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Is there a "convenient fiction" that explains why the angle of an electron's spin axis affects the probability of it being observed in a spin up or spin down state?

By "convenient fiction", I mean a story or image that provides useful intuition to novices, even though it may not be technically accurate. For example the analogy of water flowing through a pipe is a convenient fiction used to introduce the concepts of current and voltage.

I imagine the electron being sent through a Stern-Gerlach device. It makes sense that the closer the spin axis is to vertical, the more strongly the electron is drawn up or down. But, I don't see what would induce the electron to ever move in the "unexpected" direction. For example, if the axis is 5 degrees off vertical, what ever induces it to move down?

Watching this Veritasium2 video leads me to imagine that the electron is constantly flipping its spin axis; but, that doesn't seem to explain how the angle of the axis affects the probability of being measured in the up or down position.

https://www.youtube.com/watch?v=v1_-LsQLwkA&t=334s

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  • $\begingroup$ Do you know about the Bloch sphere? $\endgroup$ – PM 2Ring Mar 30 at 16:23
  • $\begingroup$ Yes. I understand how it mathematically describes the expected observation. I'm having trouble developing a mental picture of how it describes what the electron is actually doing and why. (I know such a picture is probably not completely accurate --- hence the term "convenient fiction".) $\endgroup$ – Zack Mar 30 at 16:28
  • $\begingroup$ Penrose has a nice description of electron spin in relation to the Riemann sphere in The Emperor's New Mind, p. 342. $\endgroup$ – PM 2Ring Mar 30 at 16:44
  • $\begingroup$ I understand the mathematics; but, I'm still not seeing the intuition. $\endgroup$ – Zack Mar 30 at 17:02
  • $\begingroup$ What makes you think such a "convenient fiction" exists? I'm leaning towards not. $\endgroup$ – Elio Fabri Mar 30 at 20:07
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I am afraid there is no way to get an intuition. The Stern-Gerlach experiment implies a description of the reality in terms of a superposition of states. In case of the electron spin you have Up and Down states related to the direction in which you measure. A surrogate of intuition could be the coefficients of the base kets (base states) specifying the superposition. In fact the probability of finding the electron Up or Down is the modulus squared of the related coefficient.

However the image that the electron moves up or down is not justified. In quantum mechanics you speak of collapse of the wave function during the measurement process.

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  • $\begingroup$ Is there a way to visualize the wave function? $\endgroup$ – Zack Apr 12 at 12:31
  • $\begingroup$ The wave function is a representation on a certain basis of the state of a quantum system. In case of a spin, the wave function is simply given by the coefficients of the basis kets. In case of a particle in one spatial dimension it is a function of the coordinate $x$ in the position basis or a function of the momentum $p$ in the momentum basis. If you imagine the graph of the function, you visualize it. $\endgroup$ – Michele Grosso Apr 12 at 21:04

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