I've just started learning about angular momentum and spin theory, and when I came across the definitions of the rising and lowering operators, I noticed the inverse form looks suspiciously like the definitions of $\sin$ and $\cos$ in terms of exponentials, is there a reason for why they are so similar?

$$ l_x = \frac{l_+ + l_-}{2}, \ l_y = \frac{l_+ - l_-}{2i}$$

and $$ \cos x = \text{Re}(e^{ix}) = \frac{e^{ix} + e^{-ix}}{2},$$ $$ \sin x = \text{Im}(e^{ix}) = \frac{e^{ix} - e^{-ix}}{2i}.$$

  • $\begingroup$ Please do not post formulae as screenshots, but use MathJax instead. $\endgroup$ – ZeroTheHero May 2 at 21:55

$l_\pm$ is defined as $l_x \pm il_y$. In general, any complex number $z$, $\Re(z)=\dfrac{z+\bar z}{2}$ and $\Im(z)=\dfrac{z-\bar z}{2i}$, which in the posted case is just the equations for $l_{x,y}$ and the trigonometric functions

  • $\begingroup$ I don’t see how this answers the question... $\endgroup$ – ZeroTheHero May 2 at 21:53
  • $\begingroup$ I guess I worded it somewhat oddly but OP was asking about the similarity between the two expressions, which is because they are both the real/imaginary components of a complex quantity...I'll edit for clarity $\endgroup$ – Quantumness May 2 at 23:07
  • $\begingroup$ What are R(z) and J(z)? And, is the gist of your answer that all complex numbers are of the same form? $\endgroup$ – Name Namerson May 2 at 23:54
  • $\begingroup$ They refer to the real and imaginary parts of a number respectively (it's not a J, rather a stylized I). Concerning the second part, yes: if a number is complex then it has a real and imaginary part. $\endgroup$ – Quantumness May 3 at 0:38

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