Quantum Fluctuations in the Frustrated Antiferromagnet Sr 2 Cu 3 O 4 Cl 2

${\mathrm{Sr}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{4}{\mathrm{Cl}}_{2}$ is an antiferromagnet consisting of weakly coupled CuO planes which comprise two weakly interacting antiferromagnetic subsystems I and II which order at respective temperatures ${T}_{\mathrm{I}}\ensuremath{\approx}390$ K and ${T}_{\mathrm{II}}\ensuremath{\approx}40$ K. Except asymptotically near the ordering temperature, these systems are good representations of the two-dimensional quantum spin-1/2 Heisenberg model. For $Tl{T}_{\mathrm{II}}$ there are four low-energy modes at zero wave vector, three of whose energies are dominated by quantum fluctuations. For ${T}_{\mathrm{II}}lTl{T}_{\mathrm{I}}$ there are two low-energy modes. The mode with lower energy is dominated by quantum fluctuations. Our calculations of the energies of these modes (including dispersion for wave vectors perpendicular to the CuO planes) agree extremely well with the experimental results of inelastic neutron scattering (in the accompanying paper) and for modes in the sub-meV range observed by electron spin resonance. The parameters needed to describe quantum fluctuations are either calculated here or are taken from the literature. These results show that we have a reasonable qualitative understanding of the band structure of the lamellar cuprates needed to calculate the anisotropic exchange constants used here.