From the viewpoint of interpretation and methodology, quantum mechanics as one of the greatest achievements in the field of physics in the 20th century involves two formulations, i.e., the traditional standard formalism and non-standard formalism such as quantum mechanics in phase space, quantum trajectory formulation and path integral method.
Hibert-space formalism as the standard formalism yielded an excellent mathematical formulation for explaining the microscopic world such as atoms and subatomic objects. After that, the trajectory-based quantum mechanics such as Bohmian mechanics was proposed in order to improve mathematical formulation of quantum mechanics. Quantum mechanics in phase space also attracted a great deal of interest from the aspect of mathematical formulation, but the tasks of establishing an alternative independent formalism of quantum mechanics still remained not accomplished.
After as long as 15 years of painstaking effort, Jong Chol has proposed that it would be possible to establish an alternative autonomous formalism of quantum mechanics in phase space by means of a statistical method.
The proposed theory as a new formalism of quantum mechanics in terms of ensemble in phase space leads to obtaining within the framework of its theory the fundamental quantum-mechanical equation without recourse to other formulations of quantum mechanics, and gives the idea for operators pertaining to dynamical quantities.
The theory also demonstrates that the system of operators given by quantum mechanics in phase space is a complete system of quantum operators and it is possible to explain, using the fundamental equation, the structure of quantum mechanics in phase space and the approximation of the fundamental equation of the present formalism to the Schrödinger equation. Therefore, it concludes that quantum mechanics in phase space is the general formulation comprising the formulation in configuration space as a special case.
This formalism provides reasonable results of quantization by dealing with some simple cases such as the quantization of harmonic oscillation, the two-slit interference and the uncertainty relation, which confirms the validity and generality of this formalism.
In particular, this formalism shows that it is possible to easily obtain the relativistic wave equation by making use of the relativistic phase velocity without treating the problem of linearizing the Hamiltonian operator as in the case of the deduction of the Dirac equation.
The ultimate outcome this formalism produces is that it has been confirmed that primary and general matters of quantum mechanics can be studied reasonably within the framework of statistical mechanics.
This information is from the essay “Ensemble in phase space: Statistical formalism of quantum mechanics” presented by Jong Chol, a researcher at the Faculty of Materials Science and Technology, to the SCI Journal “Pramana Journal of Physics” 92:83(2019).
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