Problem 10.4 - Partial Wave Analysis ⇢ Low-Energy Scattering: Intro to Quantum Mechanics

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In this problem, we explore 𝘭𝘰𝘸-𝘦𝘯𝘦𝘳𝘨𝘺 𝘴𝘤𝘢𝘵𝘵𝘦𝘳𝘪𝘯𝘨 from a spherical delta-function potential, 𝘝(𝘳) = αδ(𝘳-𝘢), and aim to calculate the 𝘴𝘤𝘢𝘵𝘵𝘦𝘳𝘪𝘯𝘨 𝘢𝘮𝘱𝘭𝘪𝘵𝘶𝘥𝘦 𝘧(𝜃), 𝘥𝘪𝘧𝘧𝘦𝘳𝘦𝘯𝘵𝘪𝘢𝘭 𝘤𝘳𝘰𝘴𝘴-𝘴𝘦𝘤𝘵𝘪𝘰𝘯 𝘋(𝜃), and 𝘵𝘰𝘵𝘢𝘭 𝘤𝘳𝘰𝘴𝘴-𝘴𝘦𝘤𝘵𝘪𝘰𝘯 σ. To solve this, we apply 𝘱𝘢𝘳𝘵𝘪𝘢𝘭 𝘸𝘢𝘷𝘦 𝘢𝘯𝘢𝘭𝘺𝘴𝘪𝘴, a method in quantum mechanics where the incident wave is decomposed into spherical harmonics or 𝘱𝘢𝘳𝘵𝘪𝘢𝘭 𝘸𝘢𝘷𝘦𝘴 with definite angular momentum 𝓁. In the low-energy limit 𝘬𝘢≪1, the scattering is dominated by the 𝘴-𝘸𝘢𝘷𝘦 (𝓁=0) component, while higher partial waves are negligible. Matching the boundary conditions at 𝘳=𝘢, leads to the scattering amplitude and total cross-section. This demonstrates how partial wave analysis simplifies quantum scattering by focusing on the dominant 𝘴-𝘸𝘢𝘷𝘦 contribution at low energies.

• 𝙿𝚛𝚘𝚋𝚕𝚎𝚖 𝙱𝚛𝚎𝚊𝚔𝚍𝚘𝚠𝚗 𝚃𝚒𝚖𝚎 𝚂𝚝𝚊𝚖𝚙𝚜:
00:00 - Intro & Problem Statement.
02:19 - Background.
03:09 - Stop 1: Forming the Wavefunction.
06:31 - Stop 2: Applying the Boundary Conditions.
13:57 - Stop 3: Scattering Quantities.
19:47 - 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 10 Scattering.” 𝘐𝘯𝘵𝘳𝘰𝘥𝘶𝘤𝘵𝘪𝘰𝘯 𝘵𝘰 𝘘𝘶𝘢𝘯𝘵𝘶𝘮 𝘔𝘦𝘤𝘩𝘢𝘯𝘪𝘤𝘴, 3rd ed., Cambridge University Press, 2018, pp. 376–401.

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