(adsbygoogle = window.adsbygoogle || []).push({}); Calculate [L_{z},L_{+}]

By defintion ladder operators are:

L_{+}=L_{x}+iL_{y}

L_{-}=L_{x}-iL_{y}

Important Relations:

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

L_{x}= yp_{z}- zp_{y}, L_{y}= xp_{z}- zp_{x}, L_{z}= xp_{y}- yp_{x}

To start solving;

[L_{z},L_{+}]

L_{z}- (L_{x}+ iL_{y}) = 0

Multiply through by [tex]\hbar[/tex]:

[tex]\hbar[/tex]L_{z}- [tex]\hbar[/tex]L_{x}+ i[tex]\hbar[/tex]L_{y}

The i[tex]\hbar[/tex]L_{y}is equal to L_{z}L_{x}. From this point

I've tried varying approaches in attempt to cancel variable out, but have failed. I have a feeling this problem can be solved easier. Should I try to use spherical coordinates instead of Cartesian? From trying to figure this out I have stumbled upon the answer but I would like to know how to produce the answer.

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# Understanding Ladder Operators

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