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...
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)
show that the first derivative of the legendre polynomials satisfy a self-adjoint differential equation with eigenvalue λ=n(n+1)-2
The attempt at a solution:
(1-x^2 ) P_n^''-2xP_n^'=λP_n
λ = n(n + 1) - 2 and (1-x^2 ) P_n^''-2xP_n^'=nP_(n-1)^'-nP_n-nxP_n^'
∴nP_(n-1)^'-nP_n-nxP_n^'=(...