# Why are the angular momentum raising and lowering operator coefficients real?

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 $$J_x \pm iJ_y$$?

• What do you mean by "the coefficients"? Dont "the" coefficients depend on the basis and not just on the operators?
– jd27
Mar 4 at 7:37
• 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. Mar 4 at 7:41
• A nice explanation can be found here: en.wikipedia.org/wiki/Ladder_operator Mar 4 at 7:42

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.)