Wave Packet - Analytic Continuation To Diffusion

Analytic Continuation To Diffusion

The spreading of wavepackets in quantum mechanics is directly related to the spreading of probability densities in diffusion. For a particle which is randomly walking, the probability density function at any point satisfies the diffusion equation:

where the factor of 2, which can be removed by a rescaling either time or space, is only for convenience.

A solution of this equation is the spreading Gaussian,

and since the integral of ρ_t is constant, while the width is becoming narrow at small times, this function approaches a delta function at t=0:

again only in the sense of distributions, so that

for any smooth test function f.

The spreading Gaussian is the propagation kernel for the diffusion equation and it obeys the convolution identity:

which allows diffusion to be expressed as a path integral. The propagator is the exponential of an operator H:

which is the infinitesimal diffusion operator,

A matrix has two indices, which in continuous space makes it a function of x and x '. In this case, because of translation invariance, the matrix element K only depend on the difference of the position, and a convenient abuse of notation is to refer to the operator, the matrix elements, and the function of the difference by the same name:

Translation invariance means that continuous matrix multiplication:

is essentially convolution,

The exponential can be defined over a range of ts which include complex values, so long as integrals over the propagation kernel stay convergent,

As long as the real part of z is positive, for large values of x, K is exponentially decreasing, and integrals over K are indeed absolutely convergent.

The limit of this expression for z coming close to the pure imaginary axis is the Schrödinger propagator:

and this gives a more conceptual explanation for the time evolution of Gaussians.

From the fundamental identity of exponentiation, or path integration:

holds for all complex z values where the integrals are absolutely convergent so that the operators are well defined.

Thus, quantum evolution starting from a Gaussian, which is the diffusion kernel K,

gives the time evolved state,

This illustrates the above diffusive form of the Gaussian solutions,