# Operationally, how do I choose between measuring position or momentum?

First a simple example to illustrate the essence of the question: If we have a spin half particle in the $$S_z+$$ state and if one wants to choose from either measuring $$S_z$$ or measuring $$S_y$$, which are incompatible, one could execute one of two complementary actions: ''Orient the magnets in the z direction'' or '' Orient the magnets in the y direction''.

Now, the question: What are the corresponding complementary actions in the following case? :

At $$\textbf{time t}$$, we have a system in a state W such that the standard deviation in position observable x with respect to W is very small and is centered around a coordinate r, and the standard deviation of the incompatible momentum observable $$P_x$$ is very large(to the extent allowed by the uncertainty relation).

Now $$\textbf{at time t}$$ we want to choose either to measure x or $$P_x$$. This one seems a bit tricky to me because of the fact that most of the momentum measurements themselves involve position measurements along with time of flight measurements. Let me elaborate: If i want to execute the choice of measuring position, you might suggest: "Put the position measuring detectors on those spatial regions where there is non vanishing probability of detection at time t."

On the other hand, if I choose to measure momentum, you might say: "Put the position detectors along those directions of possible momentum at time and measure the time of flight ". It is this last suggestion I find a bit unconvincing. Why? Because when I do detect a particular value for the momentum when I execute this last suggestion, then I am really measuring the momentum at time $$t+\Delta t$$, where $$\Delta t$$ is the measured time of flight. I am not really measuring the momentum at time t as I did for position. Is there a way to resolve this dilemma? It would be very useful and more clear if the answer involves crisp experimental instruction/protocols.

• Really nice question! Your dilemma is partly related to the fact that real measurements always involve at least a partial measurement of both position and momentum. Such measurements are possible: they are described by so-called positive-operator-valued measures. See Arthurs & Kelly's seminal article On the simultaneous measurement of a pair of conjugate observables and also Leonhardt's book Measuring the Quantum State of Light, where this question is discussed for light. This kind of measurements are routine in quantum-optics labs. – pglpm Jul 9 '20 at 8:08
• Can I point out that it's not the momentum at $t+\Delta t$, rather the inferred average momentum for the entire span of $\Delta t$ after $t$ – Jim Aug 14 '20 at 17:22