Stone's Theorem

Definition

Stone's Theorem states that every closed, densely defined linear operator on a Hilbert space can be uniquely extended to a self-adjoint operator on a larger space. This theorem connects self-adjoint extensions with the broader framework of unitary operators, as it highlights how operators can be transformed and extended while preserving their essential properties.

5 Must Know Facts For Your Next Test

1. Stone's Theorem applies specifically to linear operators defined on a dense subset of a Hilbert space, emphasizing the importance of the operator's domain.
2. The theorem guarantees that for any closed linear operator, there exists a unique self-adjoint extension, which is critical in quantum mechanics for ensuring physical observables are represented by self-adjoint operators.
3. One implication of Stone's Theorem is that the spectrum of the self-adjoint extension is real, which is essential for physical interpretations in quantum mechanics.
4. The theorem also establishes a connection between the self-adjoint extensions and unitary operators through the process of spectral decomposition.
5. Applications of Stone's Theorem are found in mathematical physics, particularly in the formulation of quantum mechanics and the study of differential operators.

Review Questions

• How does Stone's Theorem connect self-adjoint extensions to the properties of linear operators in a Hilbert space?
  • Stone's Theorem establishes that every closed linear operator on a dense subset can be uniquely extended to a self-adjoint operator, which retains vital properties such as real eigenvalues. This connection highlights the importance of self-adjoint extensions in ensuring operators represent observable quantities in quantum mechanics. It also illustrates how these extensions influence the behavior of linear operators within Hilbert spaces.
• In what ways does Stone's Theorem demonstrate the relationship between self-adjoint operators and unitary transformations?
  • Stone's Theorem showcases how self-adjoint extensions can be viewed in relation to unitary operators by facilitating spectral decomposition. Self-adjoint operators can be diagonalized using unitary transformations, preserving their structure while allowing for analysis across larger spaces. This relationship is pivotal as it underlines how physical systems can be studied through different mathematical frameworks while maintaining their essential properties.
• Evaluate the implications of Stone's Theorem for physical observables in quantum mechanics, particularly concerning their representation as operators.
  • The implications of Stone's Theorem for quantum mechanics are profound, as it ensures that physical observables can be represented by self-adjoint operators. This representation guarantees real eigenvalues, which correspond to measurable quantities, providing a solid foundation for interpreting measurements in quantum systems. Furthermore, the existence of unique self-adjoint extensions allows physicists to analyze various scenarios and boundary conditions without losing the essential nature of these observables, reinforcing the stability and reliability of quantum predictions.

Related terms

Self-adjoint Operator: An operator that is equal to its own adjoint, meaning it has real eigenvalues and an orthonormal basis of eigenvectors.

Unitary Operator: An operator that preserves inner products in a Hilbert space, meaning it is invertible and its inverse is equal to its adjoint.

Closure of an Operator: The process of extending a densely defined operator to a closed operator, which can involve taking limits of sequences in the operator's domain.