An eigenvalue is a scalar value that represents the magnitude of a specific characteristic of a physical system. In the context of the Schrodinger wave equation, eigenvalues represent the allowed energy levels of a particle in a potential field.
An eigenfunction is a mathematical function that describes the behavior of a physical system at a specific eigenvalue. In the Schrodinger wave equation, eigenfunctions are used to describe the wave function of a particle at a specific energy level.
Eigenvalues and eigenfunctions are closely related in the sense that an eigenfunction corresponds to a specific eigenvalue. In other words, the eigenfunction describes the behavior of the system at a particular energy level represented by the eigenvalue.
In the Schrodinger wave equation, eigenvalues and eigenfunctions play a crucial role in determining the allowed energy states of a particle in a potential field. The eigenfunctions also help to describe the probability of finding a particle at a specific location in space.
Eigenvalues and eigenfunctions have numerous applications in various fields, including physics, chemistry, and engineering. In quantum mechanics, they are used to solve the Schrodinger equation and determine the energy levels of particles in different systems. They are also used in signal processing, data analysis, and other mathematical applications.