Commutators, Compatible Observables and the General Uncertainty Relation · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.
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When two quantities refuse to be sharp together

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

  1. Two commuting finite-dimensional Hermitian observables have a common orthonormal eigenbasis.

    Circle one:   True   False

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

    • AB plus BA
    • AB minus BA
    • A² minus B²

  3. For Hermitian operators A and B and a nonzero vector v, ABv=3iv and BAv=-3iv, where i²=-1. What is [A,B]v?

    • 6iv
    • -6iv
    • 0

  4. For a free particle, how is an ideal momentum eigenstate represented in position?

    • An extended plane wave
    • A localized narrow packet
    • A stationary point particle

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

    • Half the sum of the mean values
    • Half the product of the variances
    • Half the modulus of <[A,B]>

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

    • It contradicts the operator bound
    • It is a case of the operator bound
    • It applies only to photon states

  7. A nonzero commutator rules out every shared eigenstate of two finite-dimensional Hermitian observables.

    Circle one:   True   False

  8. Spin obeys [Sx,Sy]=i hbar Sz. In a state with <Sz>=hbar/4, what lower bound does the relation give for ΔSx ΔSy?

    • hbar²/2
    • hbar²/4
    • hbar²/8
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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 · Commuting Hermitian matrices can be diagonalized in the same orthonormal basis, including within degenerate eigenspaces.
  2. AB minus BA · The commutator measures the failure of the two to commute.
  3. 6iv · [A,B]v=ABv-BAv=3iv-(-3iv)=6iv. The stated actions are compatible with Hermitian A and B.
  4. An extended plane wave · A definite free-particle momentum corresponds to a plane wave extending through space. This ideal state is not an ordinary normalizable packet.
  5. Half the modulus of <[A,B]> · The Robertson bound is ΔA ΔB>=|<[A,B]>|/2. The expectation is evaluated in the state being studied.
  6. It is a case of the operator bound · It comes from the position momentum commutator inside the general bound.
  7. False · It rules out a full common eigenbasis, not every possible shared eigenstate.
  8. hbar²/8 · Half the modulus of i hbar <Sz> is hbar²/8.
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