The problem of interacting particles can usually be reduced in quantum mechanics to that of one particle, as can be done in classical mechanics. Normally, this problem is one of an electron orbiting around, or being affected by, a positive atomic core or a scattering center with the interaction governed by the Coulomb potential. In general, this is a multi-dimensional problem, and not one of the simpler one-dimensional problems with which we have been concerned in the previous chapters. Once we begin to treat multiple dimensions, then state degeneracy begins to arise more frequently, and the most common problem treated is that of the hydrogen atom vs. the helium atom. These extra dimensions provide more degrees of freedom and more complexity. In the hydrogen atom, there is only a single electron, and the problem is solvable. On the other hand, adding a second electron for the helium atom makes the problem extremely difficult to solve, as it has become a three-body problem in which the interactions are difficult to handle.
The problem of interacting particles can usually be reduced in quantum mechanics to that of one particle, as can be done in classical mechanics. Normally, this problem is one of an electron orbiting around, or being affected by, a positive atomic core or a scattering center with the interaction governed by the Coulomb potential. In general, this is a multi-dimensional problem, and not one of the simpler one-dimensional problems with which we have been concerned in the previous chapters. Once we begin to treat multiple dimensions, then state degeneracy begins to arise more frequently, and the most common problem treated is that of the hydrogen atom vs. the helium atom. These extra dimensions provide more degrees of freedom and more complexity. In the hydrogen atom, there is only a single electron, and the problem is solvable. On the other hand, adding a second electron for the helium atom makes the problem extremely difficult to solve, as it has become a three-body problem in which the interactions are difficult to handle.
The problem of interacting particles can usually be reduced in quantum mechanics to that of one particle, as can be done in classical mechanics. Normally, this problem is one of an electron orbiting around, or being affected by, a positive atomic core or a scattering center with the interaction governed by the Coulomb potential. This simple problem of a particle in a quadratic potential energy is one of the typical problems of quantization. In two dimensions, the problem is more interesting, but not particularly more complicated. In three dimensions, people could extend the treatment to a three-dimensional harmonic oscillator, but a more important problem has spherical symmetry in the potential, and the latter is not the quadratic potential of the harmonic oscillator. It is of more interest to turn our attention to the motion of an electron about the nucleus of an atom.
Chapter 12 uses the results presented in the previous chapter to deal with systems consisting of two particles interacting via a central potential. It therefore begins by transforming the Hamiltonian of the two-particle system into one that separates the center-of-mass motion from the relative motion. When the potential is spherically symmetric, the relative motion is conveniently described in spherical coordinates to take advantage of the conservation of angular momentum and the solution of the corresponding eigenvalue problem. The chapter contains a detailed discussion of the rigid rotor and the free particle, as well as the selection rules for angular momentum, in preparation for the study of the most important problem of atomic physics, the hydrogen atom. The H spectrum is discussed in detail, and the introduction of minimal coupling to the electromagnetic field allows for a discussion of the normal Zeeman effect as well as the Aharonov-Bohm effect.
So far, we have dealt with one- and two-body systems in a variety of settings. These are quite simple, at least a priori, since the equations needed to solve one- and twoparticle scattering problems, as well as bound states, are usually easy enough to handle. The situation becomes much more intricate in the three-body problem, as we shall see in this chapter. Firstly, while two particles in free space can either scatter or bind together, three bodies have extra channels (for attractive interactions) corresponding to two bound particles scattering off a third. Moreover, the energies of these extra channels can be degenerate with those of three unbound particles, and the total momentum of a pair within the triplet is not conserved. Fortunately, we will see that getting our hands dirty with the three-body problem pays off, since there are a number of exotic phenomena associated with the quantum three-body problem, for both bound and scattering states. In this chapter, we introduce the formalism necessary to investigate the three-body problem and study in detail some of the most striking phenomenology known to date associated with, at least, the simplest interactions.
In the previous chapter we saw that for a particle in a one-dimensional potential well, physically acceptable solutions to the one-dimensional Schro¨dinger equation are possible only for particular discrete values of the total energy. Moreover, in the case of an electron in a well of atomic dimensions, the spacings between these energy levels are in qualitative agreement with the separations experimentally observed in atoms; we also interpreted the square of the wave function as a probability distribution for the position of the particle. These ideas led us to phenomena, such as quantum-mechanical tunnelling, that have been observed experimentally. The real world, however, is three-dimensional and, although one-dimensional examples often provide useful insights and analogies, we shall have to extend our theory into three dimensions before we can make quantitative predictions of most experimental results. In the present chapter, therefore, we shall set up the three-dimensional Schro¨dinger equation and obtain its solutions in a number of cases, culminating in a discussion of the hydrogen atom where we shall find that theory and experiment agree to a remarkable degree of accuracy.
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.
In this chapter we study the energy levels of two-electron atoms, principally helium. A two-electron atom is a system of three interacting particles, a problem extremely difficult to analyze within classical mechanics. Since the mass of the nucleus is much larger than that of the electrons, the nuclear motion may be ignored and the problem reduces to two interacting particles moving in an external potential. This is still difficult to solve within classical mechanics. The old Bohr-Sommerfeld quantum mechanics also had little success with this problem. Thus it was a critical triumph for wave mechanics to provide correct calculations of the energy levels of two-electron atoms. Indeed it is remarkable that very little effort is required, as we shall see, to give a good estimate for the ground state energy. Further calculations for two-electron atoms provided six-figure agreement between theory and experiment, a striking proof of the validity of wave mechanics.