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Physics · University

Operators, commutators and uncertainty

Observables are represented by suitable self-adjoint operators. In finite dimensions they are Hermitian matrices with real eigenvalues.

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Prerequisites

Linear algebra and quantum expectation values.

Study notes

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.

Worked examples

An observable has mean 3 and mean square 11 in matching units. Find its variance.

Variance=11−9=2.

2

Which expression defines [A,B]?

The commutator compares multiplication in the two orders.

AB−BA

Check your understanding

  1. For a finite-dimensional Hermitian observable, its eigenvalues are what?

    1. Always zero
    2. Real
    3. Always positive
    4. Always imaginary

  2. Using ℏ=2 in model units and Δx=4, find the minimum Δp allowed by ΔxΔp≥ℏ/2.

  3. Explain the difference between a state's quantum variance and calibration error in an instrument.

  4. An observable takes values −1 and +1 with equal probability. Find its variance.

Answers and marking guidance

Answers and marking notes
  1. Real

    Hermiticity guarantees reality, not positivity.

  2. 0.25

    Δp≥1/4 in those model units.

  3. Quantum variance is intrinsic outcome spread for the given state and observable; calibration error concerns how an apparatus reports results. Both can matter, but they are not identical.

    Credit the distinction without claiming instruments are irrelevant.

  4. 1

    Mean is zero and mean square is one.

Further reading and review references

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