Problem in derivation or momentum as generator of translation

Question

A derivation of momentum as generator of translation

In this page, part 2 “Momentum as generator of translations”, I don’t understand this step:

$$T(x)=\lim_{N \rightarrow \infty}(T(x/N))^N =\lim_{N \rightarrow \infty}\left(1-\frac{ixp}{N\hbar}\right)^N$$

How is the 2nd equality derived?

My attempt:

$$p_x=i\hbar \lim_{a\rightarrow 0} \frac{T(ax)-\mathbb{I}}{a}=i\hbar \lim_{N \rightarrow \infty} \frac{T(x/N)-\mathbb{I}}{1/N}$$

$$T(x/N)=\lim_{N\rightarrow \infty} \left(\mathbb{I}- \frac{ip }{N\hbar}\right)$$

I don’t know where the x in the numerator comes from.

Answer 1

When you change your limits from $a\to 0$ to $N\to\infty$ you need to take care of your dimensions. What you are physically doing is slicing a length $x$ into $N$ parts such that each infinitesimal segment has length $a$. This also makes it clear that $a$ has dimensions of length. Thus the correct variable substitution is $$a\to\frac{x}{N}$$ This should fix your error.