Susskind QM Lecture 4 - Time and Change
One-line takeaway
Closed quantum systems change deterministically by unitary evolution, even though individual measurement outcomes remain probabilistic.
Plain-language map
Susskind begins with the classical “minus first law”: good time evolution should not erase distinctions between states. In quantum mechanics, distinguishable states are orthogonal, so time evolution must preserve inner products. The operator that does this is unitary.
If the state at time zero is |Ψ(0)⟩, then the state at time t is:
|Ψ(t)⟩ = U(t)|Ψ(0)⟩
U†U = IFor continuous time, the Hamiltonian H generates this unitary motion. The result is the general Schrödinger equation:
iħ d|Ψ⟩/dt = H|Ψ⟩This is deterministic for the state-vector. It does not mean measurements become classically deterministic. It means that if you know the present state, you can compute future probabilities.
Core links
- Quantum time evolution
- Schrödinger equation
- Observables and eigenvalues
- Commutators and compatible observables
Useful expectation-value rule
For an observable L with no explicit time dependence:
d⟨L⟩/dt = (i/ħ) ⟨[H, L]⟩This is the bridge back to classical dynamics: commutators in quantum mechanics play a role reminiscent of Poisson brackets in Hamiltonian mechanics.
Common pitfalls
- Do not confuse deterministic state evolution with deterministic measurement outcomes.
- Do not use non-unitary evolution for a closed system.
- Do not forget that the Hamiltonian is an operator, not merely an energy number.