Eigenfunctions are functions that, when operated on by a linear operator, return a constant multiple of the original function. In other words, the function is unchanged except for a scaling factor.
Eigenstates are the quantum mechanical equivalent of eigenfunctions. They are the states of a system that, when measured, give a definite value for a physical property (such as energy) with a corresponding eigenvalue.
Eigenvalues are the constants by which an eigenfunction is scaled when operated on by a linear operator. In quantum mechanics, they correspond to the measurable values of physical properties that are obtained when the system is in a specific eigenstate.
Eigenfunctions, eigenstates, and eigenvalues are important concepts in quantum mechanics because they provide a mathematical framework for understanding and predicting the behavior of quantum systems. They allow us to calculate the probabilities of different outcomes in experiments, and they form the basis for many important quantum mechanical principles and equations.
Eigenfunctions, eigenstates, and eigenvalues are used in a wide range of real-world applications, such as in quantum computing, spectroscopy, and materials science. They are also used in many mathematical and scientific fields outside of quantum mechanics, such as signal processing and differential equations, to describe the behavior of systems and phenomena.