Principles of Quantum Mechanics · Problem 12.4.1

Rotational Invariance and Angular Momentum — R. Shankar

Physics — Quantum Mechanics · angular-momentum-three-dimensions

Problem

(1) Verify that Eqs. (12.4.9) and Eq. (12.4.8) are equivalent, given the definition of εijk\varepsilon_{ijk}. (2) Let U1U_1, U2U_2, and U3U_3 be three energy eigenfunctions of a single particle in some potential. Construct the wave function ψA(x1,x2,x3)\psi_A(x_1, x_2, x_3) of three fermions in this potential, one of which is in U1U_1, one in U2U_2, and one in U3U_3, using the εijk\varepsilon_{ijk} tensor.
Intuitive explanation

The Levi-Civita symbol εijk\varepsilon_{ijk} is the mathematical tool that encodes antisymmetry—the fundamental requirement that fermion wave functions must change sign under particle exchange. In part (1), you're checking that two different notations for the same physics (likely a determinant form versus an explicit sum) are truly identical. In part (2), you'll use εijk\varepsilon_{ijk} to build a Slater determinant, where the antisymmetric combination automatically ensures no two fermions occupy the same state and respects the Pauli exclusion principle.

Intuitive explanation

The Levi-Civita symbol εijk\varepsilon_{ijk} is the mathematical tool that encodes antisymmetry—the fundamental requirement that fermion wave functions must change sign under particle exchange. In part (1), you're checking that two different notations for the same physics (likely a determinant form versus an explicit sum) are truly identical. In part (2), you'll use εijk\varepsilon_{ijk} to build a Slater determinant, where the antisymmetric combination automatically ensures no two fermions occupy the same state and respects the Pauli exclusion principle.

Related theory

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