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At its most fundamental level, quantum mechanics (QM) is based on a mathematical abstraction, referred to as the wave equation.  The wave equation mathematically defines a superposition of harmonics possessing varying frequency and phase.  Phase coherence within this superposition of harmonics results in an energy envelope defining elementary particles and photons.

The wave packet equation is shown to the right in discrete summation form, where ci is a complex number representing amplitude:
The above linear superposition may be written in simpler form, as follows:
A quantum state vector formed by a set of quantities ψ is expressed using the ket-vector, which is a column vector comprised of complex values, i.e.:

The adjoint to the ket is the bra-vector, which is a row vector comprised of complex conjugate coefficients:

 
   

The quantum state vector that defines the learning process across a set of patterns indexed by i, may be expressed in Dirac notation as follows:

   

In quantum mechanics, the amplitude coefficient cn (or in the case of HNeT - the desired response for pattern n) is expressed by evaluating the following product :