Problem 7.20 - Fine Structure of Hydrogen TIPT ⇢ First-Order Corrections: Intro to Quantum Mechanics

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The problem involves deriving the fine structure formula in quantum mechanics, specifically for a hydrogen-like atom. This derivation is rooted in two key physical concepts: relativistic correction and spin-orbit coupling. The relativistic correction accounts for the adjustments needed when considering the electron's speed, which approaches a significant fraction of the speed of light. Spin-orbit coupling, on the other hand, arises from the interaction between the electron's spin and its orbital motion around the nucleus.

The problem challenges us to start with the relativistic correction formula (Equation 7.58) and the spin-orbit coupling formula (Equation 7.67), and to combine them into a comprehensive expression known as the fine structure formula (Equation 7.68). This formula reveals the slight energy level shifts due to these relativistic and quantum mechanical effects, providing deeper insight into atomic spectra and the fine details of electron behavior in atoms.

The process involves considering the total angular momentum 𝘫, which can take the values 𝓁±½ (with 𝓁 being the orbital angular momentum quantum number) and calculating the corresponding energy contributions. The task emphasizes understanding the interplay between different quantum mechanical phenomena and how they collectively influence the energy structure of atoms.

• 𝙿𝚛𝚘𝚋𝚕𝚎𝚖 𝙱𝚛𝚎𝚊𝚔𝚍𝚘𝚠𝚗 𝚃𝚒𝚖𝚎 𝚂𝚝𝚊𝚖𝚙𝚜:
00:00 - Intro & Problem Statement.
00:39 - Background.
01:36 - Stop 1: 𝘫=𝓁+½.
09:53 - Stop 1: 𝘫=𝓁-½.
16:24 - Concluding Remarks.
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Science is a phenomenal exploration of nature. We hope to hone our skills of problem solving by exposing ourselves to multiple contexts. In doing so, it can sometimes be challenging to see the connection between topics. I yearn to understand 𝙝𝙤𝙬 these aspects of physics, unite together. To accomplish this, I'll cover all of my old textbooks through QFT; the convergence point of the many modern scientists! These posts are very much in a "𝘯𝘰𝘵𝘦𝘴 𝘵𝘰 𝘴𝘦𝘭𝘧" style. 𝙈𝙮 𝙝𝙤𝙥𝙚 is that by sharing this exploration, I can help others navigate the beautiful world of mathematics & physics through problems and examples, connecting the mathematical tools to their physical ramifications.

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◉ ☞📚📖📓= Griffiths, David J., and Darrell F. Schroeter. “Chapter 7 Time-Independent Perturbation Theory.” 𝘐𝘯𝘵𝘳𝘰𝘥𝘶𝘤𝘵𝘪𝘰𝘯 𝘵𝘰 𝘘𝘶𝘢𝘯𝘵𝘶𝘮 𝘔𝘦𝘤𝘩𝘢𝘯𝘪𝘤𝘴, 3rd ed., Cambridge University Press, 2018, pp. 279–326.

◉ ☞ 🖼 📸 = http://tinyurl.com/4v9nef5k
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