Wave equations for determining energy-level gaps of quantum systems

An differential equation for wave functions is proposed, which is equivalent to Schr\"{o}dinger's wave equation and can be used to determine energy-level gaps of quantum systems. Contrary to Schr\"{o}dinger's wave equation, this equation is on `bipartite' wave functions. It is shown that those `bipartite' wave functions satisfy all the basic properties of Schr\"{o}dinger's wave functions. Further, it is argued that `bipartite' wave functions can present a mathematical expression of wave-particle duality. This provides an alternative approach to the mathematical formalism of quantum mechanics.