We saw in Chapter 8 that time-dependent quantum motion could be studied in its own right, but some features such as quantum transitions are connected to the stationary states of time-independent quantum systems where the potential has no explicit dependence on time. Transitions between these states give rise to *discrete* emission lines which could not be explained by classical physics when they were discovered around the turn of the twentieth century. In fact, one of the most spectacular successes of quantum mechanics was its full explanation of the hydrogen spectra, helping to establish quantum physics as the correct theory at the atomic scale.

In this chapter, we discuss the structure of time-independent quantum systems. We are interested in seeking solutions to the bound states of the time-independent Schrödinger equation (see Eq. (9.1) below). We will discuss several methods, beginning with shooting methods for 1D and central-field problems. They are the most direct methods with a high degree of flexibility in choosing the core algorithm, requiring no more than an ODE solver and a root finder at the basic level. We use it with the efficient Numerov algorithm to study band structure of quasi-periodic potentials, atomic structure, and internuclear vibrations.

We also apply the 1D finite difference and finite element methods introduced previously for boundary value problems to eigenvalue problems, including point potentials in the Dirac ...

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