Stern-Gerlach

Everyone remembers the Stern-Gerlach experiment; we usually recall it in the context of the discovery of quantized intrinsic spin.  However, I like very much the treatment of Sakurai in his Quantum Mechanics text, which uses Stern-Gerlach as the introduction to shock us into a quantum mechanical way of thinking.

The setup is familiar: Heat up silver atoms which escape through a hole in the oven.  Collimate the beam and send it though an inhomogeneous magnetic field; the force in the z direction on the atom is .  Since the atoms in the oven were randomly oriented, there's no preferred orientation of , so classically we expect to see a vertical splay of particles coming out of the magnetic field.  Of course, we instead see two distinct components, scattered up and down, which leads us to the idea of quantized spin angular momentum.

The fun part begins with sequential Stern-Gerlach setups, adding the ability to rotate the subsequent apparatus on its side to act in the x direction instead of the z. The picture below, stolen from Wikipedia which obviously it stole from Sakurai, shows the results.

The first result makes sense; we removed the minus z portion of the beam, so it doesn't recur.  The second, in which the plus z portion is now split 50-50 into plus and minus x portions, is interesting; maybe 50% of the split beam was z plus and x plus while the other 50% was z plus and x minus?  The third result is the doozy.  We get z minus out at the end, but didn't we remove it in the first splitting?  Apparently the x measurement in the middle destroys our previous information on the z direction of the spin!  We can't know both the spin in z and in x simultaneously; this is obviously not a spinning top.

Sakurai then calms our confusion with an analogy to polarized light, with the x and z spin directions above corresponding to zero and 45 degree polarized light.  In that context, this result would be right at home.  In E&M we write the 45 degree polarization in terms of a linear combination of the 0 degree coordinate vectors, x and y.  For spin, we must turn our thinking to an abstract spin space, where the base vectors Sz+ and Sz- take the place of base vectors x and y in physical space. The Sx spin directions can then be swapped into the idea of the 45 degree polarized light, so should be expressed in terms of our Sz+ and Sz- base vectors:



The missing piece is Sy, the last spin direction, which we correspond to circularly polarized light in this analogy.  Circularly polarized light is expressed as the same combination of base x and y vectors, but with the y portion 45 degrees out of phase.  This now brings imaginary numbers easily into play, as we express a light wave in exponential notation instead of cosines and pull out .  In our corresponding spin vector space we have now been lead to:



So, in a few paragraphs we've seen not only the weirdness of quantum mechanical phenomena, but we've constructed a complex vector space directly out of our observations in Stern-Gerlach and an analogy with our understanding of polarized light.

Source: Sakurai's Modern Quantum Mechanics.