I have a thought experiment that has been puzzling me regarding a spin-$\frac12$ particle placed in an extremely powerful magnetic field (say 1000 T for the sake of hyperbole) directed along the positive $z$-axis. Now, we take a measurement of spin along the $x$-axis, which forces the particle to collapse into one of the states $|\rightarrow\rangle=\frac1{\sqrt2}\left(|\uparrow\rangle+|\downarrow\rangle\right)$ or $|\leftarrow\rangle=\frac1{\sqrt2}\left(|\uparrow\rangle-|\downarrow\rangle\right)$ with equal probability. Now, if we take a measurement along the $z$-axis, the particle will be found in state $|\uparrow\rangle$ or state $|\downarrow\rangle$ with equal probability. But this means that just via a sequence of two measurements, we have discovered how to give a particle a large amount of potential energy by aligning it antiparallel to the strong magnetic field with $50\%$ probability. Clearly, there is something wrong with my logic here, but can anybody tell me what it is? Ignore for a moment the experimental difficulties with measuring spin along a direction other than that of the external magnetic field, because this experiment could also be carried out with other kinds of conjugate variables such as position and momentum.
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$\begingroup$ How is this different from asking, "I have a compass and I put it in a magnetic field, it aligns with the field, now it has some potential energy??" $\endgroup$– MauricioCommented Mar 24, 2023 at 17:59
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$\begingroup$ Because half of the wavefunction will now be antiparallel to the magnetic field without putting work in, and the antiparallel spins have much more potential energy. $\endgroup$– slithy_toveCommented Mar 24, 2023 at 23:05
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$\begingroup$ The phrase "be antiparallel" is doing a lot of work here (potentially inappropriately). In what sense do you think the particle "is" parallel or antiparallel? $\endgroup$– hftCommented Mar 25, 2023 at 4:53
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$\begingroup$ It might be helpful to review what the wavefunction means and what it doesn't mean. Grab an intro textbook like this one: amazon.com/Introduction-Quantum-Mechanics-David-Griffiths/dp/… $\endgroup$– hftCommented Mar 25, 2023 at 4:56
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1$\begingroup$ What is it that OP doesn't understand? Presumably 'parallel' means "in the low-energy eigenstate of the Hamiltonian $H=\gamma B S_z$" and antiparallel means the high-energy eigenstate. The system potentially ends in a state of well-defined energy which is higher than the (well-defined) energy that it started with, hence the apparent paradox. The question's phrasing seems perfectly fine to me. $\endgroup$– user34722Commented Mar 25, 2023 at 16:45
1 Answer
Measurement requires an interaction between the measured system and some apparatus/external system. In this case you're discovering that measurement would cost some energy to perform, with the cost being paid by the apparatus.*
In the 1000 T magnetic field, the $|\rightarrow\rangle$ state precesses very rapidly into the $|\leftarrow\rangle$ state, at a rate of $\omega=\Delta E/\hbar$, where $\Delta E$ is the energy difference between the $|\uparrow\rangle$ and $|\downarrow\rangle$ states. So the measurement apparatus needs to perform its measurement comparable to or faster than $1/\omega$. To do so, it needs to couple to the spin with a strength comparable to or larger than $\omega\hbar=\Delta E$ (so that the action of that term in the Hamiltonian over a time interval ~$1/\omega$ is non-negligible). That coupling provides the missing energy.
*Including the interaction term between system and apparatus as part of the apparatus.
