The important question at this moment is whether quantum gradient can define spatial frequency? But first the meanings of these words must be clarified. Similar to a unit of energy quantum gradient is simply a unit of a gradient. However, gradient has two definitions in both math and physics. The first is its equivalence to the derivative as defined in the differential calculus. The second as found in vector analysis is that of a vector obtained by applying the differential Cartesian operator del: Ñ = i/x+j/y+k/z to a scalar function of position: f(r). The first implies tangency and the existence of a slope at a point of a space-time coordinate system while the second specifies a given direction of the point. The first is absolutely dependent on the given coordinate system while the second is independent of any system. The first can define spatial frequency as the quantized states of slope between positive rational infinity and negative rational infinity: ±∞. The second defines spatial frequency as the constant rate of change of direction but which is the proper definition requires further detailed investigations.

Consequently, if a quantum gradient has a proper definition then this can be used to define a quantum divergence and also a quantum curl operator resulting into a quantum Gauss’s theorem and a quantum Stokes’ theorem. Furthermore, quantum these and quantum that would certainly redefine classical Maxwell’s equations of electromagnetic fields into tensors of rank zero which are basically scalar quantities signifying magnitudes alone. These assert that the spatial frequency (q) is simply fractal multiples of the irrational transcendental number p: q= ±pd where d is a rational number. In this definition, q is topologically equivalent to the circumference of an arbitrary circle. Moreover, by rational normalization (dividing by p) gives all possible values for q= ±d . This is equivalent to the intrinsic angular momentum (spin) of quantum mechanics thus establishing a principle of equivalence between spin and spatial frequency.