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Discussion: Heisenberg's Microscope

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

Note by Steven Zheng
2 years, 3 months ago

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