The Role of Eigenstates in Quantum Mechanics

Introduction

Linear algebra includes finding eigenvalue solutions to eigenequations, which have applications in many fields, including quantum mechanics. We describe the core role in quantum mechanics of eigenstates, which involve both eigenvalues and eigenfunctions.

Quantum mechanics is a theory that describes the behavior of microscopic entities such as light, subatomic particles, atoms, and molecules— yielding a crucial part of our understanding in branches of physics, chemistry, biology, materials science, and engineering. Quantum mechanics is used in the development and understanding of computer chips, light-emitting diodes (LEDs), lasers, atomic clocks in global positioning systems, nuclear power plants, solar cells, x-ray and magnetic resonance imaging, and electron microscopes.

To learn the role of eigenstates in quantum mechanics, we employ a toy model: an electron in a box (i.e., in an infinitely-deep potential well). In a toy model, the complexities of the real world are stripped away or greatly simplified so as to reveal underlying basic principles that we want to understand. Our electron is a limit case for an electron in a finite potential well, which “has a close correspondence with actual problems in the design of semiconductor quantum well structures” [Miller 2008].

The confinement of the electron by the walls of the box leads to quantization of the electron’s possible energy values. Electrons in nature are often confined to some degree, so their possible energy values are also quantized. The electron-in-a-box toy model helps us understand this phenomenon.