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When an argument like 'measure the spin along the $x$ axis', 'observe the position of a particle' and so on is made, what is the implied experimental procedure? Since laboratory equipment is macroscopic, I have a hard time relating this to measuring an observable.

For example, if I have a QM harmonic oscillator, how do I actually determine which eigenmode it is in?

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Usually it is a figure of speech, used to illustrate a theoretical fact or hypothesis in a heavily idealized setting.

Actual measurements of single quantum phenomena are always complicated. For example, to measure the position of a particular atom you need to isolate it in an ion trap whose position you know, or you need to mount it on an atomic force microscope

For example, harmonic oscillators are always idealizations of real oscillators, such as the outer electron of an atom in an ion trap, or an electromagnetic field in a cavity.

To measure a particular eigenmode of a quantum system, you have to bring the system to be measured into resonance with a macroscopic system, whose effect you can observe.

Or you have myriads of quantum systems of the same sort, and you measure their collective response. This is how you can find out the eigenmodes of hydrogen, by observing its absorption and emission spectrum and measuring the position of the lines (aka resonant frequencies of the incident light).

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