Bound States in Three Dimensions: The Atom

In this chapter, we go to three dimensions in space and look at the solution of the time independent Schrödinger equation for the hydrogen atom. The Hamiltonian is then the kinetic energy plus the potential energy due to the Coulomb coupling between the positively charged nucleus and the electron. We construct the angular momentum operator and find that the partial differential equation for the angular momentum eigenfunctions of the spherical coordinates θ,ϕ is the same as the angular part of the ∇2 operator in spherical coordinates. The angular momentum eigenfunctions are the spherical harmonics, with two quantum numbers, l and m, and the solution to the radial part of the Hamiltonian including the Coulomb potential are Laguerre polynomials with one quantum number, called the principle quantum number, n. The hydrogen wave function is the product of a Laguerre polynomial and a spherical harmonic with three quantum numbers. Since these are two- and three-dimensional functions for angular momentum and hydrogen respectively, they are best understood in a series of plots. The chapter concludes by giving the historical letter names to specific orbitals, since they continue to be used today.

Animated Atom Boy

The concept of the atom dates back to the speculations of Democritus and other Ancient Greek philosophers, but it was only in modern times that atoms were shown to exist and physicists began to investigate their structure. Rutherford’s team in Manchester discovered the atomic nucleus and Rutherford proposed that atoms consist of a tiny positively charged nucleus surrounded by negatively charged orbiting electrons. Rutherford and his colleagues went on to discover the nucleus is composed of protons and neutrons. It is now possible to produce pictures of atoms using the scanning tunnelling microscope (STM) invented by Binnig and Rohrer.

The Nuclear Atom

Chapter 5 describes how the concept of quantization (discretization) was first applied to atoms. This was done in 1913 by Niels Bohr, using Ernest Rutherford’s paradigm-changing, solar-system model of atomic structure, wherein the positively charged nucleus occupies a tiny central space, much smaller than the known sizes of atoms. Bohr, postulating a quantized version of this model for hydrogen, was able to explain previously inexplicable experimental features of that atom. He did so via an ad hoc quantization procedure that discretized the single electron’s energy, its angular momentum, and the radii of the orbits it could be in around the nucleus, formulas forwhich are presented, along with a diagram displaying the quantized energies. Despite this success, Bohr’s model failed not only for helium, with its two electrons, but for all other neutral atoms. It left some physicists hopeful, ready for whatever the next step might be.

Relativistische Verallgemeinerung des Lenzschen Vektors (I) / Relativistic generalisation of the Lenz vector (I)

The non-relativistic motion of a particle in a central field with 1/r potential, e.g. the motion of an electron in the Coulomb field of a charged nucleus at rest, is described by the equation of motion (non-relativistic Kepler problem) m x″ = α · x /r3 with α = ez e (product of the charges of the central body ez and the electron e). From this equation of motion, three statements of conservation can be derived: in respect of the energy E, of the angular momentum L and of the Lenz vector Λ = m {x′× L + α ·x/r}. The geometric meaning of Λ is that of a vector pointing in the direction of the perihelion of the particle orbits (conic sections). It will be demonstrated that also at the relativistic Kepler problem, which is based on the equation of motion an analogous Lenz vector exists. It represents a quantity of conservation - in the same way as the relativistic energy and the relativistic angular momentum. For the transitional case → ∞, where the relativistic problem turns into the non-relativistic problem, the relativistic Lenz vector also turns into the non-relativistic Lenz vector. The generalised (relativistic) Lenz vector has also a geometric meaning. Its direction coincides with the oriented axis of symmetry of the orbits (rosettes, spirals, hyperbola-type curves etc.). The quantity of conservation Λ occupies a special position in respect of the quantities of conservation energy and angular momentum. Whereas the energy and the angular momentum correspond with a symmetry of time and space, the Lenz quantity of conservation corresponds with a symmetry of the orbits. The fact that the Lenz vector can relativistically be generalised touches thereby on principal aspects.

The internal conversion coefficient for radium C

It is well known that the γ-rays emitted from a radio­active nucleus are often partially absorbed by the atomic system, giving rise to secondary β-rays. From observations of the resultant γ-ray intensity, and that of the β-rays, it is possible to infer the proportion of γ-rays reabsorbed in the atomic system. This factor is called the “internal conversion co­efficient.” Its theoretical value has been discussed by Miss Swirles and R. H. Fowler. Miss Swirles treats the nucleus as an oscillating Hertzian doublet, radiating classically, and considers the radiation field as producing photoelectric transitions in the planetary electrons, according to the Schrodinger theory. The rate of emission of γ-rays from the nucleus is taken to be the classical rate of radiation of energy by the dipole, divided by hv . The values obtained in this way were about 10 times too small, except for the γ-ray of energy 14.26 x 10 5 e. v., which has an internal conversion coefficient several hundred times that given by the theory. This special case has been discussed by Fowler ( loc . cit .), and we shall not consider it here. An obvious defect in the theory is the use of Schrödinger’s equation, which may not be expected to hold so near the nucleus, or for electrons of such high energy. It therefore seemed possible that the more correct, relativistic equation of Dirac might give results in accordance with experiment in the majority of cases, and the calculation has been carried out by Casimir. The same model is used, and, for purposes of calculation, the interaction of the other electrons is neglected, so that we have a single electron in the field of a charged nucleus. For the β-rays emitted from the K-shell, we may take the actual nuclear charge in carrying out the calculation. In the case of extremely hard γ-rays, whose energies may be considered large compared with mc 2 , it is legitimate to use the asymptotic expansion for the wave function repre­senting the β-ray. If we apply this theory to the range covered by experiment, we obtain results (Casimir, loc. cit. ) which are still much too small, so that we were tempted to attribute the bulk of the conversion to some special type of interaction with the nucleus. It seems fairly certain that this must be the case for the γ-ray with hv = 14.26 x 10 5 e. v., which has an abnormally high internal conversion coefficient.

Photo-electric absorption in hydrogen-like atoms

If light of a frequency which corresponds to an energy greater than the ionisation potential falls on an atom, an electron may be ejected and energy absorbed. To calculate the absorption coefficient, or the rate of absorption of energy per unit intensity of incident radiation for a given frequency, one must first choose a model for the atom. If we confine ourselves to the inner K electrons there will be two electrons in this shell for the heavier atoms, and a fairly good model of the atom is obtained by considering each electron to be moving independently in a central field of force due to the charged nucleus: i.e. we neglect electronic interaction and assume that the wave functions for the system are hydrogenic. Some writers make a partial correction for this neglect of interaction by modifying the central charge through the introduction of a screening factor which is so chosen that the minimum calculated energy required to remove one of the K electrons will agree with the experimental value provided by the K absorption edge. In general, however, the approximation is fairly good, and this is particularly so in the interior of a star where the atoms are highly ionised. It is not so good when the atom is bound as in a metal, and, of course, most of the laboratory work has been carried out on atoms in this bound state.