= Solution
Orthogonal projection gives the unique components
$$
|\phi\rangle=P_g|\phi\rangle+P_b|\phi\rangle.
$$
Choose
$$
a=\|P_g\phi\|,quad b=\|P_b\phi\|,quad
|\psi_1\rangle=\frac{P_g|\phi\rangle}{a},quad
|\psi_2\rangle=\frac{P_b|\phi\rangle}{b}.
$$
Then $a,b\geq0$, $a^2+b^2=1$, and the required decomposition follows. If one projection vanishes, its coefficient is zero and the corresponding normalized <vector> may be chosen arbitrarily in that subspace; the nonzero projected components remain unique.
Back to article page
