Commutators, Compatible Observables and the General Uncertainty Relation · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

When two quantities refuse to be sharp together

Science · Quantum & Modern Physics · ages 22-23
Name ______________________   Date ____________

  1. Commuting observables share a full set of eigenstates, so both values can be sharp at once.

    Circle one:   True   False

  2. What is the commutator of two operators A and B?

    • A plus B
    • AB minus BA
    • A divided by B

  3. How do you test whether a new pair of observables can be known together?

    • Compute their commutator; zero means compatible
    • Measure both at once and see
    • Check whether their names sound similar

  4. What happens when you pin down the momentum of a particle?

    • Its position spreads out
    • Its position sharpens too
    • Its energy becomes zero

  5. What sets the uncertainty bound for a general pair of observables?

    • The sum of their eigenvalues
    • The number of basis states
    • Half the size of the average commutator

  6. How does the position momentum relation connect to the general uncertainty relation?

    • It contradicts the general relation
    • It is one case of the general rule
    • It applies only to light

  7. Two observables with a nonzero commutator can both be perfectly sharp at once.

    Circle one:   True   False

  8. Two spin components have a nonzero commutator. What follows?

    • Both can be sharp together on special states
    • Their commutator must be recomputed until it vanishes
    • Their spreads obey a tradeoff with no shared eigenbasis
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Answer key

For grown-ups. Fold this page away before handing over the rest.

When two quantities refuse to be sharp together W1-mt_TKoUwu74K9-s1

  1. True · A zero commutator means a shared eigenbasis exists.
  2. AB minus BA · The commutator measures the failure of the two to commute.
  3. Compute their commutator; zero means compatible · The commutator is the compatibility test for any pair.
  4. Its position spreads out · Definite wavelength means an extended wave, spread over position.
  5. Half the size of the average commutator · The commutator average floors the product of the two spreads.
  6. It is one case of the general rule · It comes from the position momentum commutator inside the general bound.
  7. False · Nonzero means no shared eigenstate, so joint sharpness is impossible.
  8. Their spreads obey a tradeoff with no shared eigenbasis · Nonzero means incompatible: the spread product has a positive floor.
Worksheet · LightMySky