Dimensional analysis - quantum and classical lengthscale ratio

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SUMMARY

The discussion centers on using dimensional analysis to approximate length scales in classical and quantum mechanics, specifically calculating classical length scale \( l_c \approx \frac{e^2}{4\pi m_e c^2 \epsilon_0} \approx 2.8 \times 10^{-15} m \) and quantum length scale \( l_q \approx \frac{h}{m_e c} \approx 2.4 \times 10^{-12} m \). When considering the scenario where \( l_q \approx l_c \), it is established that electromagnetic effects must be included in the analysis, as the ratio \( \frac{l_c}{l_q} \) is small and relates to the fine structure constant. This necessitates a functional relationship \( l_q = f\left(\frac{l_c}{l_q}\right) \) to accurately estimate \( l_q \).

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  • Study the fine structure constant and its implications in quantum mechanics
  • Explore the role of electromagnetic effects in quantum-classical transitions
  • Learn about dimensional analysis applications in various physics problems
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Students and educators in physics, particularly those focusing on quantum mechanics and classical mechanics, as well as researchers interested in the interplay between electromagnetic effects and length scales.

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Homework Statement


I was supposed to use dimensional analysis to approximate the length scale (in classical and quantum mechanics). The results I got(same as those in the answer sheet): classicaly[tex]l_c≈\frac{e^2}{4πm_ec^2ε_0}≈2.8*10^-15m[/tex] In quantum mechanics[tex]l_q≈\frac{h}{m_ec}≈2.4*10^-12m[/tex]


Homework Equations


The next question is: how would your analysis in case of [tex]l_q[/tex]have to change if [tex]l_q≈l_c?[/tex]


The Attempt at a Solution


I have the answer to question b), but I don't think I understand it. The answer they give is:
'If[tex]l_q≈l_c[/tex] it would not be possible to estimate l_q without considering electromagnetic effects and write [tex]l_q=f(\frac{l_c}{l_q})[/tex] In fact ratio l_c/l_q is is small (apart from a factor of 2π it is just the fine structure constant).'

Could anyone please explain or comment on the given answer?
 
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Could anyone explain why the electromagnetic effects need to be considered and how it influences the ratio?
 

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