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There was some debate in my modern physics course about the statement

"The spin of an electron can only be measured in the presence of an external magnetic field".

So far in the course, intrinsic spin has only come up in two contexts, namely the Stern-Gerlach experiment and fine structure. I'm aware that there are other methods which do measure spin without an external magnetic field, but since they haven't been discussed in the course, we shouldn't consider them. Also, we're only considering Bohr model atoms with fine structure corrections for now.

I argue that the statement is true (considering only SG apparatuses or spectrometers as methods of detection) since a SG apparatus can definitively measure the spin of an electron in a specific basis. I have no doubt that spin can be detected without the presence of an external magnetic field by observing fine structure, but I'd argue that detection of the phenomenon of spin is not the same as measuring the spin in a chosen basis. So really I think the debate hinges on how we're defining "measured". If we take measurement to mean any interaction which forces the system into a specific state, I would lean more toward the initial statement being false, but I interpret "measurement" more as meaning projecting the system onto an eigenstate of a chosen basis, which I'm not sure a spectrometer can do. However, I have a limited understanding of how spectrometers work, so please correct me if I'm wrong.

Given that we have only covered Stern-Gerlach and fine structure, and considering a Bohr model atom, what is the correct answer here?

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Your title question and your quoted statement are different questions with different answers.

Does spectroscopy count as a measurement of electron spin?

Spectroscopy measures the energies of transitions within atoms (or even molecules), which consist of a nucleus, some number of electrons, and their combined electric field. Some atomic transitions can be useful modeled as transitions of "a" valence election interacting with a core of inert inner electrons. For example, sodium looks a lot like ten electrons in the inert formation of neon, with an eleventh electron that's mostly independent. But it's more common that spectroscopic observations involve coordinated changes to entangled properties of many electrons in the same atom. It's the entire atom whose spin is well-defined before and after a spectroscopic observation.

So ... sometimes.

The spin of an electron can only be measured in the presence of an external magnetic field.

This isn't correct. Electron spin can, among other techniques, be measured using Mott scattering.

An early effort to observe Mott scattering led to the discovery, published in 1927 by Cox and collaborators, that beta radiation electrons tend to be polarized, spinning so that their north poles point back towards their point of origin. This observation was forgotten for thirty years. I learned about it from Allan Franklin's book "Are there really neutrinos," where he calls it "the nondiscovery of parity nonconservation."

Reading that book — which might be a big swallow for a student in an introductory modern physics course — will reveal that the electron spin can also be measured using weak interactions. Though it's usually the other way: we use electromagnetic techniques to measure the electron spin, and then use known spins to learn new things about the weak interaction.

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  • $\begingroup$ Interesting, I'm assuming the same is true about measuring the spin of all the entangled electrons in a silver atom with a Stern-Gerlach apparatus. I've heard of Mott scattering, and so in general the statement is false, I don't disagree. Within the scope of my class however, we've only covered the Stern-Gerlach experiment and fine structure, so my contention is with whether fine structure can be considered a measurement in the same way as the latter. $\endgroup$ Commented yesterday
  • $\begingroup$ Right. The Stern-Gerlach apparatus measures the spin of the atom, which is a coherent superposition of many electrons and a nucleus. There are many types of spectroscopic measurement, which may or may not conform to your definition. $\endgroup$ Commented 12 hours ago
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The quotation is manifestly nonsense because electron spin can also be measured using an external electric field in what is known as the Stark effect. Although this is nowhere near as nice as the Stern–Gerlach experiment, there is no good reason for the quotation to be as it is.

Yes, spectroscopic measurements are measurements. If they indicate the existence of electron spin, then so be it.

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    $\begingroup$ But can spectroscopic measurements differentiate between spin up/down? If not, we need to be more clear about what is meant by measurement, because apparently you can measure the spin of an electron and not know its spin state $\endgroup$ Commented yesterday
  • $\begingroup$ Well, yes, you can sometimes deduce that after the emission of the photon, the spin of the electron is then fixed to be so and so. $\endgroup$ Commented 8 hours ago
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First of all, none of these measurements are purely of electron spin. These are measurements of atomic spin or total atomic angular momentum. You cannot use a SG apparatus or similar setup to measure the spin of a free electron, this is an old problem that has been discussed in the literature since the times of Bohr and Pauli.

The basic treatments assume hydrogen-like atoms, and it that case then the electron's total angular momentum is the total angular momentum of the atom. For SG experiments the classic atom of choice is silver, this is because it has a single $5s$ electron that grants the whole system the equivalent total angular momentum equal to the spin of a single electron. Since QM predicts that the total angular momentum will be that of a single electron, and the experiment confirms this prediction, you then associate those values with the spin of a single electron.

After clearing this up, from your post and comments essentially your question boils down to whether or not one can use spectroscopic measurements to measure the angular momentum in the same way one would for SG experiments, and without using a magnetic field.

The answer isn't quite that straightforward. If we're considering atoms left to their own devices, without being influenced by magnetic or electric fields or other external agents, then we will just have the fine structure. And the fine structure does not differentiate between $m_s$ and $m_l$ (therefore also not for $m_j$), so while it can tell you $j$ and $l$, it's strictly not a measurement of the same form as the SG experiment that differentiates atoms according to $m_s$ (or $m_j$).

With a magnetic field you could get the Zeeman splitting which tells you $m_j$, but that's "cheating" in this context. It would give you just as much information as you would get via SG, though.

With an electric field you could make use of the Stark effect, which, to second order, does differentiate between different $m_j$ values. So this would constitute a spectroscopic measurement without using a magnetic field. But, particularly for silver, the shift is the same for both $m_j$ values and therefore you would not get three separate lines.

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