(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Is [itex] z|lm\rangle [/itex] an eigenstate of [itex] L^{2} [/itex]? If so, find the eigenvalue.

2. Relevant equations

[tex] L_{z}|lm\rangle = \hbar m|lm\rangle[/tex]

[tex] L^{2}|lm\rangle = \hbar^{2} l(l+1)|lm\rangle[/tex]

3. The attempt at a solution

So since [itex] L_{z}[/itex] and [itex]L^{2}[/itex] are commuting observables, they have are simultaneously diagonalizable and hence share the same eigenkets. Now, since [itex] z [/itex] and [itex] L_{z}[/itex] commute [itex] z|lm\rangle [/itex] is an eigenstate of [itex] L_{z} [/itex] and hence of [itex]L^{2}[/itex]. Now I am just having some issues calculating the eigenvalue.

I have derived that [tex] [x_{i},L_{j}] = i\hbar \epsilon_{ijk}x_{k}[/tex] and that

[tex][x_{i},L^{2}_{j}] = i\hbar\epsilon_{ijk}(x_{k}L_{j}+L_{j}x_{k})[/tex].

Now [tex]L^{2}z|lm\rangle = ([L^2,z]+zL^{2})|lm\rangle[/tex].

So it remains to calculate [tex][L^2,z] = [L^{2}_{x}+L^{2}_{y},z] [/tex]

I have proceeded using the relations I derived above, but I can't seem to get this commutator to give me some constant times [itex]z[/itex], which is what I need to extract an eigenvalue from the whole thing.

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# Homework Help: Eigenvalues of commuting observables (angular momentum)

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