Suppose you are trying to convince a classical physicist (who is too stubborn to accept QM) about the Uncertainty Principle. Get him to walk through his arguments for measuring the change of momentum for a very microscopic object. Can he disprove the Uncertainty Principle using classical mechanics?

**Note that there is nothing peculiar about Heisenberg's Uncertainty Principle – it is a result of the mathematics that describe waves (Fourier Analysis).**

How do microscopes work? Well, even the most advanced, high-resolution, microscopes follow the principle of diffracting light. We note that the angular resolution of some microscopic object is characterized by the Rayleigh criterion \({\theta}_{min} \approx \frac{\lambda }{d}\), where \(\lambda\) is the wavelength of light and \(d\) is the diameter of the objective lens.

To calculate the spatial resolution (uncertainty in x), we multiply the angular resolution \({\theta}_{min}\) by the focal length \(l\): \[\Delta x = l\frac{\lambda}{d}.\]

The x-component momentum of a photon hitting the microscopic object is uncertain by at least \({p}_{y}sin \theta\), where \({p}_{y} = h/\lambda \). For small angles \(sin (\theta) \approx \frac{d}{l}\); therefore,

\[\Delta {p}_{x} \ge \frac{h}{\lambda } \frac{d}{l}.\]

Hence, if we multiply the spatial uncertainty with the uncertainty of momentum, we get

\[\Delta x \Delta {p}_{x} \ge h.\]

This is not the actual Uncertainty Principle, but the point is: even in classical mechanics, there is a limit to how accurate the microscope can resolve!

To see the proof of the actual Uncertainty Principle, visit UP Proof and scroll down to **Proof of the Kennard inequality using wave mechanics**.

Visit my set Lectures on Quantum Mechanics for more notes.

Check out my other notes at Proof, Disproof, and Derivation

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