Introduction to Quantum Mechanics · Problem 5.9
Identical Particles — David J. Griffiths
Physics — Quantum Mechanics · identical_particles_with_spin
Problem
In Example 5.1 and Problem 5.5(b) we ignored spin (or, if you prefer, we assumed the particles are in the same spin state). (a) Do it now for particles of spin 1/2. Construct the four lowest-energy configurations, and specify their energies and degeneracies. Suggestion: Use the notation , where is defined in Example 5.1 and in Section 4.4.3. (b) Do the same for spin 1. Hint: First work out the spin-1 analogs to the spin-1/2 singlet and triplet configurations, using the Clebsch–Gordan coefficients; note which of them are symmetric and which antisymmetric.Intuitive explanation
When you include spin, you're adding a new degree of freedom that multiplies the number of available quantum states—each spatial configuration can now pair with multiple spin arrangements. The key insight is that the total wavefunction must be antisymmetric under particle exchange (for fermions like spin-1/2, or for any identical particles under the right statistics), which means symmetric spatial states must pair with antisymmetric spin states and vice versa. For spin-1/2, this gives you singlet (antisymmetric) and triplet (symmetric) spin combinations; for spin-1, you'll use Clebsch–Gordan coefficients to build symmetric and antisymmetric spin states, then match them with the appropriate spatial wavefunctions to respect overall antisymmetry.
Intuitive explanation
When you include spin, you're adding a new degree of freedom that multiplies the number of available quantum states—each spatial configuration can now pair with multiple spin arrangements. The key insight is that the total wavefunction must be antisymmetric under particle exchange (for fermions like spin-1/2, or for any identical particles under the right statistics), which means symmetric spatial states must pair with antisymmetric spin states and vice versa. For spin-1/2, this gives you singlet (antisymmetric) and triplet (symmetric) spin combinations; for spin-1, you'll use Clebsch–Gordan coefficients to build symmetric and antisymmetric spin states, then match them with the appropriate spatial wavefunctions to respect overall antisymmetry.