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

In the case of a spatially flat Friedmann–Lema^{\i}tre–Robertson–Walker universe in $f\left(R\right)$-gravity we write the Wheeler-DeWitt equation of quantum cosmology. The equation depends on the functional form of $f\left( R\right) $\mathrm{.}Weselecttoworkwithfourspecificfunctionsof$f\left( R\right) $, 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 determine Lie-B"{a}cklund transformations. In addition we show how the classical limit is recovered by the similarity solutions of the Wheeler-DeWitt equation.