Unified Description of Matrix Mechanics and Wave Mechanics I

TL;DR

This paper simplifies quantum mechanics by removing bra-ket notation, unifying matrix and wave mechanics through direct matrix representations of operators and wave functions, and providing differential equations for solutions.

Contribution

It introduces a unified framework for matrix and wave mechanics without bra-ket notation, simplifying concepts and enabling direct conversion between operator and matrix relations.

Findings

Unified description of matrix and wave mechanics.

Direct correspondence between operators, matrices, and wave functions.

Provided differential equations for solving quantum systems.

Abstract

In this article, we discard the bra-ket notation and its correlative definitions, given by Paul Dirac. The quantum states are only described by the wave functions. The fundamental concepts and definitions in quantum mechanics is simplified. The operator, wave functions and square matrix are represented in the same expression which directly corresponds to the system of equations without additional introduction of the matrix representation of operator. It can make us to convert the operator relations into the matrix relations. According to the relations between the matrices, the matrix elements will be determined. Furthermore, the first order differential equations will be given to find the solution of equations. As a result, we unified the descriptions of the matrix mechanics and the wave mechanics.