Question

In the book of Sakurai & Napolitano, the Canonical Commutation relation is derived using the unitary translation operator:

I agree with the derivation of the book until i reach this step, and sakurai simply approximates the RHS' translation operator to identity (1.206):

[x^,T^(δx)]=δxT^(δx)≈δxI^

I also understand the rest of the reasoning from there, but this approximation step does not sit right with me. As far as I get it, the approximation means setting the limit as δx→0
but then this should be taken over the whole equation and all displacements δx
should tend to zero right?

Any help would be much appreciated.

Answer 1

The approximation that the translation operator T^(\delta x) is approximately equal to the identity for small \delta x is used to simplify the derivation of the Canonical Commutation relation. It's based on considering only first-order terms in the expansion of T^(\delta x), which is sufficient for revealing the underlying physics without complicating the mathematics unnecessarily.

Step-by-step explanation:

In quantum mechanics, the approximation made with regards to the unitary translation operator T^(\delta x) is indeed an approximation for very small but nonzero values of \delta x. You're right that in the limit as \delta x approaches zero, T^(\delta x) approaches the identity operator I. However, we must also consider the behavior of operators and observables in this limiting process.

In the derivation of the Canonical Commutation relation, the approximation [x^,T^(\delta x)]=\delta xT^(\delta x)≈\delta xI is valid in the sense that we are considering first-order effects. This means we are looking at the linear terms in an expansion of the translation operator in powers of \delta x. The exact expression for the infinitesimal translation operator can be represented as T^(\delta x)=I-i\delta x p^/\hbar + O((\delta x)^2), where p^ is the momentum operator and \hbar is the reduced Planck constant.

When \delta x is small, the higher-order terms become negligible, and we mainly consider the first-order term. The approximation simplifies the algebra and allows us to extract the physical implications of the Canonical Commutation relations without diving into the complexities introduced by the higher-order terms, which become relevant only for finite displacements.