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Simon Bridge

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How would you measure it?

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When we say that intrinsic spin has a specific value for a specific type of particle, I guess we mean with the particle itself chosen as an origin. This becomes a little subtle in the case of a massless particle, which doesn't have a rest frame.

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Simon Bridge

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If we measure angular momentum in one orientation, subsequent measurements get the same thing - but if we rotate the axis we measure against, the new statistics will come from a mixture of different angular momentum states ... so the expectation (= average) can change continuously ... but a subsequent measurement still gets the same lumps doesn't it?

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Bill_K

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Quantitatively the coefficients in this superposition can be obtained from the commutator [L

An infinitesimal translation along the z-axis corresponds to applying a derivative, ∂/∂z. For example for ℓ = 2, F

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Simon Bridge

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Oh fair enough ... though if I measure angular momentum, then use identical apparatus in a different position (i.e. translated, but not rotated) on the same particles, doesn't that just give the same measurement? In fact, would we normally expect a mere translation of axis to change the projections of a vector? Wouldn't a rotation, instead, be more illustrative for OPs question? Or maybe I have the wrong question?We're not talking about a rotation, the question posed was, what happens to L under a translation?

The z-axis projection of a "regular" vector depends on how we've drawn the axes after all.

I understood that the different axis were defined (in a meaningful way) by the

eg. The Stern-Gerlach apparatus has an important axis, which gets called "z" a lot. The angular momentum is quantized with respect to this axis, no matter how the apparatus is oriented in space. You can put random x-y position particles (the beam) in with random orientation of spins and still get quantized results out.

I was guessing that this was the sort of thing OP was asking about.

So it seems reasonable to say that the angular momentum is said to be quantized because that is what we get when we measure it. It gets that way (the "how" part), regardless of orientation, because of the interaction with the measuring apparatus. We can (very accurately) account for this in the mathematics by considering superpositions of orthogonal angular momenta.

In a way, what OP seems to be wrestling with is a less obvious manifestation of "wave particle duality" ... the math is continuous but the resulting measurement is discrete. OR I have been overthinking this and looked for deep meaning where there is none ;)

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I'm also not sure what you gain from "changing origin", you would just be changing the theoretical description of the states, not changing the states themselves. It would be harder to see the quantization because you're using a non-intuitive basis, but it wouldn't remove the actual quantization in any way.

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