Factoring Overall Phases

Given

$$ \vert \psi(t)\rangle=c_{1}(0)e^{\frac{-iE_{1}t}{\hbar}}\vert E_{1}\rangle+c_{2}(0)e^{\frac{-iE_{2}t}{\hbar}}\vert E_{2}\rangle \, , $$

factor out an overall phase of $e^{\frac{-iE_{1}t}{\hbar}}$ from the quantum state.

Previously, students have been shown that relative phases are the important phases in the context of time-independent systems. For this activity, students will factor out an overall phase from a time-dependent expression and find that, even if the overall phase is time-dependent, the overall phase does not matter probabilistically.

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