Establishment of the conserved operators using variational principle

TL;DR

This paper uses a variational principle to derive all possible forms of conserved operators in quantum mechanics, revealing only three fundamental types and suggesting quantum properties stem from conservation laws.

Contribution

It introduces a variational approach to systematically derive conserved operators, identifying three fundamental forms and linking quantum nature to conservation laws.

Findings

Only three forms of conserved operators are possible.

All operators include the same constant, indicating their quantum nature.

Quantum characteristics arise from conservation laws, not quantization assumptions.

Abstract

Operators play a substantial role in mathematical formalism of quantum mechanics. However, explicit forms of the operators are usually postulated, based on the intuitive assumptions. In this study, variational principle was applied to the basic equation for expectation value to vary a generalized form of operator while keeping psi-function invariable. A restriction of being expectation value invariable, allowed one to derive all possible forms of the operators corresponding to the conserved physical entities. As a result, it was found that only three distinctive forms of the conserved operators are possible, tentatively assigned to be angular momentum-like, momentum-like and total energy-like operators. Surprisingly, all operators included constant, the same one for each of the operators, therefore, making operators the quantum ones. Absence of the quantization in original assumptions suggests that quantum character of the operators and, therefore, the physical entities, is a direct consequence of the existence of the conservation laws.