Sequential Stern-Gerlach devices - realizable experiment or teaching aid?
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At least one textbook [1] uses sequential Stern-Gerlach devices to introduce to students that the components of angular momentum are incompatible observables. Viz., the zz-up beam from a SG device with magnetic field in the z^z^direction (an SGzz device) is passed through an SGxx device, and is found to split into two beams. Passing say, the xx-up beam through an SGzzdevice, it too splits.
Of course, knowing quantum mechanics this is exactly what we expect.
But to someone who does not know quantum mechanics, is this convincing that there is no ∣+x,+z⟩∣+x,+z⟩ state? I am not so sure it is if we consider it as a real experiment, with finite precision. We know that the beam entering the SGxx device has Sz=ℏ/2Sz=ℏ/2, we do not know anything about its SxSx. We know that the beams leaving the SGxx device have Sx=±ℏ/2Sx=±ℏ/2, respectively. By adding the second SGzz we wish to test if SxSx and SzSz can have definite values simultaneously, but there is then an assumption that the SGxx device does not disturb the value of SzSz, or at least does so with a very small spread. But already in the classical picture the Stern-Gerlach device is not such a device.
In the SGSGz device the BB-field has a large homogeneous component B0z^B0z^, such that the angular momentum around z^z^ is approximately conserved while the other components average to 0, and the force, on average, has only a z^z^component [2]. But in the SGxx device the angular momentum precesses around x^x^, with a period that is quite short, T=10−9T=10−9 s or less.
If the particle beam has a spread of velocities vvsuch that the spread in times-of-flight tt is not small compared to TT, we should not expect the second beam to be zz-polarized, even classically. The relation between the spreads is Δt=tΔv/vΔt=tΔv/v. In the original experiment [2] we can estimate vvand tt as being on the order of 102102 m/s and 10−410−4s, requiring Δv/vΔv/v on the order 10−510−5. This seems entirely unreasonable for a thermal source, considering the finite width of the collimator and if nothing else the force component neglected initially seems liable to produce a spread of at least this order.
I tried to search the literature to see if the sequential experiment has actually been carried out, but could not find anything. I did find Ref. 3 that seems to talk about two-spinors, but I cannot access it.
References
Townsend, J.S. (2000). A Modern Approach to Quantum Mechanics. University Science BooksStern, O. (1988). A way towards the experimental examination of spatial quantisation in a magnetic field. Zeitschrift für Physik D Atoms, Molecules and Clusters, 10(2), 114-116.Darwin, C. G. (1927). The electron as a vector wave. Proceedings of the Royal Society of London A, 227-253.
Of course, knowing quantum mechanics this is exactly what we expect.
But to someone who does not know quantum mechanics, is this convincing that there is no ∣+x,+z⟩∣+x,+z⟩ state? I am not so sure it is if we consider it as a real experiment, with finite precision. We know that the beam entering the SGxx device has Sz=ℏ/2Sz=ℏ/2, we do not know anything about its SxSx. We know that the beams leaving the SGxx device have Sx=±ℏ/2Sx=±ℏ/2, respectively. By adding the second SGzz we wish to test if SxSx and SzSz can have definite values simultaneously, but there is then an assumption that the SGxx device does not disturb the value of SzSz, or at least does so with a very small spread. But already in the classical picture the Stern-Gerlach device is not such a device.
In the SGSGz device the BB-field has a large homogeneous component B0z^B0z^, such that the angular momentum around z^z^ is approximately conserved while the other components average to 0, and the force, on average, has only a z^z^component [2]. But in the SGxx device the angular momentum precesses around x^x^, with a period that is quite short, T=10−9T=10−9 s or less.
If the particle beam has a spread of velocities vvsuch that the spread in times-of-flight tt is not small compared to TT, we should not expect the second beam to be zz-polarized, even classically. The relation between the spreads is Δt=tΔv/vΔt=tΔv/v. In the original experiment [2] we can estimate vvand tt as being on the order of 102102 m/s and 10−410−4s, requiring Δv/vΔv/v on the order 10−510−5. This seems entirely unreasonable for a thermal source, considering the finite width of the collimator and if nothing else the force component neglected initially seems liable to produce a spread of at least this order.
I tried to search the literature to see if the sequential experiment has actually been carried out, but could not find anything. I did find Ref. 3 that seems to talk about two-spinors, but I cannot access it.
References
Townsend, J.S. (2000). A Modern Approach to Quantum Mechanics. University Science BooksStern, O. (1988). A way towards the experimental examination of spatial quantisation in a magnetic field. Zeitschrift für Physik D Atoms, Molecules and Clusters, 10(2), 114-116.Darwin, C. G. (1927). The electron as a vector wave. Proceedings of the Royal Society of London A, 227-253.
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Hey !
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The mechanism of the theory Stern-Gerlach devices realized that the reaction of metal carbonates and metal bicarbonates with acids produces carbon dioxide.
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