Slater Determinant: A convenient and compact way of expressing multiparticle states of antisymmetric character for many fermions is the Slater determinant:
Ψn1(x1)ms1 | Ψn2(x1)ms2 | Ψn3(x1)ms3 | ΨnN(x1)msN |
Ψn1(x2)ms1 | Ψn2(x2)ms2 | Ψn3(x2)ms3 | ΨnN(x2)msN |
Ψn1(x3)ms1 | Ψn2(x3)ms2 | Ψn3(x3)ms3 | ΨnN(x3)msN |
… | … | … | … |
Ψn1(xN)ms1 | Ψn2(xN)ms2 | Ψn3(xN)ms3 | ΨnN(xN)msN |
It is based on the fact that for N fermions there must be N different individual-particle states, or sets of quantum numbers. The ith state has spatial quantum numbers (which might be ni; ℓi and m ℓi) represented simply by ni*and spin quantum number msi. Were it occupied by the Ψn1(xj)msi A column corresponds to a given state and a row to a given particle. For instance, the first column corresponds to individual-particle state Ψn1(xj)ms1 where j progresses (through the rows) from particle I to particle N. The first row corresponds to particle 1, which successively occupies all individual-particle states (progressing through the columns).
(a) What property of determinants ensures that the multiparticle state is 0 if any two individual-particle states are identical?
(b) What property of determinants ensures that switching the labels on any two particles switches the sign of the multiparticle state?
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