Over the next few videos, we're going to develop an alternative method to solving scattering problems. And it's going to involve reformulating the scattering problem through an integral equation. So to recall the problem that we're considering. We have some incident particles, which have kinetic energy given by h bar square k square over 2m. And they are traveling towards a target, which we're going to place at the origin of our coordinate system. So here, we're going to take Z as the axis along which the particles are coming. So then this would be X and this would be Y. So we have some target over here. And we modeled this target as a potential V of R. Once the incident particles hit this target, they will scatter in some direction R and give a scattered component to our wave function. And we're interested in determining the distribution of particles that a detector around here would capture. At the moment, we're only considering elastics scattering in which the kinetic energy is conserved in this scattering process. And we came up with a general form for the stationary scattering states, which we're given by the incident particles, described by a plane wave traveling in the Z direction, which is a spherical wave propagating radially outwards from the target. And to take into account any angular dependence, we added this scattering amplitude. And the goal in our scattering problem is to figure out what form this scattering amplitude will take. We're going to introduce a few vectors over here. So we'll call the vector KI as having magnitude K in the Z direction. And the vector KS, so it gives the direction of the scattered wave. This is elastics scattering. It has the same magnitude as the incident particles. And we'll call the direction N hat. So N hat is the direction in which we're looking at. In that case, we can rewrite this as KI dot R plus scattering amplitude KS dot R. And this R over here is just a magnitude of the position vector. And these stationary scattering states were created so that they would satisfy the time independent Schrodinger equation for some general potential. Not necessarily central this time. We're considering any potential in general. And the energy here is given by the energy of the free incident particles. For convenience, we're going to define a new quantity. So we're going to say PR is equal to h bar square over 2m UR. And this lets us rewrite the time independent Schrodinger equation by factoring out the h bar square over 2m from all of the factors and cancelling everything out. Or in more convenient notation. Bringing this over to the other side. And get this as our new form of the time independent Schrodinger equation for the scattering problem. The term on the right hand side is typically thought of as source of scatter waves that's due to some complicated potential, which is implicitly inside of this U of R. So to solve this, we're first going to solve the problem for the case of a point source, which we're going to model by a direct delta function. So instead of solving this problem, we're going to look to solve this problem. Here this delta 3 of R stands for the three-dimensional delta function. And the solution to this equation we're going to denote by g of R. So assuming we know what this g of R is, we can then build up the solution to this equation from this one by superimposing the response of several delta functions together to form this more complicated potential over here. So if we call this equation one, then the solution to one is given by the solution to the homogeneous differential equation. So when the right hand side is equal to zero, this one is the solution to this partial differential equation. And the solution to non-homogeneous term is a superposition of the solutions to this problem. So we're assuming here that this function g is known. So it satisfies this condition and you can verify for yourself that this solves our new Schrodinger equation over here. So our task now is to figure out what this function g is for our particular scattering problem. In the next video, we'll sort of in a hand-wavy manner develop a form for this function g.