Where does this expression come from? Why is the $N$ an exponent?
$J(\Delta x' \hat x) = lim_{N \to \infty} (1-\frac{ip_x\Delta x'}{N\hbar})^N $
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1$\begingroup$ where did you get that formula from? did you read it in a book? which one? $\endgroup$– AccidentalFourierTransformCommented May 27, 2017 at 19:54
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$\begingroup$ Modern Quantum Mechanics by Sakurai & Napolitano (page 46, 2nd Edition) $\endgroup$– Math12345Commented May 27, 2017 at 19:55
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$\begingroup$ Through binomial expansion that expression can be shown to be the exponential function as $n\rightarrow \infty$. So it is $e^{i p_x \Delta x /\hbar}$. See answers to this q: math.stackexchange.com/questions/900921/… $\endgroup$– CDCMCommented May 27, 2017 at 20:01
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1 Answer
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First, consider the expression
$$ e^{\alpha D}f(t) = f(t + \alpha) \tag{1} $$
where $D$ is the derivative operator. Eq. (1) states that the operator $D$ generates translation in the coordinates. You can find the justification for this expression here. Now, also recall that
$$ e^{\alpha D} = \lim_{N\to \infty}\left(1 + \frac{\alpha D}{N} \right)^N $$
and that the representation of momentum in coordinates space is $p = -i\hbar D$, putting everything together you get that the operator
$$ P(\alpha) = \lim_{N\to \infty}\left(1 + \frac{i\alpha p}{N\hbar} \right)^N $$
generates displacements on the coordinates
$$ P(\alpha)\psi(x) = \psi(x + \alpha) $$
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$\begingroup$ Is there a name for the second expression? $\endgroup$ Commented May 27, 2017 at 21:26
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