In the preceding chapter we found that particles sometimes behave like waves and that they can be described by wave functions. Now we’re ready for a systematic analysis of particles in bound states (such as electrons in atoms), including finding their possible wave functions and energy levels. Our analysis involves finding solutions of a fundamental equation called the Schrodinger equation. The wave functions for any system must be solutions of the Schrodinger equation for that sy tern. Each solution corresponds to a definite energy, so solving the Sc odinger equation automatically gives the possible energy levels for a ystem. We’ll discuss several simple one-dimensional applications the Schrodinger equation.
Besides energies, solving the Schrodinger equati n also gives us the probabilities of finding a particle in various regions. ne surprising result of solving the Schrodinger equation is that there i a nonzero probability that microscopic particles will pass through thin barriers, even though such a process is forbidden by Newtonian mechanics. Finally, we’ll generalize the Schrodinger equation to three dimensions. This will pave the way for describing the wave functions for the hydrogen atom in Chapter 43. The hydrogen-atom wave functions in turn
form the foundation for our analysis of more complex atoms, of the periodic table of the elements, of x-ray energy levels, and of other properties of atoms.