In general, the wave function obeys the Schroedinger Equation.

For a free particle, the solutions of the Schroedinger Equation are just the deBroglie matter waves.

For a particle subject to forces, the solutions are more complicated.

For an electron bound to an atom, the Schroedinger Equation has well-behaved solutions only for certain energies.

Here is the wave function of the electron in the lowest energy state of Hydrogen.

Examples --- What this is an example of

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