Similarity solutions for the Wheeler–DeWitt equation in $$f\left( R\right) $$-cosmology

In the case of a spatially flat Friedmann–Lemaître–Robertson–Walker Universe in $$f\left( R\right) $$fR-gravity we write the Wheeler–DeWitt equation of quantum cosmology. The equation depends upon the functional form of $$f\left( R\right) $$fR. We choose to work with four specific functions of $$f\left( R\right) $$fR in which the field equations for the classical models are integrable and solvable through quadratures. For these models we determine similarity solutions for the Wheeler–DeWitt equation by determining Lie–Bäcklund transformations. In addition we show how the classical limit is recovered by the similarity solutions of the Wheeler–DeWitt equation.