How do I find the eigenvalues for the angular momentum ladder operators?

Question

I am trying to calculate the normalising constants for the angular momentum ladder operators but am stuck when I need to calculate expected values.

How can I find the expected values

Answer 1

First, one can show that given any state $|j,m\rangle$, the raising and lowering operators $J_\pm$ do what their names suggest; $$ J_\pm |j,m\rangle = A_{j, m}|j, m\pm 1\rangle, \qquad A_{j,m} = \sqrt{j(j+1) - m(m\pm1)} $$ This can be done by noting that $$ J_\pm J_\mp = \mathbf J^2 - J_z^2 \pm \hbar J_z $$ and that $J_-^\dagger = J_+$. Once you know what the ladder operators do to the states $|j,m\rangle$, one computes $$ \langle j',m'|J_\pm|j,m\rangle = \langle j',m'|A_{j,m}|j,m\pm 1\rangle = A_{j,m}\delta_{j',j}\delta_{m',m\pm 1} $$ Let me know if this is not what you were referring to or if you were looking for more detail!

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