Exact solution in the Heisenberg picture and annihilation–creation operators

The annihilation–creation operators of the harmonic oscillator, the basic and most important tools in quantum physics, are generalised to most solvable quantum mechanical systems of single degree of freedom including the so-called ‘discrete’ quantum mechanics. They admit exact Heisenberg operator solution. We present unified definition of the annihilation–creation operators (a(±)) as the positive/negative frequency parts of the exact Heisenberg operator solution.