Susskind QM Lecture 5 - Uncertainty and Time Dependence
One-line takeaway
A complete quantum description often requires a complete set of commuting observables; noncommuting observables lead directly to uncertainty.
Plain-language map
A single spin is too small to illustrate every structure in quantum mechanics. Larger systems may require several labels to specify a basis state. The labels must come from observables that can be known at the same time. In operator language, that means the observables commute.
If L and M share a complete basis of simultaneous eigenvectors, then:
[L, M] = LM - ML = 0This motivates the phrase complete set of commuting observables. Once such a basis is chosen, a state can be represented by its components in that basis. Susskind calls that component list the wavefunction.
ψ(a,b,c,...) = ⟨a,b,c,...|Ψ⟩The squared magnitude gives the probability for the corresponding set of commuting measurement values.
Uncertainty
When observables do not commute, the state generally cannot have sharp values for both. The uncertainty relation makes this quantitative:
ΔL ΔM ≥ (1/2) |⟨[L, M]⟩|The familiar position-momentum version comes later from [X, P] = iħ.
Core links
- Commutators and compatible observables
- Wavefunction
- Uncertainty principle
- Measurement and state preparation
Common pitfalls
- Do not say “uncertainty” when you only mean practical measurement error.
- Do not assume every pair of observables can label the same basis.
- Do not forget that a wavefunction is basis-relative.