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I've done basic or introductory mechanics at the level of Resnick and Halliday. I'm currently studying calculus of variations and the Lagrangian formulation of mechanics on my own. I read somewhere that an understanding of classical mechanics is necessary for appreciating quantum mechanics. Can someone please give me the big picture, where I can see the connections between classical and quantum mechanics? (I am fairly new to quantum mechanics).

Is it the case that the Lagrangian and Hamiltonian formulations are what one uses, primarily, in quantum mechanics?

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An update: I came across a lecture video where it was mentioned that classical mechanics underlies conservation laws like that of energy and momentum through symmetry considerations. So it essentially affects all of physics. –  Joebevo Mar 11 '12 at 9:23
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2 Answers 2

The transition from classical to quantum is characterized by dropping the commutative law for the multiplication of physical quantities.

Quantum mechanics is the noncommutative version of classical Hamiltonian mechanics.

Quantum field theory is the noncommutative version of classical Lagrangian mechanics.

[Edit: This is just the big picture, explaining the most common stuff in a few lines. But the borderline between QM and QFT is fuzzy, and you can treat both QM and QFT with the main tools of the other field. For example, 1+0-dimensional QFT is equivalent to QM, essentially in the same way as classically, and QFT can be done via Hamiltonians (e.g., via similarity renormalization).]

My book Classical and Quantum Mechanics via Lie algebras is all about the connections between classical and quantum mechanics (but very little about QFT, sofar).

You'll find the big picture carefully motivated in Chapter 1, with the remainder of the book showing in increasing detail how it works. Most of the mathematics and physics you'll need to follow the book is introduced along the way, but knowledge of linear algebra and multivariate differentiation is assumed.

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Do you really mean to say, or is it accidentally implied, that the distinction between QM and QFT can be characterized as respectively between quantizations of Hamiltonian and Lagrangian classical mechanics? It's a characterization that I find very doubtful in any case, but all the more so at the level of the question. –  Peter Morgan Mar 9 '12 at 14:37
    
Note that this is just the big picture, leaving out all details, which would make drawing borders difficult. For example, 1+0-dimensional QFT is equivalent to QM, in the same way as classically, and QFT can be done via Hamiltonians (similarity renormalization!) - but if you look at the most common stuff, it fits my big picture. If you want to discuss this more, we'd go to a chat room. –  Arnold Neumaier Mar 9 '12 at 15:07
    
I'm good with that qualification. –  Peter Morgan Mar 9 '12 at 15:22
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It's been awhile since I've studied QM, but I recall it often being stated that classical mechanics was the limit of quantum mechanics as $\hbar \to 0$. Yes, the Hamiltonian approach to mechanics is often used in both. The Poisson bracket of classical Hamiltonian mechanics has it's quantum mechanical analog in the quantum mechanical commutator. IIRC, this is one way to quantize classical theories. For classical or quantum field theory, one typically works in the Lagrangian framework, where one tries to minimize some action.

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