Quantum measurement in a strong magnetic field

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.

• 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??" Mar 24, 2023 at 17:59
• 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. Mar 24, 2023 at 23:05
• 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?
– hft
Mar 25, 2023 at 4:53
• 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/…
– hft
Mar 25, 2023 at 4:56
• 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. Mar 25, 2023 at 16:45

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.