Finiteness of wave functions

Francis E. Low

Declan Traill

The images [fig 1-9] in this paper are graphical representations of the fields generated by these wave functions using a 3D finite element vector modeling program [ref 1] I wrote to aid in the visualization and testing of proposed wave function solutions.

Geoffrey Hunter

Abstract The nodes where the wave function changes sign in conventional quantum theory can alternatively be regarded as places where the wave function descends to zero but rises again without changing its sign. This behavior is accomplished by adding delta function barriers to the potential energy wherever a node occurs in the wave function. This model is supported by the recognition that the wave functions of a subsystem, which is not truly isolated, are really marginal amplitudes which implicitly average out interactions of the system with its environment. The finite effect of these interactions is to replace the delta function barriers by barriers of finite width and height.

F. Selleri

Hidetsugu Sakaguchi, Boris Malomed

Quantum collapse (QC) occurs under the action of the inverse-power potential U(r)=-U_{0}/(2r^{n}) in the framework of the Schroedinger equation. The corresponding stationary wave functions feature an infinite number of spatial oscillations at $r\rightarrow 0$. The ground state (GS) is missing in the presence of the QC, but the available bound states at finite energies may be physically relevant, as they are normalizable. This work aims to produce singular stationary wave functions. The analytical findings represent an asymptotic form of the stationary wave functions at $r\rightarrow 0$, which fill a continuous spectrum, and some exact solutions for the wave functions, obtained under appropriate conditions. As a regularized version, we also consider the potential truncated at a finite difference from the center. The regularized wave functions feature a finite number of spatial oscillations, the missing GS reappears, and the spectrum of the bound states is discrete.

Toshikatsu Koga, Ajit J. Thakkar

Abstract Finiteness is usually imposed as a condition for physical admissibility of a wave function. Examination of this condition in both position‐ and momentum‐space shows that finiteness of the wave function at the origin in one space implies “linear” integrability in the reciprocal space, except in some pathological cases. Some implications of this result are discussed.