A Hamiltonian effective potential is defined and calculated at the tree and one loop levels in a ${\phi }^4$ scalar field theory. The loop expansion for eigenfunctionals is equivalent to the combination of WKB expansion and an expansion around constant field configurations. The results are compared with those obtained from the Lagrangian effective potential. While at the tree level the potentials in the two methods coincide, at the one loop level they differ. Nevertheless, physical predictions from the two potentials agree. The same formalism is used to calculate bound-state energies from the Schrödinger equation for excited states.

• Received 2 December 1992

DOI:https://doi.org/10.1103/PhysRevD.48.3826