Editing Wave packet (section)

== Analytic continuation to diffusion ==
{{See also|Heat equation#Fundamental solutions}}
The spreading of wave packets in quantum mechanics is directly related to the spreading of probability densities in [[diffusion]]. For a particle which is [[random walk|randomly walking]], the probability density function satisfies the [[diffusion equation]]{{sfn|Kozdron|2008|loc=chpt. 3 Albert Einstein's proof of the existence of Brownian motion}}
<math display="block"> {\partial \over \partial t} \rho = {1\over 2} {\partial^2 \over \partial x^2 } \rho ,</math>
where the factor of 2, which can be removed by rescaling either time or space, is only for convenience.

A solution of this equation is the time-varying [[Gaussian function]]
<math display="block"> \rho_t(x) = {1\over \sqrt{2\pi t}} e^{-x^2 \over 2t},</math>
which is a form of the [[heat kernel]]. Since the integral of ''ρ<sub>t</sub>'' is constant while the width is becoming narrow at small times, this function approaches a delta function at ''t''=0,
<math display="block"> \lim_{t \to 0} \rho_t(x) = \delta(x) </math>
again only in the sense of distributions, so that
<math display="block"> \lim_{t \to 0} \int_x f(x) \rho_t(x) = f(0) </math>
for any [[test function]] {{mvar|f}}.

The time-varying Gaussian is the propagation kernel for the diffusion equation and it obeys the [[convolution]] identity,
<math display="block"> K_{t+t'} = K_{t}*K_{t'} \, ,</math>
which allows diffusion to be expressed as a path integral. The propagator is the exponential of an operator {{mvar|H}},
<math display="block"> K_t(x) = e^{-tH} \, ,</math>
which is the infinitesimal diffusion operator,
<math display="block"> H= -{\nabla^2\over 2} \, .</math>

A matrix has two indices, which in continuous space makes it a function of {{mvar|x}} and {{mvar|x}}'. In this case, because of translation invariance, the matrix element {{mvar|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:
<math display="block"> K_t(x,x') = K_t(x-x') \, .</math>

Translation invariance means that continuous matrix multiplication,
<math display="block"> C(x,x'') = \int_{x'} A(x,x')B(x',x'') \, ,</math>
is essentially convolution,
<math display="block"> C(\Delta) = C(x-x'') = \int_{x'} A(x-x') B(x'-x'') = \int_{y} A(\Delta-y)B(y) \, .</math>

The exponential can be defined over a range of ''t''s which include complex values, so long as integrals over the propagation kernel stay convergent,
<math display="block"> K_z(x) = e^{-zH} \, .</math>
As long as the real part of {{mvar|z}} is positive, for large values of {{mvar|x}}, {{mvar|K}} is exponentially decreasing, and integrals over {{mvar|K}} are indeed absolutely convergent.

The limit of this expression for {{mvar|z}} approaching the pure imaginary axis is the above Schrödinger propagator encountered,
<math display="block"> K_t^{\rm Schr} = K_{it+\varepsilon} = e^{-(it+\varepsilon)H} \, ,</math>
which illustrates the above time evolution of Gaussians.

From the fundamental identity of exponentiation, or path integration,
<math display="block"> K_z * K_{z'} = K_{z+z'} \,</math>
holds for all complex ''z'' values, where the integrals are absolutely convergent so that the operators are well defined.

Thus, quantum evolution of a Gaussian, which is the complex diffusion kernel ''K'',
<math display="block"> \psi_0(x) = K_a(x) = K_a * \delta(x) \,</math>
amounts to the time-evolved state,
<math display="block"> \psi_t = K_{it} * K_a = K_{a+it} \, .</math>

This illustrates the above diffusive form of the complex Gaussian solutions,
<math display="block"> \psi_t(x) = {1\over \sqrt{2\pi (a+it)} } e^{- {x^2\over 2(a+it)} } \, .</math>