How to prepare the Stern-Gerlach experiment for a particular state?

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.

One Answer

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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