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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

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

Jürg Riess

Published Aug 15, 1970• 45 citations
Publisher

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.

DOI
10.1103/physrevd.2.647

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The nodal structure of the Schr"odinger wave function is used to describe an NNN-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=0s=0s=0 and mass m0{m}_{0}m0​ is equal to hm0\frac{h}{{m}_{0}}m0​h​. In the case of charged particles, the field (cq)N(r)(\frac{c}{q})\mathbf{N}(\mathbf{r})(qc​)N(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 hc2q\frac{\mathrm{hc}}{2q}2qhc​ is possible.
The flux of probability density corresponding to a complex wavefunction, ψ=ψr+iψi, can form vortices which rotate about a nodal region of ψ, and have integer values of the circulation numbers. For one particle wavefunctions (or for approximate natural spin molecular orbitals), the vortices are either axial or toroidal. The axial vortices have angular momentum dipole moments which are usually quantized only if the wave function is an eigenfunction of Lz. The axial vortices interact with homogeneous magnetic fields whereas the toroidal interact with inhomogeneous magnetic fields. Toroidal vortices are easily created or annihilated by a perturbation which moves the nodal surface of ψi relative to the nodal surface of ψr so that these surfaces intersect or become separated. Thus, toroidal vortices are unstable and may be important only near resonance. However, axial vortices are stable since their creation requires (a) photon absorption or emission, (b) a perturbation (such as a magnetic field) which converts a real into a complex wavefunction, or (c) conversion of a toroidal into an axial vortex by expanding the nodal loop until a part of it reaches the boundary of configuration space. Thus, axial vortices should play an important role in energy transfer, photochemical processes, etc., and their circulation numbers should be good quantum numbers. A method for determining generalized first-order density matrices and natural spin–orbitals for use in scattering problems is proposed.
In previous work on the phase-space representation of quantum mechanics, we have presented detailed pictures of the electronic one-particle reduced Wigner function for atoms and small molecules. In this communication, we focus upon the nodal structure of the function. On the basis of the simplest systems, we present an expression which relates the oscillatory decay of the Wigner function solely to the dot product of the position and momentum vector, if both arguments are large. We then demonstrate the regular behavior of nodal patterns for the larger systems. For the molecular systems, an argument analogous to the {open_quotes}bond-oscillatory principle{close_quotes} for momentum densities links the nuclear framework to an additional oscillatory term in momenta parallel to bonds. It is shown that these are visible in the Wigner function in terms of characteristic nodes.
In this work, we start from the problem of quantizing the relativistic Hamiltonian of a free massive particle (rest mass different from 0), a problem exceptionally difficult in the standard approaches to quantum mechanics. In fact, in tempered distribution state space, we find the natural way to define the relativistic Hamiltonian operator and its associated Schrödinger equation The existence of a linear continuous Hermitian operator associated with the Einstein's Hamiltonian of a free particle, defined on the entire tempered distribution space, automatically implies the conservation of Born probability flux (which doesn't mean the conservation of particles number, rather it implies the conservation of the total relativistic energy of the solution wave). We, then, deduce the continuity equation for the Born probability density and study some its different (but equivalent) expressions. We determine some possible forms of probability currents and flux velocity fields associated with the particle evolution. We provide the relativistic invariant expression for both Schrödinger equation and probability flux continuity equations.
In 1972, P. Pechukas [J. Chem. Phys. 57, 5577 (1972)] proposed "classical states" associated with Miller-Good transformations, semiclassical quantization, and the Hamilton-Jacobi equation. In this paper we use some of the concepts established by Pechukas and extend them to define generalized curvilinear coordinates $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\gamma}}$ associated with nonlinear dynamical systems with Hamiltonians of the form $H=T(\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}})+V(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})$. For reasons discussed in the paper we call these coordinates "nodal" coordinates. We show that a transformation to nodal coordinates is to a very good approximation dependent only on Cartesian variables. Using this approximation we demonstrate that, within the regular regime of phase space, canonical transformation to $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\gamma}}$ coordinates simplifies the analysis of the classical and quantum mechanics of dynamical systems. Our fundamental conclusions are as follows: (1) The solution of the Hamilton-Jacobi equation is separable in $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\gamma}}$ coordinates; (2) the WKB wave function ${\ensuremath{\Psi}}^{\mathrm{WKB}}$ is separable in $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\gamma}}$ coordinates; and (3) if the transformation to $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\gamma}}$ coordinates is made conformal (which we show we are allowed to do within the approximation stated above), then the Schr\"odinger equation and wave functions become separable. Finally, the concepts in this paper are discussed for systems of two degrees of freedom but can be generalized to more degrees of freedom.
Abstract In excited states of atoms and molecules, as well as in time‐dependent situations, the one‐electron density no longer suffices to completely characterize the electronic state; in addition, one now requires information about the electronic phase or the current density. We show that, for a stationary electronic state, the continuity equation of quantum fluid dynamics represents a differential equation for the electronic phase, which must be solved subject to certain periodicity conditions. These periodicity conditions arise from the nodal topology of the wave function and give rise to quantized vortices of current. The consequences of writing an electronic “wave function” for a many‐electron system directly in terms of the single‐particle density and phase have been investigated. We have shown that such a procedure leads to the appearance of an “internal magnetic vector potential.” We also establish the connection between the electronic phase and the geometrical (“Berry”) phase accompanying the adiabatic transport of a quantal system around a closed loop in parameter space. This leads to a generalization of the current density concept and allows us to discuss the geometrical phase in terms of the circulation of this current in parameter space.
The orbit equation for the motion of a quantum in a potential is proposed by defining uniquely the quantum mechanical momentum from the Schrodinger wave equation of a one-particle system and using the Hamilton-Jacobi formalism. On this occasion it is necessary to introduce a new dynamical postulate, “initial condition”, which is independent of the quantization condition as well as of the Schrodinger wave equation. The momentum is determined in such a way that it is approximately equal to the corresponding classical mechanical momentum in the classical region of validity of the coordinates. The momentum in the non-classical region is in general very small in comparison with that in the classical region. The quantum-mechanical motion well corresponds to the classical motion. If a system is macroscopic, the motion of the particle is described by Newtonian mechanics and the quantization condition is of no validity. The “quantization of magnetic flux” derives from the quantization condition. It is closely related to the energy level structure of the atomic system. The above argument is applied to the scalar electromagnetic and sound wave equations to give the dynamical motion of a photon and a phonon without recourse to the so-called second quantization method.