eigenstateiksi

In quantum mechanics, an eigenstate is a particular state of a quantum system that, when a measurement of a certain observable is performed, yields a definite, unchanging result. This result is known as the eigenvalue. Mathematically, an eigenstate is an eigenvector of an operator corresponding to a physical observable. When the operator acts on its eigenstate, the result is simply the eigenstate multiplied by a scalar constant, which is the eigenvalue.

For example, consider the observable of energy. The eigenstates of the energy operator are called stationary states. In these states, the probability distribution of the system's position and momentum remains constant over time, hence "stationary." If a system is in an eigenstate of a particular observable, measuring that observable will always produce the corresponding eigenvalue, regardless of how many times the measurement is repeated.

However, if a quantum system is not in an eigenstate of an observable, a measurement will yield one of the possible eigenvalues, but the specific outcome cannot be predicted with certainty. Instead, the probabilities of obtaining each eigenvalue are determined by the state of the system. The concept of eigenstates is fundamental to understanding the behavior of quantum systems and the nature of measurement in quantum mechanics.