Simplifying Taylor series

45 Views Asked by Bumbble Comm At 28 Mar 2026 - 1:31

I am solving a Quantum Mechanics problem and I came across these Taylor's series.

$e^{a/2}(\psi(x)+ax\frac{\mathrm{d} {\psi(x)}}{\mathrm{d} x} +\frac{a^2x^2}{2!}\frac{\mathrm{d^2} \psi(x)}{\mathrm{d} x^2}+ ... )= e^{a/2}\psi(x+ax)$

So I was wondering, if I have simplified them correctly or is there some other option to simplify these series? I would be extremely grateful, if someone could help, as I am not mathematician myself.

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Bumbble Comm On 07 Oct 2019 - 4:45 BEST ANSWER

In a general manner, if you consider $f(x+a x)$ for small value of $a$, let $ax=h$ to have $$f(x+h)=f(x)+h f'(x)+\frac 12 h^2 f''(x)+\cdots$$ Now replace $h$ by $a x$ to get what you wrote which is correct.

Simplifying Taylor series

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