A New Derivation of the Time-Dependent Schr\"odinger Equation from Wave and Matrix Mechanics

TL;DR

This paper presents an alternative derivation of the time-dependent Schrödinger equation from wave and matrix mechanics, treating time as a classical variable, avoiding ad hoc assumptions and complex wavefunction postulates.

Contribution

It introduces a mixed classical-quantum derivation method that does not rely on the traditional exponential wavefunction assumption.

Findings

Derivation avoids second-order differential equations.

Method treats time as a classical variable.

No ad hoc assumptions are needed.

Abstract

An alternative method is proposed for deriving the time dependent Schroedinger 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 Schroedinger equation only postulating a complex wavefunction with an exponential time dependence, as did by Schroedinger in its original paper of 1926.