An effective hamiltonian for heavy quarkonia is derived from QCD by separating gluonic fields in background and quantum fields and neglecting anharmonic contributions. Mesonic states with nonperturbative gluonic components are constructed. These states are invariant under gauge changes of the background fields and form an orthogonal basis. The effective hamiltonian is diagonalized in this basis in a systematic $1/m$- and short distance expansion. For very heavy quarkonia, we obtain an effective potential similar to the phenomenological funnel potential. We compare our method to $2^{\rm nd}$ order perturbation theory in the background fields and demonstrate its applicability even for the relatively light charmonium system. The results to order $1/m$ for pseudoscalar meson masses and wave functions are shown and compared with those of the Cornell model.

Same results, but some conceptual ramifications and explanations. 32 pages. RevTex