Angular Momentum from Its Commutation Relations · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Ladders built from commutators

Science · Quantum & Modern Physics · ages 22-24
Name ______________________   Date ____________

  1. By how much does one ladder step change m?

    • Exactly j
    • Exactly one half always
    • Exactly one

  2. What feeds the whole derivation of the spectrum?

    • A solved differential equation
    • The commutation relations
    • A measured line spectrum

  3. Half-integer j values belong to spin and fall out of the same algebra.

    Circle one:   True   False

  4. For j equal to 3/2, which m values appear?

    • 3/2, 1, 1/2, 0
    • 2, 1, 0, minus 1
    • 3/2, 1/2, minus 1/2, minus 3/2

  5. A system has j equal to 2. Type how many m states its ladder holds.

    Answer: ______________

  6. Why would integer-only values be a problem for physics?

    • Spin-half would have no home
    • Spherical harmonics would disappear
    • Commutators would stop closing

  7. You are given only commutators and asked for allowed values. What is the first move?

    • Separate variables in spherical coordinates
    • Build ladders and climb from the top
    • Measure the spectrum in a magnetic field

  8. A student says spherical harmonics disprove half-integer j. What is wrong?

    • They cover orbital motion, not spin
    • Harmonics forbid all integers too
    • Ladders cannot produce any harmonics
LightMySky · lightmysky.comW1-mt_u0y20XJh-v-s1

Answer key

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

Ladders built from commutators W1-mt_u0y20XJh-v-s1

  1. Exactly one · Each application of J-plus or J-minus shifts by one.
  2. The commutation relations · The algebra starts from the brackets alone.
  3. True · The ladder logic allows them, and spin lives there.
  4. 3/2, 1/2, minus 1/2, minus 3/2 · Unit steps from 3/2 down to minus 3/2.
  5. 5 · The count is 2j plus 1, which is 5.
  6. Spin-half would have no home · Spin needs the half-integer ladders.
  7. Build ladders and climb from the top · Ladders turn brackets directly into the spectrum.
  8. They cover orbital motion, not spin · The analytic route describes orbital states; spin is extra.
Worksheet · LightMySky