Quantization Postulates for a Particle

[kilojoules]

Show that the operators x^2 p_x^2+p_x^2 x^2 and 〖 (xp_x+p_x x)〗^2/2 differ only by terms of order ℏ^2.






The attempt at a solution is attached (Postulates.pdf)

 

[bp_psy]

I don't know what you are trying to do in your solution ,you should explain it better. The first line equality is not correct,keep in mind that x,p momentum do not commute. I suggest expanding the second term first and see how it differs from the first.

 

[kilojoules]

I first found the quantum mechanical operator corresponding to the classical quantities xP_x, and according to the information I found on a downloaded file ("Dry2ans.pdf"), can't remember the source, I found that:
xP_x → xP_x + P_x x

As per your suggestion, bp_psy, I don't know which second term you are talking about. Is it of the first expansion or which one?

 

[bp_psy]

You initial post does not say that x,$p_x$ are classical observables but operators.Which one is it?
The classical observable $xp_x$ is represented by hermitian operator $\hat{x}\hat{p}_{x}+\hat{p}_{x} \hat{x}$ as they say in that document but the operator $\hat{x}\hat{p}_{x}$ is very different from $\hat{x}\hat{p}_{x}+\hat{p}_{x}\hat{x}$. Sometime people do not hat their operators so you shouldn't always assume that no hats mean classical observables.
What I meant by the second term is $\frac{(\hat{x}\hat{p}_{x}+\hat{p}_{x}\hat{x})^2}{2}$.