Rarely, the authors of quantum mechanics books have discussed Dirac - Jordan transformation theory in abstract and pure form . Mostly, in the topics of mathematical tools, quantum mechanics assimilates in a manner with matrices theory that and its operability and ability differentiates as a pure theory is difficult. The subject of this study is that to show the ability of this theory in different discussions and particular differences of its solution methods with other theories. Principally, applied mathematics in Dirac - Jordan transformation theory is particular and differs with the mathematics present in theorems and relations of wave and matrices theories . Encountering with wave and matrices theories maybe implies, at least, applied mathematics in these theories gives certain relation between them, but there is not the case of Dirac - Jordan transformation theory. Quantum state of a particle in a given time, in Schrödinger's wave theory, was defined by wave function . Probabilistic interpretation of this wave function requires that its square could be integrated, and this leads to study Hilbert, H space.