Quantum Mechanics for Engineers 

© Leon van Dommelen 

A.13 Integral Schrödinger equation
The Hamiltonian eigenvalue problem, or timeindependent Schrödinger
equation, is the central equation of quantum mechanics. It reads
Here is the wave function, is the energy of the state
described by the wave function, is the potential energy, is
the mass of the particle, and is the scaled Planck constant.
The equation also involves the Laplacian operator, defined as
Therefore the Hamiltonian eigenvalue problem involves partial
derivatives, and it is called a partial differential equation.
However, it is possible to manipulate the equation so that the wave
function appears inside an integral rather than inside partial
derivatives. The equation that you get this way is called the
integral Schrödinger equation.
It takes the form,
{D.31}:

(A.42) 
Here is any wave function of energy in free space. In
other words is any wave function for the particle in the
absence of the potential . The constant is a measure of
the energy of the particle. It also corresponds to a wave number far
from the potential. While not strictly required, the integral Schrödinger
equation above tends to be most suited for particles in infinite
space.