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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 we represent phase? Or it is something other?

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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

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

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).

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