Quantization of the Hamilton Equations of Motion

TL;DR

This paper explores the correct quantization rule for Hamilton's equations in quantum mechanics, modifying existing definitions to ensure consistency across different quantizations and demonstrating that several quantization methods satisfy the quantum equations of motion.

Contribution

It introduces a modified definition for the differential quotient of the first type, ensuring consistency with the second type, and analyzes various quantization schemes in relation to quantum Hamiltonian dynamics.

Findings

Modified differential quotient definition for operator differentiation.

Weyl, symmetric, and Born-Jordan quantizations satisfy quantum Hamiltonian equations.

Established rules for operator differentiation, including negative powers.

Abstract

One of the fundamental problems in quantum mechanics is finding the correct quantum image of a classical observable that would correspond to experimental measurements. We investigate for the appropriate quantization rule that would yield a Hamiltonian that obeys the quantum analogue of Hamilton's equations of motion, which includes differentiation of operators with respect to another operator. To give meaning to this type of differentiation, Born and Jordan established two definitions called the differential quotients of first type and second type. In this paper we modify the definition for the differential quotient of first type and establish its consistency with the differential quotient of second type for different basis operators corresponding to different quantizations. Theorems and differentiation rules including differentiation of operators with negative powers and multiple differentiation were also investigated. We show that the Hamiltonian obtained from Weyl, simplest symmetric, and Born-Jordan quantization all satisfy the required algebra of the quantum equations of motion.