Why aren't there two discrete infinitely narrow peaks in the Stern-Gerlach experiment?

The Stern-Gerlach experiment is a monumental experiment in quantum physics. It breaks the classical idea of spin and gives validity to the quantum nature of spin.

But, if the spin of an electron can have only two quantized values, then why aren't there two discrete infinitely narrow peaks evident in the experiment? Why are there two blobs?

Is it because of HUP? Or, is it due to some experimental error?

We need a full, mathematically rigorous answer.

Look at the right hand of that diagram. The part that says "Furnace". See, the beam of silver atoms (or potassium, as I did it as an undergrad) comes from heating up metal so hot it evaporates. Then, because you have a pretty good vacuum outside of the furnace and in the apparatus, the metal gas comes rushing out of the oven. The only reason it makes a beam is because of the hole in the side of the furnace. You could get a narrower beam by narrowing the hole, and making the channel it goes through before reaching vacuum longer, but then you're also going to be prone to clogging as the metal condenses after leaving the furnace. Even if you can avoid clogging, narrower aperture and longer pipe constrict the flow of atoms, making the beam less intense.

In short, the metal starts as a gas in a furnace that is going in every direction with a Maxwell-Boltzmann distribution in velocities, and you use some method to select only those atoms going in the direction you want, roughly. There will always be some spread in the velocities you can pick out, and practical concerns limit how narrow you can make that spread. If you turn off the magnet (or remove it) you'd see one beam with finite width (as you see on the left). Then the magnet with the field gradient operates on that beam that has both a spatial spread and a velocity spread to produce the two beams you identified as blobs on the right. The kind of mouth-like shape of the right hand pattern has to come from the details of the magnetic field the beam passes through. Obviously, the field gradient is not uniform in the $y$-direction.

• Lake I first studied the experiment in HRW; where it was said that Heisenberg uncertainty principle has to do something with the blobs. Is it true? – Mockingbird Oct 22 '17 at 2:03
• @Mockingbird It's relevant for if you try to narrow the blobs to sharp peaks completely, because to do that you'd need to remove all transverse velocity (no spread in $\mathbf{p}_\perp$) and make the beam narrow (no spread in $\mathbf{r}_\perp$). I highly doubt that they bumped up against that limit in this experiment. More likely they were just dealing with thermodynamic messiness, as I outlined. – Sean E. Lake Oct 22 '17 at 2:29
• @Mockingbird HRW? HUP? WTF? Google says Human Rights Watch... – Vladimir F Oct 22 '17 at 12:20
• I say Halliday Resnick Walker! – Mockingbird Oct 22 '17 at 13:27

This is because of "experimental errors". The beam was not completely collimated so the atoms were not in a pencil-thin beam but rather distributed about the beam axis, with varying but small amounts of velocity perpendicular to the beam. The field gradient was not uniform either: as a result, the field gradient affected differently different atoms in the beam.

Since neither the gradient profile nor the beam profile was measured exactly, one cannot supply a mathematically rigorous.

This experiment was redone in a careful way, with mapping of the magnetic field, at the University of Warwick. The results are published in Porter, J., R. F. Pettifer, and D. R. Leadley. "Direct demonstration of the transverse Stern–Gerlach effect." American Journal of Physics 71.11 (2003): 1103-1108 with a link to the paper (behind paywall)

• Isn't there an Arxiv Link? – Mockingbird Oct 22 '17 at 2:58
• @Mockingbird not according to Google. – ZeroTheHero Oct 22 '17 at 3:18