Hamiltonian operator

The Hamiltonian operator gives us the energy of a wavefunction. Generally the Hamiltonian is

where $\pp$ is the momentum operator and $V$ is the potential.

In position space the Hamiltonian is

The eigenfunctions $u_n(x)$ of the Hamiltonian are the eigenstates of the system, the eigenvalues are their respective energies. A wave equation can be constructed a superposition of eigenstates and their time-dependent functions (see Schrödinger equation).

The coefficients $C_n$ are given by the projection of the wavefunction onto the eigenstate $C_n = \braket{u_n|\psi}$.