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We calculate the chameleon field profile, confined between two parallelplates, filled with air at pressure $P = 10^{-4}\,{\rm mbar}$ and roomtemperature and separated by the distance $L$, in the chameleon field theorywith Ratra--Peebles self--interaction potential with index $n = 1$. We give theexact analytical solution in terms of Jacobian elliptic functions, depending onthe mass density of the ambient matter. The obtained analytical solution can beused in qBounce experiments, measuring transition frequencies between quantumgravitational states of ultracold neutrons and also for the calculation of thechameleon field induced Casimir force for the CANNEX experiment. We show thatthe chameleon--matter interactions with coupling constants $\beta \le 10^4$ canbe probed by qBounce experiments with sensitivities $\Delta E \le10^{-18}\,{\rm eV}$. At $L = 30.1\,{\rm \mu m}$ we reproduce the result $\beta< 5.8\times 10^8$, obtained by Jenke {\it et al.} Phys. Rev. Lett. {\bf 112},151105 (2014)) at sensitivity $\Delta E \sim 10^{-14}\,{\rm eV}$. In thevicinity of one of the plates our solution coincides with the solution,obtained by Brax and Pignol (Phys. Rev. Lett. {\bf 107}, 111301 (2011)) (seealso Ivanov {\it et al.} Phys. Rev. D {\bf 87}, 105013 (2013)) above a plate atzero density of the ambient matter.

Exact Solution for Chameleon Field, Self-Coupled Through the Ratra-Peebles Potential with n = 1 and Confined Between Two Parallel Plates