Eigenzustände

Eigenzustände, also known as eigenfunctions, are fundamental concepts in quantum mechanics and linear algebra. In quantum mechanics, an eigenstate of an observable, such as position or momentum, is a state of the system for which the measurement of that observable yields a definite value. When a system is in an eigenstate of an observable, the act of measuring that observable will always result in the same specific eigenvalue, regardless of how many times the measurement is repeated. The eigenstate itself represents the specific configuration or property of the quantum system associated with that eigenvalue.

Mathematically, eigenstates are solutions to eigenvalue equations. For a linear operator A, representing an observable, and a state vector |ψ⟩, an eigenstate |ψ⟩ satisfies the equation A|ψ⟩ = λ|ψ⟩, where λ is the corresponding eigenvalue. The eigenvalue λ is a scalar value representing the result of measuring the observable A when the system is in the eigenstate |ψ⟩. Eigenstates form a complete basis in the Hilbert space of the system, meaning that any arbitrary state can be expressed as a linear combination of these eigenstates. This completeness allows for the decomposition of any quantum state into its constituent eigenstate components, each associated with a specific eigenvalue.