Problem 9.9 - The Connection Formulas ⇢ WKB Approximation Accuracy: Intro to Quantum Mechanics

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This problem explores key concepts in quantum mechanics using the example of a particle in a quantum harmonic oscillator. You'll find the turning points of the oscillator, where the total energy equals the potential energy, marking the boundaries of the particle's classical motion. The problem then challenges you to quantify how far beyond this turning point you can go before a linear approximation of the potential introduces significant errors. Finally, you'll determine the minimum quantum number
𝑛 where both the linearized potential and the Airy function's asymptotic form remain accurate to within 1%. This exercise highlights the critical intersection of quantum mechanics with classical concepts and the importance of approximation methods in solving complex quantum problems.

• 𝙿𝚛𝚘𝚋𝚕𝚎𝚖 𝙱𝚛𝚎𝚊𝚔𝚍𝚘𝚠𝚗 𝚃𝚒𝚖𝚎 𝚂𝚝𝚊𝚖𝚙𝚜:
00:00 - Intro & Problem Statement.
01:47 - Background.
02:59 - Part (a): Turning Point 𝘹₂.
04:23 - Part (b): Error Analysis for 𝘥.
09:24 - Part (c): Minimum 𝘯.
14:01 - 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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⍟ 𝐂𝐫𝐞𝐝𝐢𝐭𝐬 ⍟
◉ ☞📚📖📓= Griffiths, David J., and Darrell F. Schroeter. “Chapter 5 Identical Particles.” 𝘐𝘯𝘵𝘳𝘰𝘥𝘶𝘤𝘵𝘪𝘰𝘯 𝘵𝘰 𝘘𝘶𝘢𝘯𝘵𝘶𝘮 𝘔𝘦𝘤𝘩𝘢𝘯𝘪𝘤𝘴, 3rd ed., Cambridge University Press, 2018, pp. 198–231.

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