# Hadamard direct mapping of input to output in $\theta$ and $\varphi$ form

I was wondering what would be an equation for Hadamard operation for a single qubit, given the input as the current $$\theta$$ (0 to $$+\pi/2$$) and $$\varphi$$ ($$-\pi$$ to $$+\pi$$) and output expected in $$\theta$$ and $$\varphi$$ with same ranges. Most expressions of Hadamard that I saw use Cartesian transformation, but not $$\theta/\varphi$$ transform.

I could convert the input to Cartesian form, and convert the output back into $$\theta/\varphi$$ form well, but I'm looking for an equation that does it without using Cartesian conversion step? The goal is to understand direct relation between input and output. I tried interpreting Hadamard as subtracting $$\pi/4$$ from $$\theta$$ and adding $$\pi$$ to phase, but I see that it doesn't quite work for arbitrary input.

Note: here $$\varphi$$ refers to relative phase ($$-\pi$$ to $$+\pi$$) and $$\theta$$ refers to component amplitude factor (0 to $$+\pi/2$$).

I think I got it. The two rotations with angles $$-\pi/4$$ and $$\pi/2$$ should do it. The first rotation is about Y-axis and the second one is about Z-axis. So it still not what I was looking for. I was looking for change in the angle that determines the amplitudes instead of Y-rotation (the two are not same). But the Y-rotation is enough for my intuition. How I got this answer? Well, I just observed the change through multiple trials and learned the pattern :)