I have a density matrix defined as
$\rho(t) = \int_{-\infty}^{+\infty} \sigma(t,x) \otimes |x\rangle\langle x|$$$\rho(t) = \int_{-\infty}^{+\infty} \sigma(t,x) \otimes |x\rangle\langle x|$$
I want to show that $\hat{P} \rho(t) \hat{P} - \frac{1}{2} (\hat{P^2} \rho(t) - \rho(t) \hat{P^2}) = 2\frac{\partial^2 }{\partial x^2} \sigma(t,x)$$$\hat{P} \rho(t) \hat{P} - \frac{1}{2} (\hat{P^2} \rho(t) - \rho(t) \hat{P^2}) = 2\frac{\partial^2 }{\partial x^2} \sigma(t,x)$$ where $\hat{P}$ is the momentum operator.
The question is similar to Density Operator, Expectation Value, Coherent States, but this doesn't help me in getting the answer. $\hat{P} = -i \frac{\partial}{\partial x}$ in the position basis, but I am unable to get the final answer. Can someone help me?
