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In this work, we consider the muonic LiH as a new stage of the $\mu$catalyzed fusion. We calculate the bound states of the muonic LiH and LiH$^+$and their wave functions based on the three and four body approximation. Inorder to solve the Schr$\ddot{\rm o}$dinger equation, we apply the Gaussianexpansion method which gives us both the eigen energies and the wave functions.The existences of the bound states in muonic LiH and LiH$^+$ are confirmed. Wefind the dynamical behaviors between the $^7$Li and proton in muonic LiH andLiH$^+$ are similar as the ones in dt$\mu_2$ and dt$\mu$, respectively. The$\alpha$ sticking probability in the muonic LiH$^+$ after the nuclear fusion isstudied. The calculated $\alpha$ sticking probability of being captured to theground state of He$\mu$ is 5 times smaller than the one in dt$\mu$.

Investigating the muon catalyzed fusion in muonic lithium hydride