Identical Particles and the Symmetry of a Many-Body State · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.
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When swapping changes nothing

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

  1. For distinct orthonormal states a and b, which operation produces a symmetric two-particle state before normalization?

    • Take the difference of swapped products
    • Take a single product with no partner
    • Take the sum of swapped products

  2. You swap two identical particles. What stays the same?

    • Every prediction stays the same
    • The wave always flips sign
    • The particles gain tracking labels

  3. What does swapping do to an antisymmetric state?

    • Nothing at all
    • It flips the sign
    • It doubles the amplitude

  4. How does antisymmetry forbid two fermions from occupying the same complete one-particle state?

    • Same-state addition produces a forbidden factor of two
    • Exchange changes only one particle's occupied state
    • Same-state subtraction cancels to zero

  5. What occupation of one photon mode is allowed by bosonic exchange symmetry?

    • Exactly one photon; exchange forbids a second occupant
    • Multiple photons may occupy that same mode
    • At most two photons; exchange imposes a two-occupant cap

  6. Which pair of electrons can occupy the same spatial orbital without violating exclusion?

    • Two electrons with opposite spins
    • Two electrons with identical orbital and spin states
    • Two electrons whose full one-particle states are identical

  7. Two identical electrons can occupy the same complete one-particle state, including the same spin state.

    Circle one:   True   False

  8. Use fixed normalized orthogonal spatial orbitals and neglect spin-dependent terms. E_s=E0+D+K and E_t=E0+D-K. If E0=-40 eV, D=8 eV and K=0.5 eV, what is E_s-E_t?

    • 0.5 eV
    • 8 eV
    • 1 eV
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Answer key

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

When swapping changes nothing W1-mt_hns-eEGeHU-s1

  1. Take the sum of swapped products · Adding the exchanged products gives a sum that stays unchanged under the swap.
  2. Every prediction stays the same · Indistinguishability means no measurement can tell the particles apart.
  3. It flips the sign · For an antisymmetric state, exchanging the two particle arguments multiplies the full state by -1.
  4. Same-state subtraction cancels to zero · The exchanged products become identical. Their difference is zero, and a zero vector cannot be normalized into a physical state.
  5. Multiple photons may occupy that same mode · Bosonic symmetry permits multiple photons in one mode. It imposes neither fermionic single-state exclusion nor a two-photon cap.
  6. Two electrons with opposite spins · Opposite spins give different complete one-particle states even when the spatial orbital is shared.
  7. False · The full fermionic state must be antisymmetric. Repeating one complete state makes the antisymmetric combination vanish.
  8. 1 eV · E_s=-31.5 eV and E_t=-32.5 eV. Their difference is 1 eV, equal to 2K.
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