#### samalkhaiat

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Continue reading...The Classical Limit of Commutator (without fancy mathematics)

Quantum mechanics occupies a very unusual place among physical theories: It contains classical mechanics as a limiting case, yet at the same time it requires this limiting case for its own formulation.

Many textbooks on elementary QM show you how the Hamilton-Jacobi [tex]\frac{\partial S}{\partial t} + H \left( x , \frac{\partial S}{\partial x}\right) = 0,[/tex] shows up as the classical limit, ##\hbar \to 0## , of the Schrodinger wave equation [tex]i\hbar \frac{\partial \Psi}{\partial t} – H \left(x , – i \hbar \frac{\partial}{\partial x}\right) \Psi = 0.[/tex]Textbooks also state, but without proof, that the equation of motion of a classical observable [itex]A(x,p)[/itex],[tex]\begin{equation}\frac{d}{dt}A(x,p) = – \big\{ H , A \big\}(x,p) = \frac{\partial H}{\partial p}\frac{\partial A}{\partial x} – \frac{\partial H}{\partial x}\frac{\partial A}{\partial p},\end{equation}[/tex]is the classical limit of Heisenberg...