I thought that I had angular momentum very well understood, but something has been giving me problems recently.(adsbygoogle = window.adsbygoogle || []).push({});

It is often stated in textbooks and webpages alike, that the angular momentum ladder operators defined as

[tex]J_{\pm} \equiv J_x \pm i J_y[/tex]

Then the texts often go on to say that these operators satisfy the following crucial commutation relation:

[tex]\left[ J_z , J_\pm ] = \pm J_\pm[/tex]

The problem I have is that if the above commutation relation holds perfectly, then the Clebsch Gordan coefficients would never arise. Applying the [itex]J_z[/itex] operator to a raised/lowered eigenstate should perfectly give m-1 or m+1, assuming the above commutation relation to be correct. Instead, we are told that there is some factor that creeps in.

To be honest, there is a similar thing with SHO ladder operators, we normally get factors of the form [itex]\sqrt{n}, \sqrt{n+1}[/itex].

If there is something obvious I have missed, then can someone let me know. Also, if anyone knows the derivation of the above commutation rule, or a link to it, that'd be great. I tried to derive it, but had some trouble.

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# Angular Momentum Ladder Operators

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