Poisson Brackets and Constants of the Motion · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

One bracket for motion and conservation

Science · Forces & Motion · ages 22-23
Name ______________________   Date ____________

  1. How does a position coordinate change with time?

    • As {H, H}
    • As the Poisson bracket of q with q
    • As {q, H}

  2. The bracket {A, H} vanishes. What does that say about A?

    • A is conserved
    • A is zero
    • A is the Hamiltonian

  3. The quantum counterpart of a Poisson bracket is a commutator.

    Circle one:   True   False

  4. You suspect energy-like quantity E is conserved. What do you compute?

    • {E, E} and check for zero
    • The numerical value of E once
    • {E, H} and check for zero

  5. A quantity has a nonzero bracket with H. What happens to it?

    • It stays exactly constant
    • It drifts with time
    • It vanishes at once

  6. What does {p, H} describe?

    • The rate of change of momentum
    • A constant of the motion always
    • The numerical value of energy

  7. Two quantities have a vanishing mutual bracket. What follows after quantization?

    • They become compatible observables known together
    • They can never be measured at all
    • Their commutator must diverge

  8. A student says a vanishing bracket means the quantity is zero. What is wrong?

    • Vanishing brackets are impossible
    • Zero bracket means constant, not zero-valued
    • Only H may appear in brackets
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Answer key

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

One bracket for motion and conservation W1-mt_UFzJyZRdOI-s1

  1. As {q, H} · The canonical equations are brackets with H.
  2. A is conserved · Zero rate of change means constant.
  3. True · Brackets become commutators over i h-bar.
  4. {E, H} and check for zero · Conservation is decided by bracketing with H.
  5. It drifts with time · Nonzero rate means ongoing change.
  6. The rate of change of momentum · It is the canonical equation for momentum.
  7. They become compatible observables known together · Vanishing brackets quantize to commuting operators.
  8. Zero bracket means constant, not zero-valued · The bracket reports change, not size.
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