Eigenstates

Definition

Eigenstates are specific quantum states of a physical system that correspond to definite values of an observable, as defined by a linear operator in quantum mechanics. When an operator acts on an eigenstate, the output is simply the eigenstate scaled by a constant known as the eigenvalue. This concept is fundamental because it links the mathematical framework of operator theory to physical measurements in quantum mechanics.

5 Must Know Facts For Your Next Test

1. Eigenstates are solutions to the equation $$A|\psi\rangle = \lambda|\psi\rangle$$, where A is an operator, |\psi\rangle is an eigenstate, and \lambda is the corresponding eigenvalue.
2. In quantum mechanics, measuring an observable will yield one of its eigenvalues, and the system will collapse into the corresponding eigenstate after the measurement.
3. An observable can have multiple eigenstates associated with different eigenvalues, which can indicate various possible outcomes for measurements.
4. The set of all eigenstates of an operator forms a complete basis for the vector space of possible states, allowing any state to be expressed as a combination of these eigenstates.
5. The concept of eigenstates is key in solving problems related to quantum systems, such as determining energy levels in quantum wells or harmonic oscillators.

Review Questions

• How do eigenstates relate to measurements in quantum mechanics?
  • Eigenstates are directly related to measurements in quantum mechanics because they represent definite values of observables. When an observable is measured, the system's wave function collapses into one of its eigenstates, corresponding to the observed eigenvalue. This relationship between measurement outcomes and eigenstates is fundamental for understanding how quantum systems behave when probed experimentally.
• Discuss the role of Hermitian operators in defining eigenstates and their importance in quantum mechanics.
  • Hermitian operators play a crucial role in defining eigenstates because they ensure that all eigenvalues are real, which is essential for measurable physical quantities. When a Hermitian operator acts on an eigenstate, it guarantees that the outcomes correspond to actual measurements. This characteristic makes Hermitian operators vital for formulating observable quantities within quantum mechanics, linking mathematical properties with physical realities.
• Evaluate how the principle of superposition utilizes the concept of eigenstates in describing complex quantum systems.
  • The principle of superposition utilizes the concept of eigenstates by allowing any quantum state to be expressed as a combination of multiple eigenstates. This means that even if a system is not in a single eigenstate, it can exist in a superposition of them, leading to interference effects and probabilistic outcomes upon measurement. Understanding how these combinations work helps explain phenomena like quantum entanglement and wave-particle duality, making it essential for advanced studies in quantum theory.

Related terms

Eigenvalues:

Eigenvalues are the constants that scale eigenstates when a linear operator acts on them, representing measurable quantities in quantum systems.

Hermitian Operators:

Operators that are equal to their own adjoint, crucial in quantum mechanics since their eigenvalues are real and correspond to observable quantities.

Superposition:

The principle that a quantum state can be expressed as a combination of multiple eigenstates, leading to complex behavior in quantum systems.