Basic Many-Body Quantum Mechanics

Abstract

The solutions of eigenvalue equations like the time-independent one-electron Schrödinger equation hw_i = ∈_iw_i form a a complete set of spin-orbitals $\left\{ {w_i \equiv \varphi _i (x)\chi _i } \right\}$, where ϕ_i(x) are normalized space orbitals and χ_i =↑ or ↓. The set can be taken orthogonal and ordered in ascending energy or in any other arbitrary way. Any one-electron state can be expanded as a linear combination of the w_i. Moreover, we can think of a state for N electrons obtained as follows. Choose in any way N spin-orbitals out of the set {w_i}, keep them in the original order but call them v₁, v₂... v_N; now let |v₁,v₂...v_N| be the state with one electron in each. Imagine labeling¹ the indistinguishable electrons with numbers 1, 2, ...N. In this many-body state one has an amplitude $\Psi (1,2,...N) \equiv \Psi ((x_1 ,\chi _1 ),(x_2 ,\chi _2 ),...(x_{\rm N} ,\chi _{\rm N} ))$ of having electron i in the one-particle state (x_i, χ_i). How to calculate Ψ? A product like $v_1 v_2 ...v_N = \prod\nolimits_k^N {v_k }$ is in conflict with the Pauli principle because it fails to be antisymmetric in the exchange of two particles. However, the remedy is easy, because anti-symmetrized products are a basis for the antisymmetric states. To this end, let $\mathcal{P} {\text{:}}\left\{ {{\text{1,2,}}...N} \right\} \to \left\{ {\mathcal{P}_1 {\text{,}}\mathcal{P}_2 {\text{,}}...\mathcal{P}_N } \right\}$ be one of the N! permutations of N objects. If N = 3, the set of 6 permutations comprises the rotations {(1, 2, 3), (2, 3, 1), (3, 1, 2)} and {(2, 1, 3), (3, 2, 1), (1, 3, 2)}.

The electrons are identical, but this does not prevent us from labeling them; rather it imposes that the wave function changes sign for each exchange of labels.