Eigenvalues and Eigenfunctions: Basic Review
Eigenvalues and eigenfunctions, or eigenvectors in linear algebra, are some of the fundamental concepts in mathematics and physics. Here is a very basic overview of these terms that are very important for one to know:
1. Eigenvalues
Eigenvalue is a scalar which represents how much the linear transformation changes an eigenfunction, or eigenvector, in a system.
Physical Significance: In quantum mechanics, eigenvalues often correspond to measurable quantities like energy, momentum, and angular momentum. For example, in the Schrödinger equation, the eigenvalues represent the possible energy levels of a quantum system.
**2. Eigenfunctions (Eigenvectors)
Definition: An eigenfunction is a non-zero function that remains proportional to itself when acted upon by a linear operator.
Physical Interpretation: In quantum theory, eigenfunctions are an expression of the state. For instance, the solution to the time-independent equation of Schrödinger is an eigenfunction expressing the probability distribution of finding a particle at a point in space.
Examples
1. Particle in Box
Imagine a particle, confined in one-dimensional potential well
Conclusion
Eigenvalues and eigenfunctions are the important components in the study of quantum systems, which give a clue to the quantization of physical properties and behavior of particles. These extend far beyond quantum mechanics; they play crucial roles in several fields of science and engineering.
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