Physics · University
Operators, commutators and uncertainty · revision notes
Observables are represented by suitable self-adjoint operators. In finite dimensions they are Hermitian matrices with real eigenvalues.
Key ideas and conditions
Operators
Observables are represented by suitable self-adjoint operators. In finite dimensions they are Hermitian matrices with real eigenvalues.
Commutator
[A,B]=AB−BA measures noncommutation. Matrix multiplication order matters.
Variance
The variance of A is ⟨A²⟩−⟨A⟩². A zero variance in a normalized state indicates a definite outcome for that observable.
Uncertainty
For canonical position and momentum, ΔxΔp≥ℏ/2. The spreads describe the quantum state, not merely careless measurement.
Explain without looking
- If a normalized state is an eigenvector of A, what is its variance in A?
- Why can one not infer a simultaneous eigenbasis solely because A and B have real entries?
Further reading and review references
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