(Learning tirelessly but sometime can’t help doubt the efficiency! Remember that some can spark my thoughts significantly while some just ranting along dull. Those sparkings are worth mulling over and over again and really improve.) now chatGPT version:

Although I continuously learn, I sometimes question the effectiveness of my efforts. I have found that certain experiences or insights have a significant impact on my understanding, while others do not. It is important to reflect on these valuable experiences and apply them to improve my learning process.

This Quantum Mechanics and the Principle of Least Time by “Physics Explained” is such an example! We observe light deflected across another media such as water, further, we notice the path light taken is the shortest/least time spent. Why is that? and later on we found it’s fundamentally explained by Quantum Electrodynamics(QED book by Richard Feynman).

The analytical way(doesn’t necessary to always seek the clever way) is to write out the below equations and compute the minimum using differentials:

then the Snell’s law from empirical evidence is rigorously proved by math:

Once you understand above, it’s easier to explain quite a lot of phenomena in life such as mileage in desert. But what’s physicsts seek is not to summarize and apply the rules but to explain the rules.

Photons take the path in QM manner:

Details on the math will be explored later but the conclusion is that it’s just boiled down to the derivative computation to reach the minimum as we did above.