The need for complex numbers in QM

This is the simplest explanation that I have been able to develop to see the need for complex numbers. I assume you are familiar with the QM formulation and the Stern-Gerlach experiments. 

 

A device \(SG_x\) (Stern-Gerlach aparatus oriented with the \(x\) axis) gives  us two outputs. We take the first one, which is associated with the eigenstate \(|r\rangle\) (the other would be \(|l\rangle\), from left and right), and introduce it in a device \(SG_z\) and obtain, again, two outputs \(|u\rangle\) and \(|d\rangle\), each of them with probability \(50\%\). We express 

\[
|r\rangle=\alpha |u\rangle+\beta|d\rangle
\]
with \(\alpha\) and \(\beta\) real values such that
\[
|\alpha|^2=|\beta|^2=1/2
\]
so \(\alpha=\beta=\pm \frac{1}{\sqrt{2}}\).

If we plug the \(|l\rangle\) output in the \(SG_z\) we obtain an analogous expression
\[
|l\rangle=\bar{\alpha} |u\rangle+\bar{\beta}|d\rangle
\]
with
\[
|\bar{\alpha}|^2=|\bar{\beta}|^2=1/2
\]

If we now perform another experiment, but using \(SG_y\) instead of \(SG_z\) we get similar expressions
\[
|r\rangle=\gamma |f\rangle+\delta|b\rangle
\]
\[
|l\rangle=\bar{\gamma} |f\rangle+\bar{\delta}|b\rangle
\]
where \(b\) and \(f\) stand for backward and forward; and such that
\[
|\gamma|^2=|\delta|^2=|\bar{\gamma}|^2=|\bar{\delta}|^2=1/2
\]

Now we could express \(|f\rangle\) in the basis \(\{|u\rangle, |d\rangle\}\) and would obtain
\[
|f\rangle=\frac{\bar{\delta}\alpha-\delta \bar{\alpha} }{\bar{\delta}\gamma-\delta \bar{\gamma} } |u\rangle+
\frac{\bar{\delta}\beta-\delta \bar{\beta} }{\bar{\delta}\gamma-\delta \bar{\gamma} } |d\rangle
\]

But a third experiment in which we plug the \(|f\rangle\) output of a \(SG_y\) in \(SG_z\) would yield a \(50\%\) of probabilities as usual. So it must be, for example,
$$
\left|\frac{\bar{\delta}\alpha-\delta \bar{\alpha} }{\bar{\delta}\gamma-\delta \bar{\gamma} }\right|^2=1/2
$$.

We have assumed all the previous parameters to be real numbers, and moreover we have concluded that they are \(\pm\frac{1}{\sqrt{2}}\), so:
\[
\bar{\delta}\alpha=\delta \bar{\alpha}=\bar{\delta}\gamma=\delta \bar{\gamma}=\pm\frac{1}{2}
\]

So in the previous quotient, the numerator can be 1, 0 or -1. And the same for the denominator. Therefore:
\[
\left|\frac{\bar{\delta}\alpha-\delta \bar{\alpha} }{\bar{\delta}\gamma-\delta \bar{\gamma} }\right|^2\neq1/2

\]

The contradiction comes from assume that they are real numbers. In fact, we can show that if we perform all the above computations with complex number we have even a degree of freedom, a phase factor.

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