Computing the commutator [X̂², P̂²] gives a non-Hermitian result

[antibrane]

I am attempting to calculate the commutator $[\hat{X}^2,\hat{P}^2]$ where $\hat{X}$ is position and $\hat{P}$ is momentum and am running into the following problem. The calculation goes as follows,

$$[\hat{X}^2,\hat{P}^2]=-\left(\underbrace{[\hat{P}^2,\hat{X}]}_{-2i\hbar\hat{P}}\hat{X}+\hat{X}\underbrace{[\hat{P}^2,\hat{X}]}_{-2i\hbar\hat{P}}\right)=2i\hbar\left(\hat{P}\hat{X}+\hat{X}\hat{P}\right)$$

and using that $[\hat{X},\hat{P}]=i\hbar$ we find that

$$[\hat{X}^2,\hat{P}^2]=2i\hbar\left[\left(\hat{X}\hat{P}-i\hbar\right)+\hat{X}\hat{P}\right]=4i\hbar\hat{X}\hat{P}+2\hbar^2$$

which is wrong because I know from a theorem that if $\hat{A}$ is Hermitian and $\hat{B}$ is Hermitian then $[\hat{A},\hat{B}]$ is anti-Hermitian, which is definitely not the case here. What am I doing wrong?

Thanks in advance for any help.

 

[strangerep]

$$[\hat{X}^2,\hat{P}^2]=2i\hbar\left[\left(\hat{X}\hat{P}-i\hbar\right)+\hat{X}\hat{P}\right]=4i\hbar\hat{X}\hat{P}+2\hbar^2$$

which is wrong because I know from a theorem that if $\hat{A}$ is Hermitian and $\hat{B}$ is Hermitian then $[\hat{A},\hat{B}]$ is anti-Hermitian, which is definitely not the case here. What am I doing wrong?

Maybe, in the last term, when checking whether it's anti-Hermitian, are you
forgetting to swap P and X ?

 

[antibrane]

Doesn't $2\hbar^2$ destroy the anti-hermicity, since

$$\left(2\hbar^2\right)^{\dagger}\neq -2\hbar^2$$

 

[strangerep]

Doesn't $2\hbar^2$ destroy the anti-hermicity, since

$$\left(2\hbar^2\right)^{\dagger}\neq -2\hbar^2$$

$$(4i\hbar XP + 2\hbar^2)^\dagger ~=~ -4i\hbar PX + 2\hbar^2<br /> ~=~ -4i\hbar(XP - i\hbar) + 2\hbar^2 ~=~ -4i\hbar XP - 4\hbar^2 + 2\hbar^2<br /> ~=~ -(4i\hbar XP + 2\hbar^2)$$

 

[Dickfore]

Use the rule:

$$[\hat{A}^{2}, \hat{B}^{2}] = [\hat{A}^{2}, \hat{B}] \, \hat{B} + \hat{B} \, [\hat{A}^{2} \, \hat{B}]$$

and

$$[\hat{A}^{2}, \hat{B}] = \hat{A} \, [\hat{A}, \hat{B}] + [\hat{A}, \hat{B}] \, \hat{A}$$

 

[antibrane]

oh you are right i was forgetting to change the order of X and P. Thanks. Thanks for your comment as well Dickfore I will use that.