Physics Asked by Paradox 101 on November 27, 2020

If we want to form a diagram or calculate the angles at which the magnets need to be positioned in the Stern-Gerlach setup, how do we proceed? For example, if I want to prepare the following state,

$$|psirangle= |uparrow_zrangle+2|downarrow_zrangle$$

how do I calculate the angles? Any help would be appreciated.

I understand that you speak of a fermion beam. Then take in consideration that a state $|psi(theta)rangle$ of polarization $theta$ i.e. at an angle $theta$ with respect to the $z$ axis, can be expressed as

$$|psi(theta)rangle = cos left(frac {theta}{2} right)|uparrowrangle + sin left(frac {theta}{2} right)|downarrowrangle$$

So the state you seek makes with the axis $z$ an angle defined by

$$cos(theta) = cos^2 left(frac {theta}{2}right) - sin^2 left(frac {theta}{2} right) = left(frac {1}{sqrt {5}} right)^2 - left(frac {2}{sqrt {5}} right)^2 = frac {4}{5}.$$

Two options now:

**A)** First of all decide which axis in the space is $z$, orient the magnetic field in your SG apparatus in the direction $-z$ (because spin $z$ fermions will go in the direction $-z$ of the field), and then rotate the apparatus by $theta$.

**B)** A more difficult option is that after orientating the magnetic field along $-z$, to select from the $|uparrowrangle$ beam, with a beam-splitter, a thinner beam with 1/4 intensity. Nex, by means of an opposite SG apparatus, to merge this beam with the one with $|downarrowrangle$. This option is rather theoretical because one should be very careful to have the beams coming to the 2nd SG ** in phase**, which is not simple, and even so, the precision of the merging is very poor.

Answered by Sofia on November 27, 2020

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