Niels Bohr constructed the first version of quantum mechanics. It has been called “old quantum mechanics” with a connotation of being obsolete. It is logically consistent, however, and deserves the name of simple quantum mechanics (SQM). It differs only from the semiclassical approximation by assuming that the average position and average velocity of an electron can be sharply defined on closed orbits. This assumption does not contradict Heisenberg’s uncertainty relations, since the quantization rule means that the electron can be anywhere on this orbit when it allows for stationary waves. This approach was remarkably efficient for one electron in hydrogen atoms and even for the electron pair in hydrogen molecules. However, dissociation of H2 and determination of the orbit of the single electron in H 2 + led to problems that remained unsolved for more than 100 years. Their solution, presented here, yields more physical insight and reveals, for instance, that mutual polarization of two hydrogen atoms can yield a metastable state.
Niels Bohr constructed the first version of quantum mechanics. It has been called “old quantum mechanics” with a connotation of being obsolete. It is logically consistent, however, and deserves the name of simple quantum mechanics (SQM). It differs only from the semiclassical approximation by assuming that the average position and average velocity of an electron can be sharply defined on closed orbits. This assumption does not contradict Heisenberg’s uncertainty relations, since the quantization rule means that the electron can be anywhere on this orbit when it allows for stationary waves. This approach was remarkably efficient for one electron in hydrogen atoms and even for the electron pair in hydrogen molecules. However, dissociation of H2 and determination of the orbit of the single electron in H 2 + led to problems that remained unsolved for more than 100 years. Their solution, presented here, yields more physical insight and reveals, for instance, that mutual polarization of two hydrogen atoms can yield a metastable state.
Many teachers like to introduce the Bohr atom toward the end of an introductory physics course. This is an excellent idea, given the historic importance of Bohr’s 1913 work, which provided the bridge from Planck’s quantized interaction of matter and radiation (1900) to the full theory of quantum mechanics (1925-28). Unfortunately, the version of the Bohr atom that appears in many textbooks and is no doubt often presented to students is more wrong than right and may leave both teachers and students wondering why, more than a hundred years later, it is still being taught. This “pedagogic” version postulates that an electron in a stationary state moves in a circular orbit with an angular momentum that is an integral multiple of h/2π (L = nh/2π = nħ)—ħ for the lowest-energy state, 2ħ for the next state, and so on. This picture of the hydrogen atom is wrong in two senses. First it doesn’t conform to our present understanding of the hydrogen atom. This, in itself, is not a reason to scrap it, for the historical development of quantum physics is certainly of interest. But second, it doesn’t conform to the essence of what Bohr actually did. That is a reason not to teach about circular orbits and L = nħ.
In the 1920s, the old quantum theory was gradually being superseded by the new quantum theory. The cornerstone of the old theory was Bohr’s model of the hydrogen atom which predicted that an electron cannot orbit the proton in the hydrogen atom in any arbitrary fashion. Orbits are “quantized”, meaning that only certain sizes, shapes and magnetic properties are allowed. The principal quantum number n determined the allowed radii of the orbits, the orbital quantum number l determined the allowed shapes, and the magnetic quantum number m determined the magnetic behavior. Additionally, there is a fourth quantum number s which denotes the fact that the electron has an additional angular momentum, loosely associated with self rotation about its own axis, and that is quantized in units of (1/2). The old quantum theory was useful to infer the existence of discrete energy levels in atoms, calculate energy spacings between these levels, and therefore allowed one to interpret atomic spectra.
Something is rotten in the state of Denmark. Shakespeare , Hamlet, Act 1 Scene 4 The rise and fall of the old quantum theory The principles of the old quantum theory were first articulated by Niels Bohr in 1913 in a three-part paper titled “On the Constitution of Atoms and Molecules.” Bohr had adopted Ernest Rutherford's model of the atom, according to which the mass of the atom is concentrated in a small central nucleus, while the electrons orbit the nucleus in planetary trajectories. The key challenge facing Rutherford's model was that it was unstable: according to classical electrodynamics, the electron, which is a moving charged body, should radiate energy and so eventually collapse into the nucleus. Bohr's solution was to incorporate Max Planck's theory of radiation, according to which “the energy radiation from an atomic system does not take place in the continuous way assumed in ordinary electrodynamics, but that it, on the contrary, takes place in distinctly separated emissions” (Bohr 1913, p. 4; BCW 2, p. 164). In this early paper, and most subsequent accounts, Bohr summarized his quantum theory by means of two assumptions or postulates. According to the first postulate, electrons cannot travel in any path they like around the nucleus, rather atomic systems can only exist in a series of discrete “stationary states,” in which the electron is in a particular allowed stable periodic orbit and is not emitting radiation.
The old quantum theory of Bohr and Sommerfeld was abandonned for the wrong reason. Its contradictions were caused not by the orbit concept but by a mental barrier—the inconceivability that an electron might collide with the atomic nucleus. Removing that barrier resolves the theory’s main failures—incorrect orbital momenta, He atom, H + 2 molecule ion. The inclusion of electron oscillations through the nucleus—a concept called “Coulomb oscillator”—renders the old quantum theory consistent with quantum mechanics (although devoid of wave character). The triple success of the Bohr-Sommerfeld model is its correct description of the H atom (and one-electron ions) concerning (1) the energy levels Enl, (2) the orbital angular momenta Lnl—if corrected as L 2 = l(l + 1)~ 2 and with the Coulomb oscillator included—and (3) the orbits’ space quantization—with (Lnl)z = ml~. These achievements are succinctly represented by the principal, angular and magnetic quantum numbers (n, l, ml) and visualized by orbital ellipse geometry—major axis, vertex curvature, and tilt angle, respectively. Orbit geometry also accounts for the average orbit size. Moreover, the Coulomb oscillator provides a natural explanation of (isotropic) hyperfine interaction. The shortcomings of the old quantum theory lie in its neglect of three properties of particles—their spin, their wave nature and their quantum statistics. These deficiencies notwithstanding, the visual appeal of the Bohr-Sommerfeld model remains a pedagogical asset to complement the abstract character of quantum mechanics.
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.