Brief Tutorial for
Measurement of Quantum-mechanical Spin-1/2 Systems

In 1922 Otto Stern and Walther Gerlach sent a beam of silver atoms (a spin-1/2 system) through an inhomogeneous magnetic field (generated by a permanent magnet) in the z direction.  Since the silver atoms have an intrinsic magnetic moment (or spin angular momentum) they should be deflected by the inhomogeneous magnetic field in the z direction depending on their orientation with respect to the magnetic field.  When Stern and Gerlach performed the experiment, they expected a uniform spread of the beam in the z direction to result.  Much to their surprise, the inhomogeneous magnetic field effectively split the beam in to two parts.  This result lead to the idea of quantization of spin angular momentum such that the component of spin in a particular direction can only take on two values: +ħ/2 or −ħ/2, or what we also call spin up and spin down (even though the spin itself is not up or down, it is the particular direction's component that is up or down). 

The Ideal Stern-Gerlach Apparatus

These simulations make use of two types of ideal Stern-Gerlach apparatus to spatially separate the spin-1/2 particles: those that use a transverse magnetic field and those that use a longitudinal magnetic field.  Both are ideal in the sense that there is no experimental error associated with using them.  In other words, the outcomes are always exactly those predicted by quantum theory.  Because of this idealization, we do not distinguish between the apparatus internal mechanisms; they are all simply referred to as ideal Stern-Gerlach apparatus.

The incident beam can either be a beam of a random (or statistical) mixture of spin orientations, a beam of a particular spin eigenstate (such as |z+> or |z−>), or a beam of a particular superposition of eigenstates (such as 0.707[z+> + |z−>]).  Once through one or more ideal Stern-Gerlach apparatus, the output is detected at the counters represented by the horizontal bars and their associated numbers.

Transverse Stern-Gerlach effect

The original experiment by Stern and Gerlach made use of a transverse inhomogeneous magnetic field (generated by a permanent magnet) perpendicular to the direction of propagation of the spin-1/2 particles.  So if a particle is moving in the x direction and is subject to an inhomogeneous field in the z direction, it will experience a force, Fz ~ μz dBz/dz.  Such a force will deflect the beam either up or down due to the fact that μz = gqSz/2mc, where g is the gyromagnetic ratio, q is the particle's charge, m is the mass of the particle, and Sz is the z component of the particle's spin.  Therefore such an apparatus can be used to spatially separate particles whose z component of spin are oriented either up or down.  One can also construct an ideal Stern-Gerlach apparatus to measure the spin component in the y direction (still assuming the particle is moving in the x direction).  It is essentially the same apparatus described above with its permanent magnet, and hence its magnetic field, now aligned on the y axis. 

Longitudinal Stern-Gerlach effect

These simulations also show a different type of ideal Stern-Gerlach apparatus: one which measures spin in the direction of propagation using, in principle, the longitudinal Stern-Gerlach effect.  If we again assume that the direction of propagation is the x direction, the ideal Stern-Gerlach apparatus must have an inhomogeneous magnetic field in this direction.  It is difficult to see how this can occur with a permanent magnet, but not too difficult to imagine with magnetic field generated by a current in a coil that the beam of spin-1/2 particles passes. The spin-1/2 particles are then "deflected" either forward or backward thereby generating a spatial separation that in principle could be detected.