Nodal Structure, Nodal Flux Fields, and Flux Quantization in Stationary Quantum States

Abstract

The nodal structure of the Schr\"odinger wave function is used to describe an $N$-particle system in a stationary quantum state. In terms of nodal lines (quantized flux lines) together with their circulation numbers, two fields are defined: the microscopic, singular nodal flux field n (r) and the macroscopic, regular nodal flux density field N (r). Their definition arises in a natural way from a quantum-mechanical vector potential defined by the gradient of the multivalued phase function associated with the pattern of quantized flux lines. The flux of the fields n (r) and N (r) is quantized and shown to be proportional to the circulation of the velocity in the system. The unit circulation for a system of bosons with a particle spin $s=0$ and mass ${m}_{0}$ is equal to $\frac{h}{{m}_{0}}$. In the case of charged particles, the field $(\frac{c}{q})\mathbf{N}(\mathbf{r})$ resembles Jehle's formulation of the magnetic field of a lepton as a superposition of quantized flux lines. In our case, however, the presence of quantized flux lines, forming closed loops follows from the mere existence of a current density and does not have to be assumed. By a simple argument, using permutation symmetry, it is shown that a flux quantum of the size $\frac{\mathrm{hc}}{2q}$ is possible.