I am not very familiar with operators (as I do not study mathematics) and I have just started a Quantum Mechanics course in a university. However, I am not sure what should be the precise order of mathematical operations behind this operator:

$\hat{A}=(x^2\frac{\mathrm{d} }{\mathrm{d} x})^2$

Does it apply to function like this (1):

$\hat{A}=(x^4\frac{\mathrm{d^2} f(x)}{\mathrm{d} x^2})$

or like this (2):

$\hat{A}=(x^2\frac{\mathrm{d} f(x)}{\mathrm{d} x})^2$

I would be sincerely grateful for the insights what is the correct form of applying these type of operators to function and why, since my "Google" search regarding this question was unsuccessful. Thank you very much beforehand!

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    $\begingroup$ Maybe as $$x^2\frac d{dx}\left(x^2\frac{d f(x)}{dx}\right)?$$ $\endgroup$ – Lord Shark the Unknown Sep 22 at 19:29

It means neither of those things. The "square" here refers to functional composition, which means "do this operation twice." Hence

$$\widehat{A} f = x^2 \frac{d}{dx} \left( x^2 \frac{df}{dx} \right) = 2x^3 \frac{df}{dx} + x^4 \frac{d^2 f}{dx^2}.$$

The operators $x$ and $\frac{d}{dx}$ together generate a noncommutative algebra under addition and composition which satisfies some but not all of the rules of ordinary high-school algebra; most prominently, multiplication (composition) is noncommutative, like matrix multiplication. In fact you can think of these operators as infinite-dimensional matrices, and Heisenberg did so (see matrix mechanics).

So $\left( x^2 \frac{d}{dx} \right)^2 \neq x^4 \frac{d^2}{dx^2}$, which it would be if $x$ and $\frac{d}{dx}$ commuted. See Weyl algebra or, for a physics perspective, canonical commutation relations.

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    $\begingroup$ Thank you very much for such an insightful answer with all the links provided! $\endgroup$ – aerospace Sep 22 at 19:38
  • 1
    $\begingroup$ You're very welcome! This is subtle stuff, especially if you haven't done a lot of linear algebra; I even messed up the calculation the first time I wrote this answer. $\endgroup$ – Qiaochu Yuan Sep 22 at 20:03

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