### A.13 Integral Schrö­din­ger equation

The Hamiltonian eigenvalue problem, or time-independent 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.