QUESTION

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Problem 2. Average the tensor $n i n_{k}-\frac{1}{j} \delta_{i k}$ (where $\mathbf{n}$ is a unit vector along the radius vector of a particle) over a state where the magnitude but not the direction of the vector 1 is given (i.e. $l_{z}$ is indeterminate).


For $l=1$ the angular-momentum components can be represented by the matrices \[ \hat{L}_{x}=\hbar\left(\begin{array}{ccc} 0 & \sqrt{\frac{1}{2}} & 0 \\ \sqrt{\frac{1}{2}} & 0 & \sqrt{\frac{1}{2}} \\ 0 & \sqrt{\frac{1}{2}} & 0 \end{array}\right), \quad \hat{L}_{y}=\hbar\left(\begin{array}{ccc} 0 & -i \sqrt{\frac{1}{2}} & 0 \\ i \sqrt{\frac{1}{2}} & 0 & -i \sqrt{\frac{1}{2}} \\ 0 & i \sqrt{\frac{1}{2}} & 0 \end{array}\right), \quad \hat{L}_{z}=\hbar\left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \end{array}\right) \] (a) Confirm that these matrices fulfill the commutation relations of angular momentum (i) $\left[\hat{L}_{x}, \hat{L}_{y}\right]=i \hbar \hat{L}_{x}$ (ii) $\left[\hat{L}_{y}, \hat{L}_{z}\right]=i \hbar \hat{L}_{x},\left[\hat{L}_{x}, \hat{L}_{x}\right]=i \hbar \hat{L}_{y}$. (b) Calculate the matrix which represents the Hamiltonian \[ \hat{H}=\frac{1}{2 I} \hat{\mathbf{L}}^{2}+\alpha \hat{L}_{z} \] of a rotating molecule, where $I$ and $\alpha$ are constants and $\hat{L}^{2}=\hat{L}_{x}^{2}+\hat{L}_{y}^{2}+\hat{L}_{z}^{2}$. (c) Calculate the energy levels of the molecule.

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