Quantum dynamics and symmetries
Schrödinger and von Neumann evolution, conservation laws, Wigner and Stone theorems, Kramers degeneracy and open systems.
Schrödinger equation
Governs the deterministic evolution of an isolated system between measurements.
Dynamics ↗Unitary closed-system evolution
A self-adjoint $H$ generates unitary dynamics that preserves norm and inner products. (Time-independent $H$ shown.)
Dynamics ↗Von Neumann equation
Schrödinger dynamics for density operators, including mixed states.
Dynamics ↗Conserved observables
Commuting with the Hamiltonian makes an observable's whole distribution constant in time.
Dynamics ↗Energy conservation
For a closed system with time-independent $H$, mean energy is conserved.
Dynamics ↗Ehrenfest theorem
Expectation values follow equations resembling classical mechanics, e.g.
Dynamics ↗Wigner's theorem
Symmetries preserving transition probabilities act linearly (unitary) or antilinearly (antiunitary), up to a phase.
Dynamics ↗Stone's theorem
Strongly continuous one-parameter unitary groups have self-adjoint generators, justifying $H$ as the time-translation generator.
Dynamics ↗Quantum Noether principle
Momentum generates translations, angular momentum rotations; symmetry of $H$ means conservation of the generator.
Dynamics ↗Time reversal
Time reversal must complex-conjugate numbers, as needed for $e^{-iHt}\to e^{+iHt}$ to map consistent dynamics.
Dynamics ↗Kramers degeneracy
For odd numbers of spin-1/2 particles with time-reversal symmetry, every level has a partner; magnetic fields break it.
Dynamics ↗Adiabatic theorem
If the Hamiltonian changes slowly relative to the gap, a system stays in its instantaneous eigenstate.
Dynamics ↗Berry phase
Cyclic adiabatic evolution gives a geometric phase beyond the dynamical phase $-\int E\,dt/\hbar$.
Dynamics ↗GKSL generator
The most general generator of Markovian, completely positive, trace-preserving semigroups in finite dimension; models decoherence.
Dynamics ↗Decoherence-free subspaces
If all noise operators act as the same scalar on a subspace, with Hamiltonian invariance, encoded information is immune from that noise.
Dynamics ↗