White light is an ideal. You could argue that we never actually get truly perfectly white light. However, many processes are capable of outputting a very smooth spectrum because there's enough different energy transitions possible that we cannot distinguish any quantization.
An excellent example of this appears in the conduction bands for many metallic compounds. In these bands, energy of electrons is quantized, just as it always is. However, there are so many bands that they start to blur together (both by actual overlap and because scientific instruments just can't detect the boundary).
As for the equations, it's not actually just the vibration that matters, but the changes in states. If you can determine the difference in energy between the two states, you know the energy of a photon that would be emitted during that state change.