We introduce an atomic model that avoids some of the worst dissonances. According to this model, the electron is not point-like but extended. The square of the wave function is interpreted as the density of a fluid, the electronium. On the basis of the electronium model suggestive pictures and animations of the atom can be generated. We shall see and discuss pictures of the various states of a hydrogen atom as well as animations of transitions from one stationary state to another. We also shall represent the internal movement of the electronium, that is responsible of the angular momentum and the magnetic moment of the atom. From the pictures we shall directly read properties of the various states and the various transitions by only using arguments of classical physics.
We introduce an atomic model that avoids some of the worst dissonances. According to this model, the electron is not point-like but extended. The square of the wave function is interpreted as the density of a fluid, the electronium. On the basis of the electronium model suggestive pictures and animations of the atom can be generated. We shall see and discuss pictures of the various states of a hydrogen atom as well as animations of transitions from one stationary state to another. We also shall represent the internal movement of the electronium, that is responsible of the angular momentum and the magnetic moment of the atom. From the pictures we shall directly read properties of the various states and the various transitions by only using arguments of classical physics.
The hydrogen atom is the Rosetta stone of the early twentieth century atomic physics. The attempt to decipher its structure and properties led to the development of quantum mechanics and the unraveling of many of the mysteries of atomic, molecular, and solid state physics, and a good deal of chemistry and modern biology. Unlike the various one-dimensional model problems that we have been studying in the previous chapters, the hydrogen atom is a real physical system in three dimensions. It consists of an electron moving in a spherically symmetric potential well due to the Coulomb attraction of the positively charged nucleus. In three dimensions, the electron is not constrained to move linearly. It can execute orbital motions and, thus, has angular momentum. Not only is the total energy of the electron in the atom quantized, its angular momentum also has interesting and unexpected quantized properties that cannot possibly be understood on the basis of classical mechanics and electrodynamics. They are, however, the natural and necessary consequences of the basic postulates of quantum mechanics, as will be shown in this chapter. According to classical mechanics and electrodynamics, it is not possible to have a stable structure consisting of a small positively charged nucleus at the center of an electrically neutral atom with an electron sitting in its vicinity. For the electron not to be attracted into the positive charge, it must be orbiting around the nucleus so that the centrifugal force will counter the Coulomb attraction of the nucleus and maintain a constant electron orbit.
It is well known that the solutions of the wave-equation for hydrogen-like atoms may be represented graphically by interpreting $\ensuremath{\Psi}{\ensuremath{\Psi}}^{*}$ as a probability density. The probability density factors ${\ensuremath{\Phi}}_{m}{{\ensuremath{\Phi}}_{m}}^{*}\ifmmode\cdot\else\textperiodcentered\fi{}{[{\ensuremath{\Theta}}_{m,l}]}^{2}\ifmmode\cdot\else\textperiodcentered\fi{}{[{R}_{n,l}]}^{2}=\ensuremath{\Psi}{\ensuremath{\Psi}}^{*}$ are represented graphically and briefly discussed and compared with the electron orbits of four classical models. Graphs for s, p, d, f, g, and h electrons are given. An attempt to combine the probability density factors and form some graphical representation of $\ensuremath{\Psi}{\ensuremath{\Psi}}^{*}$ has resulted in the construction of a mechanical device or model, see Fig. 5, which when photographed, gives very closely the desired result. Photographs for the magnetic states $m=0,\ifmmode\pm\else\textpm\fi{}1,\ifmmode\pm\else\textpm\fi{}2,\ifmmode\pm\else\textpm\fi{}3,\ensuremath{\cdots}$ are given for 1s, 2p, 3d, 4f, 2s, 3p, 4d, 5f, 3s, 4p, and 5d electrons, see Fig. 6.
A point-like picture of the Schrödinger solution for hydrogen atom is worked to emphasize that "point-like particles" may describe as "probability wave function". In each case, the three-dimensional shape of the |Ψnlm(rn, cos θ)|2 is plotted and the paths of the point-like electron (it is better to say reduced mass of the pair particles) are described in each closed shell. Finally, the orbital shape of the molecules are given according to the present simple model. In our opinion, "interpretations of the Correspondence Principle", which is a basic principle in all elementary quantum text, seems to be reviewed again!
According to the prevailing belief, the spin of the electron or of some other particle is a mysterious internal angular momentum for which no concrete physical picture is available, and for which there is no classical analog. However, on the basis of an old calculation by Belinfante [Physica 6, 887 (1939)], it can be shown that the spin may be regarded as an angular momentum generated by a circulating flow of energy in the wave field of the electron. Likewise, the magnetic moment may be regarded as generated by a circulating flow of charge in the wave field. This provides an intuitively appealing picture and establishes that neither the spin nor the magnetic moment are ‘‘internal’’—they are not associated with the internal structure of the electron, but rather with the structure of its wave field. Furthermore, a comparison between calculations of angular momentum in the Dirac and electromagnetic fields shows that the spin of the electron is entirely analogous to the angular momentum carried by a classical circularly polarized wave.
The mere existence of stable atoms, and their arrangement into many-atom nanoscale structures, can only be explained by the laws of quantum mechanics. Indeed, the accumulated evidence regarding the internal structure of atoms motivated the formulation of quantum mechanics at the early twentieth century. Here we focus on single-electron (hydrogen-like) atoms. The dominance of the Coulomb central potential between the electron and nucleus, translates into a simple Schrödinger equation for their relative position distribution. Solving rigorously this equation reveals quantized energy levels, which explains the emission spectra of hydrogen-like atoms (Rydberg formula). The corresponding stationary probability density distributions for the electron around the nucleus are reminiscent of the Bohr model, where classical trajectories (orbits) are replaced by single-electron wave functions (orbitals). The degeneracies of different orbitals, associated with three quantum numbers, will later turn out to be critical for understanding many-electron atoms and the periodic table of the elements.
Chapter 12 introduces light–matter interaction at the atomic scale and at interaction energies less than, or comparable to, the chemical bond. The classical electrodynamics treatment for fields is retained, but a very simple quantum atomic structure with electric dipole transitions among atomic and molecular internal states is introduced. The correspondence between classical and quantal electron orbital angular momentum states in atoms is developed in detail for the hydrogen atom, and the difference between quantised orbital states and real, space-directed orbitals is explained. A wave mechanical quantum representation brings out the analogies between classical light waves, quantum matter waves, classical dipole radiation, and atomic radiative emission. The intrinsic electron spin, that has no classical analogue, is introduced and discussed. The chapter ends with a discussion of the rigid rotor, harmonic oscillator model to interpret rotational and vibrational diatomic molecular spectra. Finally rovibronic states complete the picture with the introduction of diatomic molecular electronic transitions.