# 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 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 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 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 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 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.