Problem 4.21 - Angular Momentum ⇢ Raising & Lowering Operators: Introduction to Quantum Mechanics

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In this video, we take a deep dive into the intriguing angular momentum ladder operators, 𝘓±. Angular momentum is a fundamental property in quantum mechanics, governing the rotational motion of particles. These operators play a crucial role in manipulating the quantum state of angular momentum, raising or lowering it by one unit.

How do we know this? From the hermiticity of these operators! Enter the Levi-Civita symbol ε, a mathematical tool that helps us demonstrate the hermiticity of in one go, remind you of another weird symbol? Cough-Cough δ By exploiting the properties of this symbol, we can establish the self-adjointness of the operators, ensuring that they possess real eigenvalues.

Moreover, we'll explore the resulting constants 𝘈_𝓁^𝘮 and 𝘉_𝓁^𝘮 obtained from the normalization of the eigenfunctions. These constants encapsulate the intricate relationship between the orbital and magnetic quantum numbers 𝓁 and 𝘮, providing valuable insights into the behavior of angular momentum in quantum systems.

• 𝙿𝚛𝚘𝚋𝚕𝚎𝚖 𝙱𝚛𝚎𝚊𝚔𝚍𝚘𝚠𝚗 𝚃𝚒𝚖𝚎 𝚂𝚝𝚊𝚖𝚙𝚜:
00:00 - Background.
↳ Note ↦ Angular Mom Diagram. https://tikz.net/spin/
↳ Note ↦ Angular Mom Diagram. https://steemit.com/physics/@pauldira...
↳ Note ↦ Angular Mom Diagram. https://www.lancaster.ac.uk/staff/sch...
00:48 - Problem Statement.
03:19 - Start of Solution - Hermiticity Part 1.
06:12 - Levi-Civita Symbol ε. Note ↦ https://en.wikipedia.org/wiki/Levi-Ci...
12:24 - Hermiticity Part 2.
15:11 - Ladder Operator - Eigen-equation Setup.
20:04 - Lowering Operator Constant.
23:20 - Raising Operator Constant.
24:38 - Limiting Cases of these Operators.
26:20 - 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 4 Quantum Mechanics in Three Dimensions.” 𝘐𝘯𝘵𝘳𝘰𝘥𝘶𝘤𝘵𝘪𝘰𝘯 𝘵𝘰 𝘘𝘶𝘢𝘯𝘵𝘶𝘮 𝘔𝘦𝘤𝘩𝘢𝘯𝘪𝘤𝘴, 3rd ed., Cambridge University Press, 2018, pp. 131–197.

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