Eigenállapotok

Eigenállapotok, often translated as "eigenstates" or "eigen-states", are fundamental concepts in quantum mechanics and related fields like linear algebra. In quantum mechanics, an eigenállapot represents a state of a quantum system that remains unchanged in its fundamental character when a particular observable is measured. The term "eigen" comes from German and means "own" or "characteristic".

When a measurement is performed on a quantum system, the observable quantity will yield a specific value, known as an eigenvalue. If the system is in an eigenállapot corresponding to that observable, then the measurement will always result in the same eigenvalue, and the system's state vector (or wavefunction) will not change after the measurement. In essence, the eigenállapot is the characteristic state associated with a particular measurement outcome.

Mathematically, this is described by the eigenvalue equation. For an operator $\hat{A}$ representing an observable and a state vector $|\psi\rangle$, if $|\psi\rangle$ is an eigenállapot of $\hat{A}$, then the equation $\hat{A}|\psi\rangle = \lambda|\psi\rangle$ holds, where $\lambda$ is the eigenvalue. The eigenvalue $\lambda$ is the value that will be obtained when measuring the observable represented by $\hat{A}$ on a system in the eigenállapot $|\psi\rangle$.

Quantum systems can often exist in superpositions of different eigenállapotok. When a measurement is made in such a case, the system "collapses" into one of the eigenállapotok, with a probability determined by the coefficients of the superposition. This is a cornerstone of quantum measurement theory and explains the probabilistic nature of quantum phenomena. Eigenállapotok are crucial for understanding the properties and behavior of quantum systems, from atomic energy levels to the behavior of subatomic particles.