- #1
Hart
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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.