Quantum size effects in Friedel oscillations inside films

► An analytical formula of Friedel oscillations is found for metallic films of finite thickness. ► The analytical formula agrees well with our DFT numeric calculations. ► Our results show that there exist quantum size effects in Friedel oscillations. ► We also find such quantum size effect in Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction for system of finite size.