## The Free Propagator

The same technique in the previous chapter can be used to solve the initial value problem for the Schro¨dinger equation given by{

∂u ∂t (x, t) = −i(∆u)(x, t), x ∈ Rn, t 6= 0, u(x, 0) = f(x), x ∈ Rn, (7.1)

where f is again a function in S. In spite of the slight modification of the heat equation by multiplying the

Laplacian by −i, the nature of the equation and the properties of the solution are changed completely. The heat kernel gives the heat flow, while the free propagator that we shall derive describes the quantum mechanical motion of a particle in a vacuum. The solution u(x, t), x ∈ Rn,−∞ < t <∞, represents the state of the particle at the position x and time t. Hence the free propagator is also known as the Schro¨dinger kernel.