Basic Mathematical Properties of Electronic Wave Functions in Configuration Space

David M. Ceperley

Jincheng Zhang

Structural Wave Function Mathematics (SWFM) is a novel framework that associates a complex‑valued wave function with every mathematical object in order to capture its full spectrum of possible structural manifestations. The wave function Ψ(X) is defined on an abstract configuration space that encodes all admissible structural attributes of X. By constructing a Hilbert‑type state space for each object, SWFM unifies disparate mathematical disciplines under a single analytic formalism. This paper presents the foundational axioms of SWFM, derives key properties such as superposition, orthogonality, and normalization, and demonstrates how the framework can be employed to analyze algebraic structures, topological spaces, and combinatorial configurations. The absence of empirical data is intentional; SWFM is designed as a purely theoretical edifice that can be expanded upon in future work.

Goong Chen, Zhonghai Ding, Alain Perronnet +3 more

Raj Bhakta

The quantum wave-function is at the heart of quantum mechanics. It is the physical state of a quantum system before a measurement is taken. This wave-function is described mathematically by the formulation of matrix mechanics and represents the evolution of the quantum wave-function. In contemporary philosophy of physics, the wave-function’s nature of space is hotly debated. Contemporaries such as David Albert, who posits an ontology which is based off the mathematical features of the wave-function, claim that the dimensionality of our physical space is proportional to 3N, where N is equal to the number of particles in the universe. Others such as Bradley Monton claim quantum mechanics to be a false theory from the outset due to its incompatibility with general relativity and posit a much more realist property-based-physics ontology. However, these ontologies have their problems. Albert’s ontology takes the mathematics of the quantum wave-function at face value and disregards the distance properties of metric space to misconstrue the phase space nature of the 3N-configuration space as a metric space. Monton, on the other hand, questionably grounds his thesis on the incompatibility of quantum mechanics with general relativity and on common sense rather than on observation and experimentation, as required by the scientific method. In this thesis, I discuss the viewpoints of Albert and Monton in detail and explore the nature of space in the quantum wave-function in an attempt to reach a better understanding of the fundamental nature of space in the quantum wave-function.

Iwo Białynicki‐Birula

This chapter describes the photon wave function, and explains the basic properties of a well-defined mathematical object—a six-component function of space-time variables—that describes the quantum state of the photon. The most essential property that does not hold for the photon wave function is that the argument of the wave function cannot be directly associated with the position operator of the photon. The position operator for the photon simply does not exist. However, one should remember that for massive particles also, the true position operator exists only in the nonrelativistic approximation. The concept of localization associated with the Newton-Wigner position operator is not relativistically invariant. The photon wave function is not restricted to the wave mechanics of photons. The same wave functions also appear as mode functions in the expansion of the electromagnetic field operators.

Ramon Carbó‐Dorca