The Schrödinger equation can be thought of as the equivalent to the Newton’s second law but in the quantum world. In the Schrödinger equation, the wave function (Ψ) is used to describe the quantum particle. The quantum Hamiltonian (H) is an operator which corresponds to the kinetic and potential energy of the system. The Hamiltonian operates on the wavefunction giving a set of energy eigenvalues which are the possible values of the energy of the system.
Now back to our harmonic oscillator system. How can we describe it quantumly? First, we develop the Hamiltonian operator into the kinetic and the potential energy parts: