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Physics · lesson 4 of 4

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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Learn the ideas

Prerequisites: Linear algebra and quantum expectation values.

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

Example 1

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

Variance=11−9=2.

2

Example 2

Which expression defines [A,B]?

The commutator compares multiplication in the two orders.

AB−BA

Check your understanding

8 questions

Decimals or fractions are accepted for numeric answers. Proofs and written responses use sample answers and require human marking. This is practice, not a protected examination.

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

Show answer and explanation

2

Variance=11−9=2.

2 Which expression defines [A,B]?

Show answer and explanation

AB−BA

The commutator compares multiplication in the two orders.

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

Show answer and explanation

Real

Hermiticity guarantees reality, not positivity.

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

Show answer and explanation

0.25

Δp≥1/4 in those model units.

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

Sample answer and marking guidance

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.

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

Show answer and explanation

1

Mean is zero and mean square is one.

7 If a normalized state is an eigenvector of A, what is its variance in A?

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Zero

All ideal measurements of A return the same eigenvalue.

8 Why can one not infer a simultaneous eigenbasis solely because A and B have real entries?

Sample answer and marking guidance

Real entries do not imply commutation or Hermiticity. A simultaneous eigenbasis requires relevant structural hypotheses, such as commuting Hermitian matrices in finite dimensions.

Credit missing commutation and observable assumptions.

Further reading and review references

These are reference links, not copied exercises, institutional endorsement or proof of independent review.