# Phase factor when deriving expressions for up/down spin in the $z$-basis

From what I understand, spin in the x direction takes the form $$|+\rangle_x = a|+\rangle + b |-\rangle$$ (in the z basis), where $$a$$ and $$b$$ are complex. You can work out the value of the constants $$a$$ and $$b$$ (which is $$\frac{1}{\sqrt{2}}$$) from using the results attained in the Stern Gerlach experiment.

During the derivation of spin in the x direction, I understand that a phase factor is added since the constants can be complex, however why is this added to only one of the terms (in this case the $$|-\rangle$$ co-efficients).

$$|+\rangle_x = \frac{1}{\sqrt{2}}(|+\rangle+e^{i\alpha}|-\rangle)$$ $$|-\rangle_x = \frac{1}{\sqrt{2}}(|+\rangle-e^{i\beta}|-\rangle)$$

(where $$\alpha$$ and $$\beta$$ are constants)

• Mixing bra and ket vectors doesn't make much sense. You probably mean $|+\rangle_x$ and $|-\rangle_x$ instead of $\langle+|_x$ and $\langle-|_x$. – Thomas Fritsch Sep 3 '19 at 18:59
• Apologises, I have corrected it. – Lol Lolling Sep 3 '19 at 19:04
• I'm not sure I understand your question correctly. $|+\rangle_x$ and $|-\rangle_x$ must be orthogonal to each other. Therefore $e^{i\alpha}$ and $e^{i\beta}$ cannot be the same. – Thomas Fritsch Sep 3 '19 at 19:17
• Yes they are not, they are different values. – Lol Lolling Sep 3 '19 at 19:26

$$\frac{1}{\sqrt{2}}\left(|+\rangle+ e^{i \alpha}|-\rangle\right)$$ is the same as $$\frac{1}{\sqrt{2}}\left(e^{i \delta}|+\rangle+ e^{i (\alpha + \delta)}|-\rangle\right)$$, up to an overall phase. So we can freely set $$\delta = 0$$.
It is kind of arbitrary if you apply the factors to the $$|+\rangle$$ or to the $$|-\rangle$$ components.
Instead of defining \begin{align} |+\rangle_x = \frac{1}{\sqrt{2}}(|+\rangle+e^{i\alpha}|-\rangle) \\ |-\rangle_x = \frac{1}{\sqrt{2}}(|+\rangle+e^{i\beta}|-\rangle) \end{align} \tag{1} you could as well define \begin{align} |+\rangle_x = \frac{1}{\sqrt{2}}(e^{-i\alpha}|+\rangle+|-\rangle) \\ |-\rangle_x = \frac{1}{\sqrt{2}}(e^{-i\beta}|+\rangle+|-\rangle) \end{align} \tag{2}
The vectors $$|+\rangle_x$$ from (1) and (2) differ only by an over-all phase factor $$e^{2i\alpha}$$. This is irrelevant to the physics. They are physically the same state.