# What is the spin of an electron along the x-axis?

I know that an electron or any other particle for that matter, has a measured spin which is either up or down. This spin is along the z-axis. But what if we do not measure it along the z-axis and do it along the x-axis instead. What will the result be?

Will it get deflected along the x-axis in a Stern Gerlach experiment? Lastly, will it always get deflected along the axis we measure its spin?

Up or down, obviously. However, the representation of the x-up state or the x-down state using z-up and z-down states depends on how you represent the spin operator. For the most favoured choice

$S_x=\frac{1}{2}\left( \begin{array}{cc} 0 & 1\\ 1 & 0 \end{array} \right)$

gives up/down state as (not normalised)

$\left|x_{up}\right>=\left( \begin{array}{c} 1\\ 1\end{array} \right), \left|x_{down}\right>=\left( \begin{array}{c} 1\\ -1\end{array} \right)$

• So will the electron get deflected in a Stern Gerlach experiment along the x-axis? And sorry I do not understand Pauli Matrices. – rahulgarg12342 Sep 2 '14 at 13:39
• Yep. They get deflected along the x-axis. Try volume 3 of Feynman Lecture series or Sakurai's Modern Quantum Mechanics. – Dexter Kim Sep 5 '14 at 4:01
• Pauli matrices are nothing more than a method of representing the spin of a particle with spin half. When distinct states are finite in number, we can use linear algebra to represent and calculate the physical system. That is what Pauli matrices do. – Dexter Kim Sep 5 '14 at 4:05

It's important that the spin need not be up or down, but may be in some superposition of the two, $|\psi\rangle = a |\uparrow\rangle+b|\downarrow\rangle$. If you choose to measure it in the $z$-direction, the squared magnitudes of $a$ and $b$ will give relative probabilities of getting each answer.

But given any direction, you can always rewrite the state as a superposition of up and down spin in that direction instead (technical bit: the up and down are the eigenstates of $\vec{n}\cdot\vec{\sigma}$ where $\vec{n}$ is the unit direction vector and $\vec{\sigma}$ the Pauli matrices). There will now be different probabilities associated to the possible outcomes of the measurement. For example, if the original state had definite spin up in the $z$-direction, $|\psi\rangle = |\uparrow\rangle$, the probability of getting 'up' when measuring the spin in a direction at an angle $\theta$ to the $z$-axis will be $\cos^2(\theta/2)$.

• But if we measure it along the x-axis, then will the deflection also be along the x-axis? I am talking about the Stern Gerlach experiment. Thanks. Unfortunately I do not understand Pauli Matrices and the measurement angle part. All I need to know at the moment is that will the electron deflect along the x-axis. – rahulgarg12342 Sep 2 '14 at 14:34
• Yes, Stern-Gerlach is one way of measuring the spin. The particle will be deflected in a direction aligned with the applied magnetic field, in whichever direction you choose to put that applied field. – Holographer Sep 2 '14 at 14:39
• So we can in a way say that the particle obtains the poles according to our measurement. Unless we measure it, it has all the poles in all directions? – rahulgarg12342 Sep 3 '14 at 9:23