In the statement |a> + |b>, both a and b are probability amplitudes (complex numbers). You obtain the probability density by squaring the probability amplitude

In the statement |a> + |b>, both a and b are probability amplitudes (complex numbers). You obtain the probability density by squaring the probability amplitude (or rather, multiplying it by it's complex conjugate aa*), which gives the same result as squaring the magnitude, so it can be written |a|^2. Since there's two possible states in our system, their probabilities have to add to unity, so |a|^2 + |b|^2 = 1. https://en.wikipedia.org/wiki/Probability_amplitude https://en.wikipedia.org/wiki/Probability_amplitude