Central Potentials and the Hydrogen Atom

Claude N. Cohen-Tannoudji, Bernard Diu, Franck Laloë

J. W. Zink

Aditi Sharma

Solutions of the Schrödinger equation for central potentials and for cubic systems are the main contents of the chapter. Motion in a central field, energy eigenvalues and wave functions for the hydrogen atom, radial probability density, solutions for the 3D spherically symmetric potential well, and free electron motion in a 3D-box are discussed with the use of algebra that clarifies the essential physics in a simple but appealing manner. Derivation of energy eigenvalues for the hydrogen atom and for a free electron cubic box clearly illustrates the concept of quantization of energy. The number of states and the density of states are introduced for the case of an electron confined to a cubic box. Carefully chosen solved examples further consolidate the understanding of solutions of the Schrödinger equation for central potentials and for a cubic system.

Andrei Yu. Khrennikov, Yaroslav Volovich

We analyze dynamical consequences of a conjecture that there exists a fundamental (indivisible) quant of time. In particular we study the problem of discrete energy levels of hydrogen atom. We are able to reconstruct potential which in discrete time formalism leads to energy levels of unperturbed hydrogen atom. We also consider linear energy levels of quantum harmonic oscillator and show how they are produced in the discrete time formalism. More generally, we show that in discrete time formalism finite motion in central potential leads to discrete energy spectrum, the property which is common for quantum mechanical theory. Thus deterministic (but discrete time!) dynamics is compatible with discrete energy levels.

Josh Izaac, Jingbo B. Wang

Andrei Yu. Khrennikov

We analyze dynamical consequences of a conjecture that there exists a fundamental (indivisible) quant of time. In particular we study the problem of discrete energy levels of hydrogen atom. We are able to reconstruct potential which in discrete time formalism leads to energy levels of unperturbed hydrogen atom. We also consider linear energy levels of quantum harmonic oscillator and show how they are produced in the discrete time formalism. More generally, we show that in discrete time formalism finite motion in central potential leads to discrete energy spectrum, the property which is common for quantum mechanical theory. Thus deterministic (but discrete time!) dynamics is compatible with discrete energy levels.