Quantum Mechanics: Concepts and Applications · Problem 5.29

Three-Dimensional Problems — Nouredine Zettili

Physics — Quantum Mechanics · angular-momentum-j-three-halves-matrices

Problem

Consider a particle of angular momentum

j = 3/2

. (a) Find the matrices representing the operators

\hat{J}^2

\hat{J}_x

\hat{J}_y

, and

\hat{J}_z

in the

\{|\frac{3}{2},\ m\rangle\}

basis. (b) Using these matrices, show that

\hat{J}_x

\hat{J}_y

\hat{J}_z

satisfy the commutator

[\hat{J}_x,\ \hat{J}_y] = i\hbar\hat{J}_z

. (c) Calculate the mean values of

\hat{J}_x

and

\hat{J}_x^2

with respect to the state

\begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \end{pmatrix}

. (d) Calculate

\Delta J_x \Delta J_y

with respect to the state

\begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \end{pmatrix}

and verify that this product satisfies Heisenberg's uncertainty principle.