Classical limit of the commutator is a derivative?

[pellman]

I just came across the following claim:

\lim_{\hbar\rightarrow 0}[\frac{1}{\hbar}(AB-BA)]

(which approaches the classical Poincare commutator) is a derivative with respect to \hbar. I know it looks like derivative, but is it really? Please elaborate.

 

[meopemuk]

I just came across the following claim:

\lim_{\hbar\rightarrow 0}[\frac{1}{\hbar}(AB-BA)]

(which approaches the classical Poincare commutator) is a derivative with respect to \hbar. I know it looks like derivative, but is it really? Please elaborate.

The commutator can be generally written as a series

[A,B] = i \hbarK_1 + i\hbar^2K_2 + i \hbar^3 K^3 + \ldots

where K_i are Hermitian operators. This follows simply from the fact that [A,B] is antiHermitian and that it must tend to zero as \hbar \rightarrow 0. So the limit

-\frac{i}{\hbar}\lim_{\hbar\rightarrow 0}[A,B] = K_1

(which is the classical Poisson bracket of A and B) can be regarded as the derivative of [A,B] with respect to \hbar at \hbar \rightarrow 0.

Eugene.

 

[pellman]

Very cool. Thanks, Eugene!