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I had a homework problem where I had to find the coefficients for the angular momentum raising and lowering operators. I know the answer is supposed to be $\sqrt{l(l+1)-m(m\pm1)}$. I have figured out how to get to the fact that the magnitude of the coefficients squared is $l(l+1)-m(m\pm1)$. However, I do not know how to deduce that they are real from their definition as $J_x \pm iJ_y$. My professor's answer key does not provide any justification for assuming they are real, nor can I find any in my textbook. I know that you could give them different phases (while retaining their raising/lowering behavior), but then you would have to change their definition from $J_x \pm iJ_y$.

How can you figure out that they are real, given that they are defined as $J_x \pm iJ_y$?

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  • $\begingroup$ What do you mean by "the coefficients"? Dont "the" coefficients depend on the basis and not just on the operators? $\endgroup$
    – jd27
    Mar 4 at 7:37
  • $\begingroup$ I think you mean the coefficients of the action of $J_{x} + i J_{y}$ on the basis states $|j, m\rangle$. However, I'm afraid that you need to use the operators $L_{x} \pm i L_{y}$ to arrive at the result you want. $\endgroup$
    – cconsta1
    Mar 4 at 7:41
  • $\begingroup$ A nice explanation can be found here: en.wikipedia.org/wiki/Ladder_operator $\endgroup$
    – cconsta1
    Mar 4 at 7:42

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Yes, OP is right: Normalized eigenstates $|j, m\rangle$ are in principle defined modulo an arbitrary phase factor. However, it is customary to chose the phase factors of neighboring eigenstates $|j, m\rangle$ and $|j, m\pm 1\rangle$ such that the matrix elements of $J_{\pm}$ are non-negative. (To be clear: the underlying $so(3)$ Lie algebra is not affected by these choices; only the representation matrices thereof.)

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