Question

Consider a particle in a Coulomb potential in three dimensional space. In Heisenberg's picture,d/dt⟨L^z⟩=0, where L^z is the z-component of angular momentum operator. Using this information which of the following choice(s) are correct:
1.H,L^z,L^x share simulatenous eigenbasis

2.L^z,L^x share simultaneous eigenbasis

3.H,L^z share simultaneous eigenbasis

4.L^.L^,L^z,L^x share simultaneous eigenbasis

Answer 1

Simultaneous eigenbasis can be shared only by pairwise commuting collections of operators. Since, in Heisenberg's picture, we have

$$\frac{d}{dt} L_z = \frac{i}{\hbar} \left[ H , L_z \right] = 0 \, ,$$

we observe that $H$ and $L_z$ commute and, therefore, share simultaneous eigenbasis. On the other hand, $L_z$ and $L_x$ do not commute, $\left[ L_z , L_x \right] = i L_y$, so they cannot share simultaneous eigenbasis. Thus, there is only one correct answer to this question.

Answer: $H , L_z$ share simultaneous eigenbasis.