Quantum mechanics is a theoretical science that deals with the study of motion of microscopic objects that have both observable wave like and particle-like properties. It specifies the laws of motion that these objects obey.
Quantum mechanics was developed independently in 1926 by Werner Heisenberg and Erwin Schrodinger . When quantum mechanics is applied to macroscopic objects(for which wave like properties are insignificant) the results are the same as those from the classical mechanics . The fundamental equation(Schrodinger equation) of quantum mechanics was developed by Schrodinger and it won him the Nobel Prize in Physics in 1933. This equation which incorportates wave-particle duality of matter, as proposed by de Broglie, is quite complex and knowledge of higher mathematics is needed to solve it.
For a system, an atom or molecule whose energy does not change with time , the Schrodinger equation is written as HΨ = EΨ , where H is a mathematical operator called Hamiltonian. Schrodinger gave a recipe of constructing this operator from the expression for the total energy of the system. The total energy of the system takes into account the kinetic energies of all the sub-atomic particles(electrons,nucleons), attractive potential between the electrons and nuclei individually. Solution of this equation gives E and Ψ.