown.