Quantum Mechanics in One Dimension

Abstract

Solid-state physics is primarily concerned with the quantum mechanics of bulk materials and surfaces. Molecular physics and quantum chemistry are similarly the application of quantum mechanics to molecular problems. Bulk materials may be described as three-dimensional objects, and their spatial dimensions have a significant influence on the allowed solutions for quantum mechanical energy states or levels. These quantum mechanical levels in three dimensions give rise to electronic band structures which are commonly used to define a material as a metal, insulator, or semiconductor. Energy bands are formed from quantum mechanical states that are nearly continuous in energy. If the states that comprise a band are only partially filled with electrons, a metal is formed. For a fully occupied band separated by a relatively small energy gap, a semiconductor is the result. If the energy gap between a filled band and an empty band is large, the material is described as an insulator. Molecules are zero-dimensional objects with vanishing of the wave function in all three spatial directions and the bound electrons do not propagate. This gives rise to a discrete energy spectrum that is characteristic of molecules; the spacing between energy levels is large and there is no corresponding band picture of the electronic spectrum.