The Mathematical Structure of Elementary Quantum Mechanics

Élliott H. Lieb, Robert Seiringer

In this second chapter we will review the basic mathematical and physical facts about quantum mechanics and establish physical units and notation. Those readers already familiar with the subject can safely jump to the next chapter. An attempt has been made to make the presentation in this chapter as elementary as possible, and yet present the basic facts that will be needed later. There are many beautiful and important topics which will not be touched upon such as self-adjointness of Schrödinger operators, the general mathematical structure of quantum mechanics and the like. These topics are well described in other works, e.g.. Much of the following can be done in a Euclidean space of arbitrary dimension, but in this chapter the dimension of the Euclidean space is taken to be three-which is the physical case-unless otherwise stated. We do this to avoid confusion and, occasionally, complications that arise in the computation of mathematical constants. The interested reader can easily generalize what is done here to the R d , d > 3 case. Likewise, in the next chapters we mostly consider N particles, with spatial coordinates in R 3 , so that the total spatial dimension is 3 N . A Brief Review of the Connection Between Classical and Quantum Mechanics Considering the range of validity of quantum mechanics, it is not surprising that its formulation is more complicated and abstract than classical mechanics. Nevertheless, classical mechanics is a basic ingredient for quantum mechanics. One still talks about position, momentum and energy which are notions from Newtonian mechanics.

Philip L. Bowers

Kamal Barley, Andreas L. Ruffing, Sergeĭ K. Suslov

We review Bohr's atomic model and its extension by Sommerfeld from a mathematical perspective of wave mechanics.The derivation of quantization rules and energy levels is revisited using semiclassical methods.Sommerfeld-type integrals are evaluated by elementary techniques, and connections with the Schr odinger and Dirac equations are established.Historical developments and key transitions from classical to quantum theory are discussed to clarify the structure and significance of the old quantum mechanics.

David A. Edwards

Borros Arneth

Modern theoretical physics is built upon several fundamental mathematical structures, including Hilbert spaces, Green functions, operator algebras, Gibbs ensembles, and spacetime geometry. These structures are traditionally introduced independently and serve as the foundations of quantum mechanics, quantum field theory, statistical mechanics, and general relativity. In this work, we propose an alternative mathematical framework based on a single Reconstruction Framework, in which these structures are generated sequentially from one primitive mathematical architecture. Beginning with an elementary transport architecture, transport operators, Green functions, Diagram Hilbert Spaces, complex quantum operators, Gibbs states, entropic projection, and finally macroscopic spacetime and gauge fields are reconstructed as successive mathematical layers. Each layer is determined by the preceding one and introduces no additional primitive mathematical assumptions. The proposed framework unifies several existing approaches to emergent physics within a common mathematical hierarchy and suggests that the diverse mathematical languages of modern theoretical physics are different representations of one underlying microscopic architecture. Rather than postulating Hilbert spaces or spacetime as fundamental, the present framework treats them as reconstructed structures arising naturally from progressively richer mathematical descriptions.