First-Order Perturbation Theory · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Small pushes, small shifts

Science · Quantum & Modern Physics · ages 21-22
Name ______________________   Date ____________

  1. What gives the first order energy shift?

    • The slope at the well center
    • The largest value of the slope
    • The average of the slope in the known state

  2. A small slope is added to a solved square well. What is the correct split?

    • Solved well plus the small slope
    • Slope alone, ignoring the well
    • A brand new unsolved system

  3. A perturbation must be tiny beside the gaps between known levels.

    Circle one:   True   False

  4. A small extra lift raises the whole well floor by the same amount. What is the shift?

    • Zero
    • Equal to the lift
    • Twice the lift

  5. Which of these is a valid perturbation?

    • A wall that doubles the well depth
    • A tiny tilt added to a solved well
    • A second well with unknown states

  6. A push is half as big as the gap to the next level. Should you trust the simple average?

    • No, it is not tiny beside the gap
    • Yes, half sized pushes are always safe
    • Yes, size never matters here

  7. Two levels sit almost on top of each other and a small push arrives. What should you do?

    • Use the simple average anyway
    • Ignore the push completely
    • Pause, because the close levels will mix and need a wider recipe

  8. A student computes the shift with a guessed new wave. What is the mistake?

    • The average must also include the solved part
    • The average must use the known unperturbed state
    • Averages are never used in this recipe
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Answer key

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

Small pushes, small shifts W1-mt_UsbiiQFOAt-s1

  1. The average of the slope in the known state · The shift is an average taken in the unperturbed state.
  2. Solved well plus the small slope · The recipe starts from a solved system plus its small extra.
  3. True · Small means small against the level spacing, not small in isolation.
  4. Equal to the lift · The average of a constant lift is the lift itself.
  5. A tiny tilt added to a solved well · Only a small extra on a solved system fits the recipe.
  6. No, it is not tiny beside the gap · Half the gap is not tiny beside the gap, so the recipe is untrustworthy.
  7. Pause, because the close levels will mix and need a wider recipe · Nearly paired levels mix, so the simple average fails.
  8. The average must use the known unperturbed state · First order means averaging in the old known state only.
Worksheet · LightMySky