The fundamental fact is revealed that in the old good quantum mechanics there is possible such unexpected inversion: potential barriers can drag in wave-particles and wells can push them off.
The famous exchange forces arise in quantum mechanics from the uncertainty at any given time as to which of the two potential wells a particle, which is capable of leaking back and forth between the two wells, is occupying. This entirely quantum-theoretical result can be given an energetic explanation, and in fact be estimated quantitatively, by finding the possible energy eigenfunctions for the particle, which correspond to nearly-equal or to nearly-opposite probability amplitudes for the wave function in each of the two wells. The variation of system energy with the separation of the wells can be interpreted as a force. The present paper is an effort to give a much more intuitive account of the origin of these important forces. The force is here interpreted purely kinematically as the result of the fact that the particle bombards the well edges both from within and from without. The net attractive or repulsive force between the wells then arises from the fact that for one type of wave function the bombardment is more frequent on the inner, facing edges, and so the wells are forced together, and conversely for the other type. The ideas are generalized to wells other than the idealized square potential well, and compared quantitatively with more familiar treatments of the problem.
We investigate the transmission and reflection of a quantum particle incident upon a step potential increase, a step potential decrease, a square well, and a square barrier, all well studied in undergraduate quantum mechanics. We are especially interested in the extreme where the change in the potential is arbitrarily large, but with the difference in the energy of the particle and the potential held fixed, if possible. We obtain the probabilities of transmission and reflection using the time-independent Schrödinger equation and also the time-dependent Schrödinger equation. In the time-dependent case, we have the particle initially in a Gaussian wave packet or a similar localized state. We obtain results that fall into three categories: results that are not surprising, results where time-dependent and time-independent agree surprisingly well, and results that are very different. We discuss the unexpected results. Our work may be of interest to instructors of and students in upper year undergraduate quantum mechanics courses.
It was an important examination to give a review talk at the previous Conference on Inverse Quantum Scattering (1996, Lake Balaton) about computer visualization of this science in front of its fathers — creators, B. M. Levitan and V. A. Marchenko. We have achieved a new understanding that the discovered main rules of transformations of a single wave function bump, e.g., for the ground bound states of one dimensional quantum systems are applicable to any state of any potential with arbitrary number of bumps from finite to unlimited ones as scattering states and bound states embedded into continuum. It appeared that we need only to repeat the rule mentally the necessary number of times. That uttermost simplification and unification of physical notion of spectral, scattering and decay control for any potential have got an obligatory praise from B. M. Levitan at the conference and was a mighty stimulus for our further research After that we have written both Russian (2002) and improved English editions of “Submissive Quantum Mechanics. New Status of the Theory in Inverse Problem Approach”1 (appeared at the very end of 2007). This book was written for correction of the present defect in quantum education throughout the world. Recently the quantum IP intuition helped us to discover a new concept of permanent wave resonance with potential spatial oscillations.2 This means the constant wave swinging frequency on the whole energy intervals of spectral forbidden zones destroying physical solutions and deepening the theory of waves in periodic potentials. It also shows the other side of strengthening the fundamentally important magic structures. A ‘new language’ of wave bending will be presented to enrich our quantum intuition, e.g., the paradoxical effective attraction of barriers and repulsion of wells in multichannel systems, etc.