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. When a ladder operator gives a nonzero new state, by how much does it change m?

    • Exactly j
    • Exactly one half always
    • Exactly one

  2. For Hermitian angular-momentum operators, which relations start the algebraic ladder derivation?

    • A separated orbital differential equation
    • The angular-momentum commutation relations
    • An observed atomic emission-line spectrum

  3. The ladder algebra permits half-integer j, which can describe intrinsic spin but not ordinary orbital spherical harmonics.

    Circle one:   True   False

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

    • 3/2, 1, 1/2, 0, -1/2
    • 2, 1, 0, -1, -2
    • 3/2, 1/2, -1/2, -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 be excluded
    • Spherical harmonics would disappear
    • Commutators would stop closing

  7. For Hermitian angular-momentum operators in a finite normalized ladder, how do you obtain the allowed m values algebraically?

    • Solve an orbital differential equation for every possible spin
    • Use ladder norms to find endpoints, then take unit steps
    • Infer every spin state from an observed atomic line spectrum

  8. A student says ordinary orbital spherical harmonics disprove half-integer intrinsic spin. Which distinction resolves the claim?

    • 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 · A nonzero action of J+ raises m by one; J- lowers it by one.
  2. The angular-momentum commutation relations · The derivation uses the component commutators and Hermiticity, rather than solving an orbital differential equation.
  3. True · A finite ladder needs integer 2j, not necessarily integer j. Orbital angular momentum has integer l.
  4. 3/2, 1/2, -1/2, -3/2 · The m values descend from 3/2 to minus 3/2 in unit steps.
  5. 5 · The count is 2j plus 1, which is 5.
  6. Spin-half would be excluded · Electron spin requires a half-integer ladder. Intrinsic spin in general may be integer or half-integer.
  7. Use ladder norms to find endpoints, then take unit steps · Nonnegative ladder norms and zero endpoint norms give m=j and m=-j. Unit steps enumerate the states between them.
  8. They cover orbital motion, not spin · Orbital spherical harmonics realize integer angular-momentum ladders. They are not intrinsic spin wave functions.
Worksheet · LightMySky