Quantum theory is one of the most revolutionary theory of all time it is the branch of physics which deals with the microscopic particles. The laws of classical physics are valid only for bodies of linear sizes they are even valid for the grains of sand but when it comes to microscopic particles like electrons,protons and neutrons the laws of classical physics fail.on the basis of dual nature of matter and radiation proposed by de broglie and heisenberg`s uncertainty principle,Australian physicist Erwin schrodinger developed this branch of physics called quantum mechanics also known as wave mechanics.the most important thing this theory gave was the schrodinger wave equation which for a system such as an atom or a molecule whose energy does not change with time is written as Hcap psi=E psi.where Hcap is the total energy operator called Hamiltonian.Hamiltonian operator is the sum of kinetic energy operator Tcap and potential energy operator Vcap ie Hcap=Vcap+Tcap.the potential energy operator Vcap for a system is normally equal to its expression for potential energy,V.psi is the wave function of the system and E is the total energy of the system.so equation Hcap psi=Ecap may be written as (Tcap+Vcap)psi=E psi.Writing schrodinger equation for a system means writing the explicit mathematical forms of Tcap and Vcap in the previous equation. The equation since it contains two variables E and psi is solved by partial differentiation method to get the values of E and psi for the system.In quantum mechanics there is an operator corresponding to every observable property.An operator is like a mathematical command which acts as a mathematical function. Hope you understood this