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1 year ago in Quantum Mechanics By Zack

Why do we need Hamiltonian or Lagrangian functions to quantize a classical system?

We successfully use Newton's laws for classical mechanics, so it seems natural to try to quantize F=ma directly. However, every textbook and course insists on first formulating the classical Hamiltonian or Lagrangian. This feels like an extra, non-intuitive step. I need to understand the deep-seated mathematical and physical rationale that makes these frameworks the essential gateway to quantum theory.

TheoreticalPhysics Quantization Hamiltonian

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By Sourabh Answered 1 year ago

This is a profound question that gets to the heart of how we build physical theories. The short answer is that quantization is a ruleset for transitioning between classical and quantum descriptions, not just equations. Newton's F=ma gives you trajectories specific solutions. The Hamiltonian/Lagrangian, however, encodes the system's entire structure: its symmetries, conservation laws (via Noether's theorem), and the full phase space of possible states. Quantization rules, like [x, p] = i?, apply to these fundamental degrees of freedom. You can't impose commutation relations on an equation of motion; you need the underlying conjugate variables that the Hamiltonian/Lagrangian provides. It's about having the correct fundamental objects to promote to operators.

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Why do we need Hamiltonian or Lagrangian functions to quantize a classical system?

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