Exercise 2.5 Discussion

$\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\mathbf{Exercise\ 2.5}$

Give the set of all values $\theta$ for which the following pairs of states are equivalent.

a) $\ket 1$ and $\frac{1}{\sqrt 2}\left(\ket{+} + e^{\i\theta}\ket{-} \right)$

b) $\frac{1}{\sqrt 2}\left(\ket{\i} + e^{\i\theta}\ket{-\i} \right)$ and $\frac{1}{\sqrt 2}\left(\ket{-\i} + e^{-\i\theta}\ket{\i} \right)$

c) $\frac{1}{2}\ket 0 - \frac{\sqrt 3}{2}\ket 1$ and $e^{\i\theta} \left(\frac{1}{2}\ket 0 - \frac{\sqrt 3}{2}\ket 1 \right)$