Visual Quantum Mechanics

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Chapter Summary:

2. Coulomb Problem

Perhaps the most important success of quantum mechanics is the explanation of the internal electronic structure of atoms on the basis of general physical laws. The Schrödinger equation with a Coulomb potential, although a crude model of a "real" hydrogen atom, can describe its properties with high accuracy and explains the spectroscopic observations that have puzzled physicists at the beginning of the 20th century.

Already the classical Coulomb problem has interesting aspects. We discuss, in particular, the conservation of the Runge-Lenz vector, which implies that the form and the orientation of the classical orbits (Kepler ellipses) is constant in time and depends only on the initial conditions. In quantum mechanics, the Runge-Lenz vector can be used for an elegant algebraic computation of the energy levels given in Section 2.3.

The following sections are devoted to various approaches to the solution of the Coulomb problem. As the Coulomb potential is spherically symmetric, we may apply the results of the previous chapter in order to reduce the problem to the solution of ordinary differential equations in the angular-momentum subspaces. In Section 2.4, we use a factorization method to solve the radial Coulomb problem in an essentially algebraic way. This approach leads to a solution of the Coulomb problem via a system of simultaneous eigenfunctions of H, L2, and L3 and exhibits clearly the structure of the energy spectrum, in particular the high degeneracy of the eigenvalues. In this approach, the l-degeneracy appears as a consequence of a supersymmetry of the radial Schrödinger operators, which is similar to the one observed for the harmonic oscillator in Book One.

We continue by outlining the traditional approach the Coulomb problem, which consists in solving the radial Schrödinger equation in terms of special functions. In Section 2.5.1, we present a solution of the Coulomb problem in two dimensions which, naturally, plays an important role for the visualizations accompanying this book. Finally, we present a method for solving the Schrödinger equation in parabolic coordinates. The separation in parabolic coordinates is important for the investigation of the Stark effect (hydrogen atom in a constant electric field).

In the major part of this chapter, we use dimensionless units in all calculations. This simplifies mathematical derivations, yields beautiful formulas, and is by no means a restriction of generality. A simple scaling transformation gives all information about the eigenfunctions and eigenvalues in SI units (or in any other system of units). Section 2.7 is devoted to the actual physical dimensions of the hydrogen atom. Moreover, we describe general scaling transformations (dilations) as unitary transformations in the Hilbert space, because they are an important tool in mathematical physics. In this connection, we present the virial theorem in Section 2.7.4.

Concerning the dynamics of states in the subspace of bound states, we investigate the behavior of simple superpositions in Section 2.4.5 and the dynamics of circular Rydberg states in Section 2.8. Circular Rydberg states are highly excited states that move in the close neighborhood of classical circular orbits. The long lifetime of these states and their quasiclassical behavior make them an interesting research topic in atomic physics.

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    • 2.1. Introduction 58
      • 2.1.1. Classical models of atoms 58
      • 2.1.2. Transitions between eigenstates 59
    • 2.2. The Classical Coulomb Problem 61
      • 2.2.1. The Coulomb force 61
      • 2.2.2. Classical motion 63
    • 2.3. Algebraic Solution Using the Runge-Lenz Vector 66
      • 2.3.1. The Coulomb problem in quantum mechanics 66
      • 2.3.2. The Runge-Lenz vector in quantum mechanics 67
      • 2.3.3. The eigenvalues of $H$ 68
    • 2.4. Algebraic Solution of the Radial Schrödinger Equation 70
      • 2.4.1. Factorization of the radial Coulomb problem 70
      • 2.4.2. The ground state of the radial Schrödinger operator 72
      • 2.4.3. Excited states of the radial Schrödinger operator 73
      • 2.4.4. Quantum numbers of the Coulomb problem 76
      • 2.4.5. Time evolution of simple superpositions 82
    • 2.5. Direct Solution of the Radial Schrödinger Equation 84
      • 2.5.1. Solution of the Coulomb problem in two dimensions 84
      • 2.5.2. Solution of the Coulomb problem in three dimensions 88
    • 2.6. Special Topic: Parabolic Coordinates 91
    • 2.7. Physical Units and Dilations 96
      • 2.7.1. The Coulomb problem in physical units 96
      • 2.7.2. Scaling transformations 99
      • 2.7.3. Energies and eigenfunctions in physical units 101
      • 2.7.4. The generator of dilations and the virial theorem 103
    • 2.8. Special Topic: Dynamics of Rydberg States 105