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The high energy spin excitation in the high T$_{c}$ cuprates is studied inthe single mode approximation for the $t-t'-J$ model. An exact form for themode dispersion is derived. When the Gutzwiller projected BCS state is used asthe variational ground state, a spin-wave-like dispersion of about 2.2$J$ isuncovered along the $\Gamma=(0,0)$ to $\mathrm{M}=(\pi,0)$ line. Both the modeenergy and the integrated intensity of the spin fluctuation spectrum are foundto be almost doping independent in large doping range, which agrees very wellwith the observations of recent RIXS measurements. Together with previousstudies on the quasiparticle properties of the Gutzwiiler projected BCS state,our results indicate that such a Fermionic RVB theory can provide a consistentdescription of both the itinerant and the local aspect of electronicexcitations in the high T$_{c}$ cuprates.

On the high energy spin excitations in the high temperature superconductors: an RVB theory