Quantum Mechanics: Concepts and Applications · Problem 5.7

Three-Dimensional Problems — Nouredine Zettili

Physics — Quantum Mechanics · angular-momentum-j1-system

Problem

Consider a system of total angular momentum j=1j = 1. As shown in (5.73) and (5.75), the operators J^x\hat{J}_x, J^y\hat{J}_y, and J^z\hat{J}_z are given by J^x=ℏ2(010101010),J^y=ℏ2(0−i0i0−i0i0),J^z=ℏ(10000000−1).\hat{J}_x = \frac{\hbar}{\sqrt{2}}\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}, \quad \hat{J}_y = \frac{\hbar}{\sqrt{2}}\begin{pmatrix} 0 & -i & 0 \\ i & 0 & -i \\ 0 & i & 0 \end{pmatrix}, \quad \hat{J}_z = \hbar\begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \end{pmatrix}. (a) What are the possible values when measuring J^x\hat{J}_x? (b) Calculate ⟨J^z⟩\langle \hat{J}_z \rangle, ⟨J^z2⟩\langle \hat{J}_z^2 \rangle, and ΔJz\Delta J_z if the system is in the state jx=−ℏj_x = -\hbar. (c) Repeat (b) for ⟨J^y⟩\langle \hat{J}_y \rangle, ⟨J^y2⟩\langle \hat{J}_y^2 \rangle, and ΔJy\Delta J_y. (d) If the system were initially in state ∣ψ⟩=114(−3223)|\psi\rangle = \frac{1}{\sqrt{14}}\begin{pmatrix} -\sqrt{3} \\ 2\sqrt{2} \\ \sqrt{3} \end{pmatrix}, what values will one obtain when measuring J^x\hat{J}_x and with what probabilities?

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