In quantum mechanics, an eigenfunction (or eigenvector) and eigenvalue are important concepts related to the mathematical representation of physical systems.
An eigenfunction is a wave function that represents a particular state of a quantum system. When an observable quantity (such as energy, momentum, or position) is measured, the eigenfunction of the corresponding operator provides the possible outcomes and associated probabilities. Mathematically, an eigenfunction is a solution to the Schrödinger equation for a given potential energy function.
An eigenvalue, on the other hand, is the corresponding value obtained when an observable is measured on an eigenfunction. It represents the result of the measurement, such as the energy level or the momentum magnitude. Eigenvalues are often quantized in quantum mechanics, meaning they can only take on specific discrete values.
The concept of eigenfunctions and eigenvalues is closely related to the concept of operators in quantum mechanics. An operator is a mathematical operation that acts on a wave function, transforming it into another wave function or a scalar multiple of itself. When an operator acts on an eigenfunction, the result is a scalar multiple of the original eigenfunction, where the scalar is the corresponding eigenvalue.
Eigenfunctions and eigenvalues have wide applications in quantum mechanics. They allow us to describe and understand the behavior of quantum systems, determine the possible outcomes of measurements, and analyze the time evolution of wave functions. Additionally, the superposition principle in quantum mechanics states that any wave function can be expressed as a linear combination of eigenfunctions, which greatly simplifies the mathematical representation of complex systems.