In Section 5.1 we found that for noninteracting particles the wave function can be expressed as a product of single-particle states (Equation 5.9)—or, for identical particles, as a symmetrized/antisymmetrized linear combination of such states (Equations 5.20 and 5.21). For interacting particles this is no longer the case. A famous example is the Laughlin wave function, which is an approximation to the ground state of N electrons confined to two dimensions in a perpendicular magnetic field of strength B (the setting for the fractional quantum Hall effect). The Laughlin wave function is
ψ(z1,z2,…,zN)=Aj<k∏N(zj−zk)qexp[−21k∑N∣zk∣2]
where q is a positive odd integer and
zj≡2ℏceB(xj+iyj).
(Spin is not at issue here; in the ground state all the electrons have spin down with respect to the direction of B, and that is a trivially symmetric configuration.)
(a) Show that ψ has the proper antisymmetry for fermions.
(b) For q=1, ψ describes noninteracting particles (by which I mean that it can be written as a single Slater determinant—see Problem 5.8). This is true for any N, but check it explicitly for N=3. What single particle states are occupied in this case?
(c) For values of q greater than 1, ψ cannot be written as a single Slater determinant, and describes interacting particles (in practice, Coulomb repulsion of the electrons). It can, however, be written as a sum of Slater determinants. Show that, for q=3 and N=2, ψ can be written as a sum of two Slater determinants.
Comment: In the noninteracting case (b) we can describe the wave function as "three particles occupying the three single-particle states ψa, ψb and ψc," but in the interacting case (c), no corresponding statement can be made; in that case, the different Slater determinants that make up ψ correspond to occupation of different sets of single-particle states.