reading Zee\'s QFT Theory in a Nutshell and on p 13 on the explanation of the Ga

Question

reading Zee's QFT Theory in a Nutshell and on p 13 on the explanation of the Gaussian integral he loses me going from (9) to (10) by jumping from a relatively straightforward derivation of ?e-1/2ax2dx
by then saying "acting on this with -2(d/da)

to get <x2n>=?e-1/2ax2x2ndx/?e-1/2ax2dx=1an(2n-1)!!

not sure how he gets to the double factorial expression - how do you know this integral would not diverge?

also not clear on the meaning of -2(d/da) as an operator

assuming the notation <x> means the expectation of some arbitrary variable

the wiki article and other internet sources on Wick's theorem refers to ordering of creation and annihilation operators, which Zee has not mentioned yet

sorry for the rambling post, but any help would be greatly appreciated

Explanation / Answer

The idea is to calculate I0(a)=?+8-8dxe-ax2 Then you observe that x2e-ax2=-??ae-ax2 Now you rewrite the integral as follows I2(a)=?+8-8dxx2e-ax2=?+8-8dx(-??a)e-ax2=-??a?+8-8dxe-ax2=-??aI0(a) Of course this is a formal trick and requires a proof in principle. From now one you can continue with I2n(a)=-??aI2n-2(a) and I2n(a)=(-??a)2I0(a)

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