Wave-Particle Duality
The double slit experiment demonstrates that light and matter can display characteristics of both classically defined waves and particles.
Interference
When coherent waves pass through two slits, they interfere. Constructive interference (peaks meeting peaks) produces bright fringes. Destructive interference produces dark fringes.
The Mystery
In particle mode, particles are fired one by one. They arrive at discrete locations, but over time, they build up an interference pattern! This implies each particle's probability wave passes through all slits simultaneously.
Intensity Distribution
The total intensity $I(\theta)$ is the product of the multi-slit interference pattern and the single-slit diffraction envelope:
$$ I = I_0 \cos^2\left(\frac{\pi d \sin\theta}{\lambda}\right) \mathrm{sinc}^2\left(\frac{\pi a \sin\theta}{\lambda}\right) $$
Where $\mathrm{sinc}(x) = \frac{\sin x}{x}$.
Minima and Maxima
Interference Maxima: $d \sin\theta = n\lambda$
Diffraction Minima: $a \sin\theta = m\lambda$
$\beta = \frac{\lambda L}{d}$
Central Max = $\frac{2\lambda L}{a}$
Path Diff $\Delta = d \sin\theta$