Kirchhoff's Diffraction Formula using the De Broglie Wavelength of Electron Can we apply Kirchhoff's diffraction formula to the matter wave of electron? Kirchhoff's diffraction formula was initially used to model the propagation of light. Similarly, is it possible to use this same formula to model the diffraction pattern of an electron by substituting the de Broglie wavelength in the formula? 
 A: Yes! Kirchhoff's diffraction theory can be applied to both light and matter waves, indeed. The original result you are referring to can be derived from a scalar theory, and we have as usual,
$$\Psi(r') = \frac{1}{4\pi}\iint_S \left[ \Psi(r_S)e_n \cdot \nabla \frac{e^{ik|r_S-r'|}}{|r_S-r'|} - \frac{e^{ik|r_S-r'|}}{|r_S-r'|}e_n \cdot \nabla \Psi(r-S)\right]d^2r_S$$
which gives an exact solution, given the value of the solution on a bordering surface $S$. I now advise you to read this paper which will show a derivation of the analogue from the Dirac equation.
The final result will give us, 
$$\Psi(r') = -\frac{ik}{4\pi}\iint_S \frac{e^{ik|r_S-r'|}}{|r_S-r'|} \left[1- \left( 1+\frac{i}{ks}\right)\gamma^j (\partial_js) \right]\gamma^n \Psi(r_s)\, d^2r_S$$
where $\Psi$ is a Dirac spinor instead, and there are many new notations introduced; for example $\gamma$ here are not quite the usual gamma matrices satisfying $\{\gamma^\mu,\gamma^\nu\} = 2\eta^{\mu\nu}\mathbb{1}$. 
The analogue to considering $\Psi(r) = \frac{e^{ikr}}{r}$ in the classical wave case applied to this formula derived from the Dirac equation will show that for a Dirac spinor, we expect,
$$\Psi(r') \propto \iint \frac{e^{ik(r_0+s)}}{r_0 s}(1+\gamma^3) \gamma^3 \Psi(r_s)\, dS$$
subject to certain boundary conditions.
