Quantum Mechanics: Concepts and Applications · Problem 3.22

One-Dimensional Problems — Nouredine Zettili

Physics — Quantum Mechanics · separable-hamiltonian-csco

Problem

Consider a particle which moves in a scalar potential V(r⃗)=Vx(x)+Vy(y)+Vz(z)V(\vec{r}) = V_x(x) + V_y(y) + V_z(z). (a) Show that the Hamiltonian of this particle can be written as H^=H^x+H^y+H^z\hat{H} = \hat{H}_x + \hat{H}_y + \hat{H}_z, where H^x=px2/(2m)+Vx(x)\hat{H}_x = p_x^2/(2m) + V_x(x), and so on. (b) Do H^x\hat{H}_x, H^y\hat{H}_y, and H^z\hat{H}_z form a complete set of commuting operators?

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