How is the quantum propagator related with Huygens principle?

Question

Usually in quantum mechanics the wave function can be propagated via the so-called Kernel or Amplitude: $\Psi(x,t) = \int K(x,t;x',t')\Psi(x',t')dx'$. I have read in some paper that this comes from Huygen's principle, and it makes somehow sense, since we are calculating the wave function at some point and time from the wave function at all the other points and times. However, I have two questions:

First of all, Huygens' principle is referred to waves, i.e., to solutions of the wave equation

$$\frac{\partial^2u}{\partial t^2} = v^2\frac{\partial^2u}{\partial x^2}.$$

However, the Schrödinger equation is not a wave equation since the time and position derivatives are not of the same order.

Second, if $\Psi(x,t) = \int K(x,t;x',t')\Psi(x',t')dx'$ is true because of Huygens' principle, could we just say that for every wave function $u(x,t)$ that satisfies the above wave equation (which is not the Schrödinger equation) something like

$$ u(x,t) = \int G(x,t;x',t')u(x',t')dx'$$

always holds? Could we say that this is the mathematical formulation of Huygens' principle?

Answer 1

Huygens' principle really just says the space a wave travels through is homogeneous. The wave equation $\partial_t^2u(x,t)-$ $c^2\partial_x^2u(x,t)=0$ is from Maxwell's equations. We can just we well consider the Schroedinger equation for a wave $\psi(x,t)=\psi(x)e^{-iEt/\hbar}$ with $i\hbar\partial_t\psi(x,t)=$ $E\psi$. The Schroedinger wave equation $i\hbar\partial_t\psi(x,t)=$ $H\psi(x,t)$ then reduces to $E\psi(x,t)=H\psi(x,t)$.

A typical propagator for Huygens' principle is found from the case of a wave passing through a slit with $\frac{e^{i\vec k\cdot\vec r}}{r}K(r)$ and the propagator would be $$ U(r)=\int\frac{e^{i\vec k\cdot\vec r'-iEt/\hbar}}{r'}K(r')dS. $$ This is an integration over the hemispherical wave front so $dS=2\pi r'dr'$ and we have $$ U(r)=2\pi\int e^{i\vec k\cdot\vec r'-iEt/\hbar}K(r')dr'. $$ The Hamiltonian operator for a free particle $\hat H=-\frac{\hbar}{2m}\nabla^2$ gives the Schroedinger equation $$ U(r)=\int e^{ikr'-iEt/\hbar}\left(\frac{\hbar}{2m}(k^2K(r')+2i\vec k\cdot\nabla K(r')+\nabla^2K(r')\right)=2\pi\int e^{i\vec k\cdot\vec r'-iEt/\hbar}E. $$ The energy can be read directly off.

The function $K(r')$ can be a Bessel function, or it can just be for the simple Huygen's-Fresnel case just $K(r')=\frac{1}{2}(1+cos\theta)$ for the angle $\theta$ the angle between a normal on the wave front and a line to the point $r$. This can be found with elementary analytic geometry.

The connection between nonrelativistic QM and Huygens' principle is that the kernel is modified for the Schroedinger equation. The full second order DE works for the Klein-Gordon equation, and again one would have to modify this for the Dirac equation.