Stone's Theorem

Definition

Stone's Theorem, in the context of quantum mechanics, establishes a connection between the time evolution of quantum states and the structure of their underlying Hilbert space. It shows that if you have a self-adjoint operator representing an observable in quantum mechanics, the time evolution operator can be expressed in terms of that operator, demonstrating how quantum states evolve over time within the Schrödinger picture.

5 Must Know Facts For Your Next Test

1. Stone's Theorem provides a framework to connect the dynamics of quantum systems through the time evolution operator, given by $$U(t) = e^{-iHt/\\hbar}$$.
2. The theorem asserts that for each self-adjoint operator, there exists a unique one-parameter group of unitary operators that corresponds to its exponential map.
3. It demonstrates that the continuous time evolution of quantum states can be rigorously derived from self-adjoint operators acting on the Hilbert space.
4. This theorem reinforces the relationship between observables and their corresponding physical evolution, linking measurement and dynamics in quantum mechanics.
5. In practical applications, Stone's Theorem allows physicists to calculate how states change over time based on the Hamiltonian operator.

Review Questions

• How does Stone's Theorem relate the time evolution operator to self-adjoint operators in quantum mechanics?
  • Stone's Theorem shows that every self-adjoint operator corresponds to a unique one-parameter group of unitary operators, which can be expressed through the time evolution operator. This means that the evolution of a quantum state over time can be described using the exponential of the self-adjoint operator associated with a system's observable. This connection is critical for understanding how measurements affect state dynamics within the Schrödinger picture.
• Discuss the implications of Stone's Theorem on the preservation of probability in quantum mechanics.
  • Stone's Theorem implies that since the time evolution operator is unitary, it preserves inner products between state vectors. This preservation is essential because it ensures that probabilities calculated from state vectors remain consistent over time. Therefore, when applying this theorem, we can be confident that even as quantum states evolve according to their Hamiltonians, total probabilities do not exceed 1, adhering to the fundamental rules of probability in quantum systems.
• Evaluate how Stone's Theorem facilitates calculations of state evolution in practical scenarios involving various Hamiltonians.
  • Stone's Theorem plays a vital role in simplifying calculations involving the evolution of quantum states by allowing physicists to use self-adjoint operators associated with different Hamiltonians. By expressing these Hamiltonians in terms of their spectral decompositions, one can derive explicit forms for the time evolution operator and thus compute how specific states evolve. This capability is crucial for solving complex problems in quantum mechanics, including those related to interactions and perturbations in various physical systems.

Related terms

Self-adjoint Operator: An operator that is equal to its own adjoint, playing a crucial role in quantum mechanics as it ensures real eigenvalues, which correspond to measurable quantities.

Unitary Operator: An operator that preserves inner products and thus the norm of state vectors, ensuring that the evolution of quantum states is consistent with probability conservation.

Hilbert Space: A complete vector space equipped with an inner product, where quantum states reside and their properties can be analyzed mathematically.