What are Eigenstates and Eigenvalues in Quantum Mechanics?

In quantum mechanics, an eigenstate (or "eigenvector") of a linear operator is a non-zero vector that, when operated on by the operator, results in a scaled version of itself. In other words, the operator acts as a scalar multiplication on the eigenstate, rather than changing its direction.

For example, consider a matrix A representing a linear transformation, and a vector v. If there exists a scalar λ such that Av = λv, then v is an eigenvector of A with eigenvalue λ. In this case, the matrix A can be thought of as "stretching" the vector v by a factor of λ, but not changing its direction.

Eigenstates and eigenvalues play a central role in many areas of quantum mechanics, including quantum computing, quantum field theory, and condensed matter physics. They are used to describe the behavior of quantum systems, and to solve problems involving quantum systems.

In summary, an eigenstate is a non-zero vector that, when operated on by a linear operator, results in a scaled version of itself, and an eigenvalue is the scalar that represents the amount of stretching or shrinking that the operator applies to the eigenstate.