Notation for differential operators and wave function math

I know that $[\frac {d^2}{dx^2}]\psi$ is $\frac {d^2\psi}{dx^2}$ but what about this one $[\frac {d^2\psi}{dx^2}]\psi^*$? Is it this like $\frac {d^2\psi\psi^*}{dx^2}$ or this like $\frac {\psi^*d^2\psi}{dx^2}$?

And what about this one $[\frac {d^2\psi}{dx^2}]\phi$?

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I think the answer is the second one you are suggesting. In words, what you are doing is multiplying the second derivative of $\psi$ by ${\psi^*}$. The double derivative (by your notation, with the way you use the brackets) is only applied to ${\psi}$, and then it's just multiplication.

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i've seen both of them in QM book! –  rabi Dec 10 '12 at 6:56
It's very typical to find what seems like (and I'm sure, sometimes really is) an ambiguity in a textbook. Besides asking your best physicist friend what he thinks, one option is always to try every possible interpretation you can think of, starting with the likeliest-sounding one, and see if it fits. Usually, some thought and some calculation will align one of the interpretations with the context and eliminate the others. –  ruggy Dec 10 '12 at 16:48
$$\left[\frac {d^2\psi}{dx^2}\right]\psi^* = \frac {\psi^*d^2\psi}{dx^2}$$
$$\left[\frac {d^2\psi}{dx^2}\right]\phi = \frac {\phi d^2\psi}{dx^2}$$