10 The ground state wave function is given by [5] The associated probability density for the particle positions is then given by It is also easy to show also that the wave function above is an eigenfunction of the following similar looking Hamiltonian ψ$_{0}$ ( x ) = exp ( − 1 2 ∑ i | ⃗ x$_{i}$ | 2 ) = exp( − ∑ i ⃗ z$_{i}$ ⃗ ¯ z$_{i}$ ) (3) p = | ψ$_{0}$ | $^{2}$µ2 = exp ( − ∑ i | ⃗ x$_{i}$ | 2 + ∑ i<j log | ⃗ x$_{i}$ − ⃗ x$_{j}$ | 2 ) (4) H = ∑ i − 1 µ 2 ∇ $_{z,i}$µ $^{2}$∇ ¯ z,i + | ⃗ z$_{i}$ | 2 (5) with the derivatives being holomorphic and anti-holomorphic in the prescribed order. To prove this, one has to use the fact that µ 2 is a scaling function under rescalings of the z variables that leave ¯ z fixed (we are treating them as independent variables, rather than complex conjugate variables for that). The complete argument, which is beyond the scope of the present paper, is very similar to the one appearing in [5, 23]. The next set of simplest wave functions in this quenched dynamics are those associated to BPS states. They are described by holomorphic wave functions of the z variables multiplying the ground state wave function and these are symmetrized over the eigenvalues (this is is a discrete gauge symmetry of permutation of eigenvalues). These are not exact wave functions of H , but they are argued to be close to exact wave functions of the system. However, it turns out that all the homogeneous holomorphic wave functions are eigenfunctions of ˜ H , with energy E − E$_{0}$ equal to the degree of the polynomial of the z . As such, the Hamiltonian ˜ H seems more appropriate to capture all BPS states exactly than H , since the holomorphic wave functions do not seem to be eigenfunctions of H , except under certain approximations in the large N limit [23]. In general, the set of BPS solutions of the classical equations of motion on S 3 are in one to one correspondence with the moduli space of vacua of superconformal field theories. The quantization of these configurations is a particular holomorphic quantization of the moduli space of vacua [22], which requires a measure. The Hamiltonian ˜ H would give the exact answer for the chiral ring state energies and has the correct measure of the ground state wave function in the classical approximation to these solutions, which is the same quenched approximation we have been studying. This is the main reason to introduce it above. However, nothing else that we do depends on that fact.