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Ok. I will try to do my best to explain you what is my doubt. I'm not a native English Speaker and the book I was reading is not in English Lang, but I've translated it to English.

In a spin-1/2 Stern Gerlach-like experiment, we can express the

I dont understand this reasoning. I thought the phase angle of complex numbers were just a form of locate them in a complex plane, but according to the book that angle determines direction (in this case, the direction along the spin component is alligned to).

Having doing that, they derived the spin component in y direction as follows:

After they derived the equations, they founded the angle to be ± π/2 and come to their conclusion:

So what can I conclude about all those things? Phase angle of a complex number determine its direction in 3 dimensional space as well its location in the complex plane? Can we represent both complex plane and "cartesian plane" of x,y,z-direction together? How could the direction of a spin component be determined only by a coefficient?

In a spin-1/2 Stern Gerlach-like experiment, we can express the

**ket**representing the spin component of the electrons in one direction as a linear combination of the ket spin components in z direction (z choosen to be standard basis), by multiplying them by a coefficient and summing the terms. My book says each coefficient is complex and thus have an amplitude and phase. Ok. To express the spin component in x direction, my book says, choose the phase angle to be zero for both coefficients:*"we are free to choose the value of the phase. This freedom comes from the fact that we have required only that the x-axis be perpendicular to the z-axis, which limits the x-axis only to a plane rather than to a unique direction.".*

I dont understand this reasoning. I thought the phase angle of complex numbers were just a form of locate them in a complex plane, but according to the book that angle determines direction (in this case, the direction along the spin component is alligned to).

Having doing that, they derived the spin component in y direction as follows:

*"having arbitrarily choosen the phase to be zero for the x states, we are no longer free to make that same choice for the y states.*

After they derived the equations, they founded the angle to be ± π/2 and come to their conclusion:

*"The two choices for the phase correspond to the two possibilities for the direction of the y-axis relative to the already determined x- and z-axes."*

So what can I conclude about all those things? Phase angle of a complex number determine its direction in 3 dimensional space as well its location in the complex plane? Can we represent both complex plane and "cartesian plane" of x,y,z-direction together? How could the direction of a spin component be determined only by a coefficient?

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