Dimensionless quantum mechanical state vector

Starting from the superposition principle, Dirac introduced complex vector space representation for quantum mechanics, leaving behind in a hash some flaws in the theory, in particular dimension discrepancy concerning the basis vector sets {|<i<x</i<<} and {|<i<p</i<<} for position operator and momentum operator, respectively. A remedy for this discrepancy was recently proposed by assigning dimension to the state vectors |<i<x</i<< and |<i<p</i<<, which is unacceptable and may bring with new confronts. Abiding by the principle of dimensionless representation for state vectors, and thus the relevant transformations, we formulate a dimensionless representation for the state vectors for position and momentum, i.e., |<i<q</i<<and |<i<p</i<<, by using the annihilation and creation operators <i<a</i<, <i<a</i<<sup<+</sup<, thus eliminate the problems caused by dimension discrepancy in quantum mechanics literature.