Dynamics Results
Dynamics result cards collect compact statements about how quantum states, expectation values, and eigenvalues respond to time evolution or parameter changes.
| Result | Main role | Key assumption to check |
|---|---|---|
| Ehrenfest Theorem | Expectation values obey force-law-like equations | domains and force expectation |
| Virial Theorem | Stationary bound states relate kinetic and potential averages | bounded stationary state and vanishing boundary terms |
| Adiabatic Theorem | Slow evolution follows instantaneous eigenspaces | spectral gap and smooth slow driving |
| Hellmann–Feynman Theorem | Eigenvalue derivatives are expectation values of Hamiltonian derivatives | exact normalized eigenbranch |
Shared Caveats
Section titled “Shared Caveats”- A dynamics theorem is not a substitute for checking a Hamiltonian’s domain and boundary conditions.
- Approximate states may require correction terms even when exact-state theorem statements look simple.
- Time-dependent Hamiltonians often need stronger hypotheses than the time-independent formulas suggest.
- Degeneracies and level crossings usually change the statement.
References
Section titled “References”- R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
- J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
- C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
- A. Messiah, Quantum Mechanics, Dover, 1999.