A New Derivation of the Time-Dependent Schrödinger Equation from Wave and Matrix Mechanics

Abstract

An alternative method is proposed for deriving the time-dependent Schrödinger equation from the pictures of wave and matrix mechanics. The derivation is of a mixed classical–quantum character, since time is treated as a classical variable, thus avoiding any controversy over its meaning in quantum mechanics. The derivation method proposed in this paper requires no ad hoc assumption and avoids going through a second-order differential equation that can be reduced to the well-known time-dependent Schrödinger equation only postulating a complex wavefunction with a time dependence given by , as did by Schrödinger in its original paper of 1926 [1].

Keywords: Schrödinger equation, wave–particle duality, Hermitian operators, commutation relations