# Magnitudes and phases of coefficients of a qubit [duplicate]

Quantum mechanics is based on the idea of waves, and waves have both a magnitude and a phase?

$$|\psi\rangle = i\alpha|0\rangle + \beta|1\rangle.$$

Does $$\alpha$$ and $$\beta$$ represent magnitude and $$i$$ represent phase?

Or how do we represent phase? Or is it something else?

It is more common to write a qubit as

$$|\psi\rangle = \alpha|0\rangle + \beta|1\rangle,$$

where $$\alpha, \beta \in \mathbb{C}$$, i.e. to omit $$i$$ and get it in complex number $$\alpha$$.

Parameters $$\alpha$$ and $$\beta$$ are called complex amplitudes, $$|\alpha|^2$$ is a probability of measuring state $$|0\rangle$$ and $$|\beta|^2$$ is a probability of measuring state $$|1\rangle$$. So, $$\alpha$$ and $$\beta$$ (or rather square of their absolute values) can be called "magnitude".

Any qubit can be rewritten as

$$|\psi\rangle = \cos(\theta/2)|0\rangle + \mathrm{e}^{i\phi}\sin(\theta/2)|1\rangle,$$

where $$\theta$$ and $$\phi$$ are coordinates on so-called Bloch sphere.

In this notation, $$\cos^2(\theta/2)$$ and $$\sin^2(\theta/2)$$ are probabilities ("magnitudes") of measuring $$|0\rangle$$ and $$|1\rangle$$, respecitively. Parameter $$\phi$$ is a phase (or to be precise, relative phase).