# The products of powers of Hermitian operators

Let's say I have two operators, $$\hat{x}^k$$ and $$\hat{p}_x^l$$, where $$\hat{x}$$ and $$\hat{p}_x$$ are the ordinary position and momentum operators. It seems fairly straight forward to show that $$\hat{x}^k$$ and $$\hat{p}_x^l$$ are Hermitian. And the product of two Hermitian operators is Hermitian if they commute. However, it seems like something funny happens with $$\hat{x}^k$$ and $$\hat{p}_x^l$$, when I try and work out the commutator. I end up with:

$$[\hat{x}^k,\hat{p}_x^l] = -\left( \frac{\hbar}{i} \right)^l\frac{\partial^l}{\partial x^l}x^k$$

If $$l=k=1$$, then this is just the plain old-fashioned commutator for position and moment. No problems. However, because of the integer powers:

$$-\left( \frac{\hbar}{i} \right)^l\frac{\partial^l}{\partial x^l}x^k=-\left( \frac{\hbar}{i} \right)^l\frac{\partial^{l-1}}{\partial x^{l-1}}kx^{k-1} = -\left( \frac{\hbar}{i} \right)^l\frac{\partial^{l-2}}{\partial x^{l-2}}(k-1)kx^{k-2}$$

etc.

So this is the first step that I'm a little unsure of. So my first question is: is the above correct?

Now if $$k>l$$, then I get:

$$[\hat{x}^k,\hat{p}_x^l] =-\left( \frac{\hbar}{i} \right)^lk(k-1)(k-2)...(k-l)x^{k-l}$$ so clearly the two operators do not commute. If $$l=k$$:

$$[\hat{x}^k,\hat{p}_x^l] =-\left( \frac{\hbar}{i} \right)^lk!$$ And again, they don't commute. However, if I let $$l>k$$, then:

\begin{align} [\hat{x}^k,\hat{p}_x^l] &= -\left( \frac{\hbar}{i} \right)^{l-k}\frac{\partial^{l-k}}{\partial x^{l-k}}x^k!x^0\\ \\ &=-\left( \frac{\hbar}{i} \right)^{l-k}\frac{\partial^{l-k-1}}{\partial x^{l-k-1}}\frac{\partial}{\partial x}k!\\ \\ &=0 \end{align}

So now they commute and $$\hat{x}^k\hat{p}_x^l$$ is Hermitian! What's happened here? Have I made some trivial error somewhere?

• The first equation (v1) is incorrect. – Qmechanic Mar 24 at 11:31
• Ah yes, thank you. Doing it again, I can see that it should be quite different, as I have only applied the product rule once, when it should be applied $l$ times. However, my situation is now worse, because I end up with a long expression that I can't simplify. – monkeyofscience Mar 24 at 20:30