The quantum relativistic oscillator may be described by the Klein-Gordon equation: (-d/dx d/dx + mo2) W = (E- .5kx2)2 W ((1)) but due to difficulties with the x4 term for large x leading to a lack of bound states, ((1)) has recently been replaced in the literature (1) by: d/dx d/dx W - (mwx2) W = (E2-mo2)2 W ((2)) ((2)) may be solved exactly to yield Hermite polynomial solutions multiplied by exp(-ax2) just as in the nonrelativistic case, although “a” for the relativistic and nonrelativistic cases differ. In this note, we wish to analyse ((2)) as it does not yield a quantum averaged conservation of energy at each point x as the Schrodinger equation (and the Klein Gordon) do. In a previous note (2), we suggested that ((1)) may be used to solve the relativistic oscillator problem between the classical turning points, but could be replaced at (or near) the turning points and out to +/- infinite with the Schrodinger equation as this region is nonrelativistic (the quantum averaged kinetic energy drops to 0).