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Understanding quantum mechanics often begins with Heisenberg's Uncertainty Principle. This fundamental concept suggests that we can't precisely determine both the position and momentum of a particle simultaneously. This principle is mathematically expressed as:\[ \Delta p \Delta x \geq \frac{\hbar}{2} \]Here, \(\Delta p\) stands for the uncertainty in momentum and \(\Delta x\) for the uncertainty in position. The symbol \(\hbar\) represents the reduced Planck constant, which is a fundamental constant in quantum mechanics.In our exercise, the slit width translates directly to the position uncertainty \(\Delta x\). It quantifies how precisely we can say where the photon is as it passes through the slit. By knowing this, we can then determine the corresponding minimum uncertainty in the photon's momentum, giving us an insight into its behavior post-filtration through the slit.

Photon momentum might not be something we think about often, but light particles carry momentum even without mass. In physics, the momentum \(p\) of a photon is given by:\[ p = \frac{h}{\lambda} \]Where \(h\) is Planck's constant and \(\lambda\) is the photon's wavelength. This equation shows that the momentum of a photon is inversely proportional to its wavelength—shorter wavelengths mean more momentum.In this case, after applying Heisenberg's principle, we find the uncertainty in this photon momentum. \[ \Delta p \geq \frac{\hbar}{2\Delta x} \]That tells us how much the momentum might waver as it squeezes through the narrow slit, causing it to spread out in different directions.

When light interacts with obstacles, it bends slightly, creating patterns. This is known as diffraction. The central diffraction maximum refers to the brightest spot on a screen where light bands form, straight opposite the opening causing the diffraction.In our task, after calculating the uncertainty in photon momentum, this \(\Delta p\) is linked to the angular spread \(\Delta \theta\) of light in the central diffraction maximum:\[ \Delta \theta = \frac{\Delta p}{p} \]Returning to simple geometry, the width \(w\) of this central band on the screen is calculated as:\[ w = 2L \Delta \theta \]Where \(L\) is the distance from slit to screen. This width accounts for the range over which the majority of the light intensity is concentrated and is an essential detail for understanding how different variables affect the diffraction pattern.