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This is a question about bridging reality to theory.

In learning QM theory, we learn a bunch of stuff about observables, like their operators, that two quantities can be measured at once if their operators commute and stuff like that.

But what we can actually measure and how?

We know from theory that we can measure position, momentum, angular momentum, spin. Is there something else we can measure?

Now the second part of this question, is how we can measure these things?

In theory, the only device we learn is the Stern-Gerlach machine, which measures spin. What other apparatus is there to enable us to actually make the observations that we have spent so many hours on paper, trying to understand.

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Basicly you can measure any observable. An observable is a hermitian operator $A$ with a complete base. So \begin{equation} \sum \nolimits_n \vert n\rangle \langle n\vert = 1 \end{equation} With \begin{equation} A \vert n \rangle = n \vert n \rangle \end{equation} I do not think that there is a generall scheme how to measure the observable in an experiment. So I guess you just have to be creative and look at the specific observable. For example you could measure the position of a charged particle by the the distribution of the charge density.

By the way the hamilton operator is an observable too. So you can measure the energy as well.

I hope I was able to help.

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Let me focus on the question how you can measure these things:

It is important to note that a measurement in general is an active process where you interact with the system of interest in order to gain some information about this system.

A general description of quantum measurements was introduced by von Neumann and considers the system under investigation and the measuring device as a quantum system such that both are described in terms of a wavefunction. The measurement interaction is described with an interaction Hamiltonian

$$ H_{int} = g(t) A \otimes P_d, $$

where the operator $A$ and $P_d$ are acting on the Hilbert space of the system and the measuring device, respectively. The time-dependent coupling constant $g(t)$ is only different from zero during the time interal of the measurement $0 \le t \le T$ such that $\int_{-\infty}^{\infty}g(t) dt = g_0$.

Now suppose the the device operator $P_d$ is the momentum operator of a pointer on a dial, such that the canonical conjugate position operator $Q_d$ obeys $[Q_d,P_d] = i \hbar$ and the interaction Hamiltonian generates translations of the pointer position in the device sub-space. Assume we now prepare our pointer state with corresponding pointer position $Q_d(0)$ and the system in a state with $\langle A \rangle$. According to the Heisenberg equation of motion, the change in $Q_d$ during the measurement is

$$ Q_d(T) - Q_d(0) = \int_0^T dt \frac{dQ_d}{dt} = \int_0^T \frac{i}{\hbar} [H_{int},Q_d] = q_0 \langle A \rangle. $$

Thus the pointer position has changed by an amount corresponding to the expectation value of the system operator of interest, whereas the coupling constant $g_0$ amplifies the signal such that if can be read of macroscopically.

Of course, this is just a very basic idea of the quantum mechanical process itself and one can go much further by asking what are the uncertainties in the pointer state and which how we define our ensemble, but I hope this give you an idea how one could, at least in principle, measure these quantum mechanical operators.

In order to make this more abstract example more concrete, we can quickly look at the Stern-Gerlach device. Here you want to measure the spin of a particle in a particular direction, say $z$-direction, then your system operator is simply $S_z$. In this experiment the pointer of your measurement device is given by an other degree of freedom of the same particle, namely, tha particles deflection. And by making your magnetic field gradient strong enough, you can have a macroscopic deflection of your particle beam such that you can read of the outcome of the shift in the pointer positon.

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A very good question. I posed a similar question about how to measure $L^2$ and got no answer - it seems the maths is easy but tying it back to reality is hard. Welcome to QM.

In the case of the double slit experiment, the final position of the photon/electron/whatever is typically recorded on a photographic plate or sensor array. The interaction represents the measurement. The momentum of the photon can be measured by it frequency, and particles are typically tracked in bubble chambers and the like.

How do we know what we can measure? and how?

How is not easy. Someone prove me wrong by telling me how to measure $L^2$. Also measurements are often necessarily indirect (e.g. measurement of neutrinos - we look at the decay products from rare collisions and work backwards)

What can we measure? My opinion is anything that ultimately can translated into differing electron states in a device (or the brain of man) can be measured since we are ultimately made of electrons (and other stuff). That definition is very wide but allows us to claim that we can indirectly "measure" even abstract quantities such as isospin.

How do we know what we measure? We measure classical quantities (position, momentum, angular momentum) according to Bohr, and (I would argue) everything else is interpreted through the eyes of theory.

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  • $\begingroup$ "It seems the maths is easy but tying it back to reality is hard." Haha you nailed it. $\endgroup$
    – Filippo
    Commented Apr 21, 2023 at 19:53

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