Quantum mechanics provides a detailed understanding of the molecular geometry and bond angles of methane CH4 through the application of molecular orbital theory and the Schrödinger equation. The Schrödinger equation is a fundamental equation in quantum mechanics that describes the behavior of electrons in atoms and molecules.In the case of methane, the molecule consists of one carbon atom C and four hydrogen atoms H . The carbon atom has six electrons, with two in the 1s orbital and four in the 2s and 2p orbitals. The hydrogen atoms each have one electron in their 1s orbital. To form a stable molecule, the carbon atom forms four covalent bonds with the four hydrogen atoms by sharing electrons.According to molecular orbital theory, the atomic orbitals of the carbon and hydrogen atoms combine to form molecular orbitals that describe the distribution of electrons in the molecule. In methane, the carbon atom hybridizes its 2s and three 2p orbitals to form four equivalent sp3 hybrid orbitals. These sp3 hybrid orbitals are directed toward the vertices of a tetrahedron, with the carbon atom at the center.Each of the four hydrogen atoms overlaps with one of the carbon's sp3 hybrid orbitals, forming four sigma bonds. These bonds are the result of the head-on overlap of the hydrogen 1s orbitals with the carbon sp3 hybrid orbitals. The bond angles between the hydrogen-carbon-hydrogen H-C-H atoms are approximately 109.5 degrees, which is the tetrahedral bond angle.The tetrahedral geometry and bond angles of methane can be explained by the principle of minimum energy. The electrons in the molecule repel each other due to their negative charges, and the tetrahedral arrangement allows for the maximum separation between the electron pairs, minimizing the electron-electron repulsion and stabilizing the molecule.In summary, quantum mechanics, through the application of molecular orbital theory and the Schrödinger equation, explains the molecular geometry and bond angles of methane by describing the formation of sp3 hybrid orbitals in the carbon atom, the formation of sigma bonds with hydrogen atoms, and the minimization of electron-electron repulsion in the molecule.