Derivation of the Schrödinger Equation: What is the Physical Meaning of Its Wave Function?

In this chapter, we show that the Schrödinger equationSchrödinger equation can be derived naturally from the quantum wave modelQuantum wave model. In order to derive the Schrödinger equationSchrödinger equation, one needs to first develop the “correspondence rulesCorrespondence rules”. Historically, these rules were developed based on the Hamiltonian framework in classical mechanics. In this work, we show that the correspondence rulesCorrespondence rules simply reflect the operation of the de Broglie relationDe Broglie relation and the Planck’s relationPlanck’s relation on a traveling waveWave. Using these correspondence rulesCorrespondence rules, one can derive the Schrödinger equationSchrödinger equation by following the analogy of using the correspondence rulesCorrespondence rules to derive the Klein–Gordon equationKlein-Gordon equation. From this derivation, one can see that the physical meaning of the quantum wave functionQuantum wave function in the Schrödinger equationSchrödinger equation is similar to the waveWave function of the Klein–Gordon equationKlein-Gordon equation; they differ only by a phase factor.