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 θminλd{\theta}_{min} \approx \frac{\lambda }{d}, where λ\lambda is the wavelength of light and dd is the diameter of the objective lens.

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

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

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

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

ΔxΔpxh.\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
5 years, 2 months ago

No vote yet
1 vote

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}

Comments

There are no comments in this discussion.

×

Problem Loading...

Note Loading...

Set Loading...