Role of Flux Density (Wavefunction Gradient Wavefunction) in Quantum Bound States

  In the literature (1) it has been noted that (d/dx W)/W - (d/dx W*)/W* represents a classical velocity, where W is the wavefunction. For bound states, this expression is zero, yet (d/dxW)/W seems to play the role of density flux as d/dx (WW) = 2W d/dxW. When multiplied by E, the energy, it becomes and energy flux as there is constant energy at each point x. In this note, we wish to consider some examples related to this flux. In particular, for the case of V(x)=0 (no potential), the flux becomes a solution of the time independent Schrodinger equation. This and other considerations may suggest that periodic type functions (possibly related to photons or phonons) when multiplied by the wavefunction W, can lead to a new, higher energy wavefunction. We also consider the role of energy flux as it relates to discrete energy levels in quantum mechanics.