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## Homework Statement

For [tex]l=1[/tex] the angular momentum components can be represented by the matrices:

[tex]

\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]

[/tex]

[tex]

\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]

[/tex]

[tex]

\hat{L_{z}} = \hbar \left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \end{array} \right]

[/tex]

Q(a). Confirm that the matrices fulfill the commutation relations of angular momentum.

Q(b) Calculate the matrix which represents the Hamiltonian:

[tex]\hat{H} = \frac{1}{2I}\hat^{

**L**}^{2} + \alpha \hat{L}_{z}[/tex]

of a rotating molecule, where [itex]I[/itex] and [itex]\alpha[/itex] are constants and:

[tex]\hat{

**L**}^{2} = \hat{L}_{x}^{2} + \hat{L}_{y}^{2} + \hat{L}_{z}^{2}[/tex]

Q(c) Calculate the energy levels of the molecule.

## Homework Equations

Commutation Relations of angular momentum:

[tex] \hat{L_{x}},\hat{L_{y}} = i\hbar \hat{L_{z}}[/tex]

[tex] \hat{L_{y}},\hat{L_{z}} = i\hbar \hat{L_{x}}[/tex]

[tex] \hat{L_{z}},\hat{L_{x}} = i\hbar \hat{L_{y}}[/tex]

Commutator Definition:

[tex]\hat{A},\hat{B} = \hat{A}\hat{B} - \hat{B}\hat{A}[/tex]

Rest as relevant within the question statement (and subsequent answers)

## The Attempt at a Solution

I have seen examples using Pauli matrices that are 2x2 but I don’t know how to go about this using these 3x3 matrices, i.e. how to adapt to these matrices from the standard Pauli ones.

A bit of help and advice to get me going in the right direction would be great, then I think I should hopefully be OK.