![]() ![]() The Pauli exclusion principal says overlapping electrons cannot all have the same state. The orbitals spread out, extending across many atoms. In many materials, the outer electrons are not confined to an orbital around a single atom. This explains nothing from the atomic/quantum point of view, but as explained in one of the comments, it is only when you have vast numbers of atoms that this classical behaviour emerges at all. The only way this can happen is if the arguments of the exponentials are all equal to each other for all values of $x$ and $t$ and this yields $\omega_i=\omega_r=\omega_t$ (frequency unchanged for the reflected and transmitted components) and $k_i \sin\theta_i = k_t\sin \theta_t$ (Snell's law) and finally that $k_i \sin\theta_i = k_r \sin\theta_r$ and since they travel in the same medium with the same frequency, speed and wavelength, then $k_i = k_r$ and hence the desired result $\theta_i = \theta_r$. Where this must be true for all values of $x$ and $t$. If we have an incoming wave travelling in the $xy$ plane, incident upon an interface plane defined by $y=0$, with amplitude $E_i$ at incidence angle $\theta_i$ that produces a reflected wave at angle $\theta_r$ and a transmitted wave at $\theta_t$ to the normal, then you can write down the following equation (where I assume the electric field is $z$-polarised with its direction also parallel to the interface plane). perpendicular to the normal to the surface) must be the same immediately either side of the interface. The alternative "textbook" electromagnetism answer is to use the boundary condition for the electric field either side of an interface - this is that the component of electric field parallel to the interface (i.e. Any optics textbook should have the calculation, or a quick Google found an example here. This is basically the Huygen's construction, and if you do the sums for a surface you can show that the overall scattering is only non-zero when the angle of reflection is equal to the angle of incidence. Add lots more to make a 2D surface, then add more layers of silver atoms below, and you're building up a system where the overall light scattering is the sum of individual scattering from huge numbers of individual silver atoms. Now add lots of atoms in a row, and you get something like a diffraction grating. (I'm oversimplifying because two atoms would be too closely spaced to act as Young's slits, but bear with me.) Now the light isn't simply isotropically scattered, but instead it's scattered into preferred directions. Each atom will scatter isotropically, so in effect we have two closely spaced emitters of light and the system behaves like a Young's slits setup. it will scatter the light equally in all directions.īut suppose we have two silver atoms side by side. The starting point it that a single silver atom is far smaller than the wavelength of light, so any scattering from it will be isotropic i.e. Without realising it you have stumbled across the Huygens-Fresnel principle. ![]()
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