# Operators, commutators and uncertainty

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

Course: [Introduction to quantum physics and relativity](https://studybrix.com/en/courses/quantum-and-relativity/index.md)
Level: university
Prerequisites: Linear algebra and quantum expectation values.

## 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.

## Practice questions

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

Answer: 2

Explanation: Variance=11−9=2.

### 2. Which expression defines [A,B]?

A. A−B
B. A/B for every operator
C. AB+BA
D. AB−BA

Answer: AB−BA

Explanation: The commutator compares multiplication in the two orders.

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

A. Always zero
B. Real
C. Always positive
D. Always imaginary

Answer: Real

Explanation: Hermiticity guarantees reality, not positivity.

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

Answer: 0.25

Explanation: 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 / 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.

Explanation: Credit the distinction without claiming instruments are irrelevant.

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

Answer: 1

Explanation: Mean is zero and mean square is one.

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

A. Always negative
B. Undefined for all eigenvectors
C. Zero
D. Always one

Answer: Zero

Explanation: 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 / 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.

Explanation: Credit missing commutation and observable assumptions.

## Further reading

- https://ocw.mit.edu/courses/8-09-classical-mechanics-iii-fall-2014/
- https://openstax.org/details/books/university-physics-volume-3

Original AI-assisted educational draft; not independently teacher-reviewed. References support further study, not copied exercises, affiliation or curriculum accreditation.