Quantum time evolution
Quantum time evolution describes how a closed system’s state-vector changes between measurements.
Plain-language picture
Susskind connects this to the classical demand that laws of motion preserve distinctions between states. In quantum mechanics, distinguishability is represented by orthogonality. A valid closed-system time evolution must preserve inner products and normalization.
That requirement leads to unitary operators:
|Ψ(t)⟩ = U(t)|Ψ(0)⟩
U†U = IHamiltonian as generator
For continuous time, the Hamiltonian H generates the unitary evolution:
iħ d|Ψ⟩/dt = H|Ψ⟩H is the energy observable, but it also plays the dynamical role of telling the state how to move through Hilbert space.
Deterministic but not classical
The state evolution is deterministic: a starting state plus Hamiltonian fixes the later state. Measurement outcomes can still be probabilistic because the state predicts distributions, not a classical list of definite outcomes.
Common pitfalls
- Do not confuse unitary evolution with measurement collapse/update.
- Do not say quantum mechanics is simply nondeterministic without specifying measurement versus state evolution.
- Do not forget that changing H changes the dynamics.