I'm trying to understand the construction of the T(ε) operator and why it is equal to I-iεG/hbar.(adsbygoogle = window.adsbygoogle || []).push({});

The textbook I'm using (Shankar) talks defines the translation operator with the phase factor:

[itex]T(ε)\left|x\right\rangle=e^{i \epsilon g(x)/\hbar}\left|x+\epsilon\right\rangle[/itex]

and translationational invariance

[itex]\langleψ| H|ψ\rangle=\langle ψ_\epsilon| H|ψ_\epsilon\rangle[/itex]

The book then says

"To derive the conservation law that goes with the above equation, we must first construct the operator T(e) explicitly. Since ε=0 correspons to no translation, we may expand T(ε) to order (ε) as

[itex]I-\frac{i ε}{\hbar} G[/itex]

Why is this so? How can you find an equation for only T without it acting on anything?

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# Expanding a translation operator

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