Bundle of tiny randomly orientated magnets through a Stern-Gerlach app

[ajw1]

When I would send a bundle of tiny randomly orientated magnets through a Stern-Gerlach apparatus I don’t think I would expect the pattern as obtained with silver atoms, displaying two separated spots. So how exactly does the electron spin effect the path of the atoms in this experiment?

 

[Doc Al]

Classically you wouldn't expect the pattern that is observed--it's something new. The component of spin parallel to the magnetic field is quantized. (The net force experienced by a magnetic dipole in a non-uniform field depends on the component of the dipole moment parallel to the field.)

 

[ajw1]

So you're saying that the magnetic field forces the spin to be quantised in the z-direction (direction of the magnetic field)?

 

[Fredrik]

The field doesn't force anything to be quantized. What it does is to put a silver atom in the state |spin-z up>|left>+|spin-z down>|right>, so that you can measure the spin-z state by measuring the position. The screen accomplishes the latter.

Quantum theory says that when you measure an observable, the result is always an eigenvalue, and the measurement either destroys the system or leaves it in the state represented by the corresponding eigenvector. It doesn't say why observables are quantized. It just defines a mathematical model in which the observables don't commute (AB≠BA), and tells you how to use it to make predictions about the results of experiments.

Heisenberg's argument (mentioned in my post here) leads to an approximate version of the uncertainty principle. This is one reason why we might expect a model with non-commutative observables to be useful. The axioms of QM (the definition of the mathematical model and the instructions on how to interpret the math as predictions about the results of experiments) imply that an exact version of the uncertainty principle must hold, which makes it plausible that QM might actually be a useful theory. After that it was up to the experimentalists to show that the predictions of QM are more accurate than the predictions of classical mechanics, and as you know they succeeded beyond anyone's expectations. Some of the predictions have been tested with an enormous accuracy.

 

[ajw1]

Articles llike the one on http://en.wikipedia.org/wiki/Stern%E2%80%93Gerlach_experiment" explain the apparatus by arguing that an electron's angular momentum is quantized. But they fail to explain why that would result in a quantized state in the z-direction.

 

[Doc Al]

When wiki states that the spin angular momentum is quantized, they mean that when measured in some particular direction (such as z) only certain values are obtained.

 

[ajw1]

You must be right, but I don't think such a formulation would hold in a formal contract ;)

Others just presume the angle between the spin and the magnetic field is constant. Anyway IMO just having the spin-up and spin-down for the electron can't be the complete explanation.

 

[Fredrik]

Anyway IMO just having the spin-up and spin-down for the electron can't be the complete explanation.

No one says it is.

If you rotate the apparatus, it would measure another observable of the form $$\vec n\cdot\vec S$$, where $\vec n$ is a vector in $\mathbb R^3$ of magnitude 1, and $$\vec S=(S_1,S_2,S_3)$$. A measurement of the z component of spin is just the special case $\vec n=\vec e_3$.

I'm not sure what else I can tell you. You should probably ask a more specific question.

 

[ajw1]

Wiki (and others) certainly try to explain the effect by the quantized spin of the electron. I think this is not enough, because it doesn't say anything about why the orientation of the spin should be in constant angle with the magnetic field. So without further information this can't be an explanation at all (the quantized spin would still be randomly orientated, and would not lead to the observed results).

So I would like to know what causes this angle to be constant.

 

[conway]

You can understand the Stern-Gerlach experiment by tracing the evolution of the wave function as it passes through the magnets. The most important thing, however, is to remember that the electron beam has a finite width, so the upper and lower portions of the beam feel a different magnetic field.

If you analyze the sideways electron as the superposition of |spin up> and |spin down>, it is fairly easy to see that each of these beam components curve differently as they pass through the field. It is a little trickier to analyze the same beam as the pure state of a sideways electron. The wave functions look quite different but they give the same result in the end: the beam is separated into an upward and downward moving componenets.

 

[Fredrik]

Wiki (and others) certainly try to explain the effect by the quantized spin of the electron. I think this is not enough, because it doesn't say anything about why the orientation of the spin should be in constant angle with the magnetic field. So without further information this can't be an explanation at all (the quantized spin would still be randomly orientated, and would not lead to the observed results).

So I would like to know what causes this angle to be constant.

I don't understand what you mean by "in constant angle with the magnetic field". Can you quote the exact words that you find confusing? I'm also not sure what your conclusion is. It sounds like you have the wrong idea about what's being claimed.

If you imagine shooting a small magnet through the apparatus, it would get pushed either to the left or to the right, depending on how it's oriented. How hard it gets pushed (and therefore how far it gets deflected) depends its orientation. So a beam of classical magnets would hit the screen all over the place. The electrons on the other hand, all hit one of two spots on the screen. What this tells us is that electrons have a property that's similar to a classical magnetic moment, but is different enough to mean that we need quantum mechanics to be able to accurately predict the results of experiments where this property affects the result.

 

[ytuab]

In Stern-Gerlach experiment, they needed a magnet to create a strongly inhomogenous magnetic field and narrowic atomic beams to get correct and sharp results.

At that time, Bohr thought that Bohr model is correct. So he explained the results of the Stern-Gerlach experiment using Bohr model as follows,

-------------------------
Niels Bohr had excluded the possibility that the spatial quantum number n1 be zero because
then the plane of the electron orbit would contain the direction of magnetic field and the electron motion might become instable under certain conditions.
Terefore in quantum theory(of Bohr model), the magnetic moment of an atom should possesses only two discrete positions, parallel or antiparallel.
----------------------------

In page 438 the Historical Development of Quantum Theory part 2 by Jagdish Mehra
(Sorry. the link was technically difficult.)

(I don't understand the effect of spin precession on the Stern-Gerlach experiment well.)