How do we know which term to attach a phase factor to in a state equation?

Question

I need to find the state of a particle in a one-dimesional harmonic oscillator where a measurement of the energy yields the values $\hbar\omega\over 2$ or $3\omega\hbar\over 2$, each with a probability of one-half at time t. I would have thought that the state would be $\big|\psi(0)\big>= {1\over \sqrt2}\big|0\big>+{1\over \sqrt2}\big|1\big>$. However the right equation is $\big|\psi(0)\big>= {1\over \sqrt2}\big|0\big>+{1\over \sqrt2}e^{-i\phi}\big|1\big>$. I know that the $e^{-i\phi}$ is a relative phase factor, but I can't figure out where it came from.

Where did the phase factor come from and when do phase factors need to be applied to the terms in state equations?

Answer 1

Your answer is correct, but there are other states that would give probability 50% and 50% if you add a phase shift to any of the states. However, any phase shift could be rewritten by a relative phase shift factor, as in the last expression in your question.