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Year 0·Month 6·Week 4

Day 165: The Hamilton-Jacobi Equation — Where Classical Meets Quantum

Day 165 of 2,016~20 min read

Learning Objectives

•Derive the Hamilton-Jacobi equation from canonical transformation theory
•Distinguish between Hamilton's principal function S and characteristic function W
•Apply separation of variables to solve the HJ equation for standard systems
•Extract trajectories from a complete integral
•Connect the HJ equation to the Schrödinger equation via S → iℏ ln ψ
•Understand Hamilton's optical-mechanical analogy

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