The Formal Framework of Quantum Mechanics

Walter Greiner

Walter Greiner

Walter Greiner

Tilak Sinha

Although, the main objective of this book has been to expose the facts that underpin formal description of quantum mechanics and to make quantum mechanics seem natural, in going about this process we have tried to provide the young student with a guideline on how to approach learning a new theory. Thus, somewhat unconventionally, we start the discourse with a discussion on the meaning of theory and its formal description. This allows us to demonstrate, in concrete terms, how quantum mechanics is just another theoretical framework that conforms to the standard definition of a theory in the natural sciences. In this first chapter, we start with a nontechnical discussion of what one means by a theoretical framework and its formal description. We clarify what qualifies as a theory, and list the questions whose answers one must seek in trying to learn it. A notion of what amounts to understanding a theory is thus conveyed. The meaning of law, axiom, postulate, theorem, falsifiability etc., are thus discussed. We try to draw out the purpose of constructing a formal description of a theory. We also outline the role of abstraction and generalization in its construction. Finally, we discuss the sense in which the terms “philosophy” and “fundamental theorem” are used in the construction of a theoretical framework.

Y.V. Przhiyalkovskiy

A proper deformation of the underlying coordinate and momentum commutation relations in quantum mechanics provides a phenomenological approach to account for the influence of gravity on small scales. Introducing the squared momentum term results in a generalized uncertainty principle, which limits the minimum uncertainty in particle position to the Planck length. However, such a deformation of the commutator significantly changes the formalism, making it separate from the canonical formalism of quantum mechanics. In this study, it is shown that the deformed algebra of position and momentum operators can be incorporated into the framework of ordinary quantum mechanics.

O. Melsheimer

Results of a previous paper are used to obtain a rigorous mathematical formulation of the transformation theory of nonrelativistic quantum mechanics within the framework of rigged Hilbert spaces.