# Classical limit of the commutator is a derivative?

• pellman
In summary, the conversation discusses the claim that the commutator \lim_{\hbar\rightarrow 0}[\frac{1}{\hbar}(AB-BA)] is a derivative with respect to \hbar, approaching the classical Poincare commutator. The discussion confirms this claim, stating that the limit can be regarded as the derivative of the commutator at \hbar \rightarrow 0.
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.

pellman said:
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.

Very cool. Thanks, Eugene!

## 1. What is the classical limit of the commutator?

The classical limit of the commutator is the behavior of the commutator when the quantum mechanical system approaches the classical regime. In this limit, the commutator tends to zero, indicating that the observables involved in the commutator become compatible and can be measured simultaneously with high precision.

## 2. How is the classical limit of the commutator related to the derivative?

The classical limit of the commutator is directly related to the derivative of the observables involved. In the classical limit, the commutator of two observables A and B tends to the Poisson bracket of A and B, which is equivalent to the derivative of A with respect to B.

## 3. Why is the classical limit of the commutator important?

The classical limit of the commutator is important because it helps bridge the gap between the classical and quantum mechanical descriptions of a system. It allows us to understand how the behavior of a quantum system approximates that of a classical system, which is crucial for many practical applications in physics.

## 4. Can the classical limit of the commutator be calculated for all quantum mechanical systems?

No, the classical limit of the commutator can only be calculated for certain systems, such as those with a large number of particles or those with high energy states. In general, the classical limit becomes increasingly accurate as the system size or energy increases.

## 5. How can the classical limit of the commutator be experimentally tested?

The classical limit of the commutator can be experimentally tested by comparing the commutator of two observables in a quantum mechanical system with the Poisson bracket of the same observables in a classical system. This can be done by performing measurements on both systems and comparing the results.

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