Adding Angular Momenta and the Coupled Basis · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.
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Adding two spins into a total

Science · Quantum & Modern Physics · ages 23-24
Name ______________________   Date ____________

  1. An angular momentum j1=1 has three projection states. A spin j2=1/2 has two. How many basis states are in their product space?

    • 5
    • 3
    • 6

  2. For angular-momentum quantum numbers j1 and j2, which total J values are allowed?

    • Only J=j1+j2, with no other allowed totals
    • |j1-j2| through j1+j2 in steps of one
    • Only J=|j1-j2|, with no other allowed totals

  3. The coupled basis adds brand-new states beyond the product ones.

    Circle one:   True   False

  4. Two full bases are meant to describe the same state space. One count is 4 and the other is 6. What follows?

    • At least one count is incorrect
    • A basis change can add extra states
    • Different bases must have different sizes

  5. For j1=1 and j2=1/2, the allowed totals are J=1/2 and J=3/2. How many coupled basis states are there altogether?

    Answer: ______________

  6. For two spins 1/2, |1,0>=(ud+du)/sqrt(2) and |0,0>=(ud-du)/sqrt(2). Which product state equals (|1,0>+|0,0>)/sqrt(2)?

    • du
    • ud
    • (uu+dd)/sqrt(2)

  7. A Hamiltonian assigns the same energy E to all four states of two spins 1/2. Must rewriting them as a triplet and a singlet split that energy?

    • No; changing basis alone cannot split energy
    • Yes; a three-state sector must have a higher energy
    • Yes; a one-state sector must have a higher energy

  8. For j1=2 and j2=1/2, a proposed list is J=3/2,5/2,7/2. Which correction does the coupling rule require?

    • Remove 3/2 because it lies below the lower endpoint
    • Remove 5/2 because half-integer totals are excluded
    • Remove 7/2 because it exceeds the upper endpoint
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Answer key

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

Adding two spins into a total W1-mt_eSBNPKm9C4-s1

  1. 6 · Every first-system state pairs with both second-system states, so the dimension is 3×2=6.
  2. |j1-j2| through j1+j2 in steps of one · The coupling rule starts at the absolute difference and ends at the sum, with unit steps.
  3. False · Both bases span the same space. Their basis vectors can differ, but a basis change adds no new physical states.
  4. At least one count is incorrect · Two bases of the same state space must have matching counts.
  5. 6 · J=1/2 contributes two states and J=3/2 contributes four, so the total is six.
  6. ud · Substitution gives [(ud+du)+(ud-du)]/2=ud. The du terms cancel.
  7. No; changing basis alone cannot split energy · The Hamiltonian is E times the identity on this space. A basis change leaves all four energies equal.
  8. Remove 7/2 because it exceeds the upper endpoint · The endpoints are |2-1/2|=3/2 and 2+1/2=5/2. Thus J=7/2 is outside the allowed range.
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