What is the commutator of position and momentum squared in quantum mechanics?

[fluidistic]

Homework Statement

Calculate $$[\hat X, \hat P^2]$$.

Homework Equations

$$[\hat A, \hat B] \Psi =[\hat A \hat B - \hat B \hat A ] \Psi$$.

The Attempt at a Solution

I am confused by $$P^2$$.
P is worth $$-i \hbar \frac{\partial}{\partial x}$$.
So I believe $$P^2= \hbar ^2 \left ( \frac{\partial}{\partial x} \right ) ^2$$.
If so, I get that $$[\hat X, \hat P^2]= \hbar ^2 \left [ x \left ( \frac{\partial \Psi}{\partial x} \right ) ^2 -\Psi ^2 - 2x \Psi \frac {\partial \Psi}{\partial x} - x^2 \left ( \frac{\partial \Psi}{\partial x} \right )^2 \right ]$$.
However if by $$\hat P ^2$$ they mean $$\hbar ^2 \frac{\partial ^2}{\partial x^2}$$, then I get $$[\hat X, \hat P^2]=-2 \hbar ^2 \frac{\partial \Psi}{\partial x}$$.
I think my first approach was correct, but the answer I get seems way too complicated. I would like a feedback.

 

[Mindscrape]

It should the the latter.

$$\hbar ^2 \frac{\partial ^2}{\partial x^2}$$

The Hamiltonian is often defined with the "P^2" operator.

While not all that physically meaningful, it's interesting to see what P^3 gives... (and so on)

 

[fluidistic]

Oh, thank you. I wasn't aware of that.
I'd like to know if I got a logical result, if it's not too much asked.
In all cases I'll redo the algebra tomorrow.

 

[Mindscrape]

Oh yeah, the result is correct, and I guess see my edit that just made if you are curious.

 

[dingo_d]

Or you could change to impulse representation where the momentum operator acts on $$\psi (k)$$ the same as the position operator acts on $$\psi (x)$$ in coordinate representation :) That way you need not mess with derivatives.

But in the end you get the same result ;) So I guess it's the matter of which way is easier to you...