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Consistent Treatment of Quantum Systems with a Time-Dependent Hilbert Space
1Departments of Mathematics and Physics, Koç University, Sarıyer, 34450 Istanbul, Turkey.
Entropy (Basel, Switzerland)
|April 26, 2024
Summary
We present a consistent framework for quantum mechanics with time-dependent Hilbert spaces. The Hamiltonian may not be an observable due to geometric aspects and gauge potentials in quantum systems.
Area of Science:
- Quantum Mechanics
- Theoretical Physics
Background:
- Quantum systems often assume a fixed Hilbert space.
- Time-dependent Hilbert spaces present unique theoretical challenges.
- Understanding these systems is crucial for advanced quantum theories.
Purpose of the Study:
- To develop a consistent mathematical framework for quantum mechanics with time-dependent Hilbert spaces.
- To investigate the role of the Hamiltonian operator in such systems.
- To explore the geometric aspects and gauge potentials in quantum mechanics.
Main Methods:
- Developed a consistent treatment for quantum systems with time-dependent Hilbert spaces.
- Analyzed the conditions under which the Hamiltonian operator represents an observable.
- Investigated quantum systems with time-dependent inner products on vector spaces.
Main Results:
- Demonstrated that the Hamiltonian operator is generally not an observable, even if self-adjoint.
- Identified a hidden geometric aspect related to operator-valued gauge potentials.
- Provided a rigorous treatment for systems with time-dependent inner products.
Conclusions:
- Quantum mechanics with time-dependent Hilbert spaces requires careful consideration of geometric phases.
- The standard interpretation of the Hamiltonian as an observable may fail in these systems.
- This work offers a foundation for studying complex quantum dynamics.
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